Shortest Distance Problem

Nancy Maple Ray Shannon Christine Bolte

Kathy Padgett Jehan Khan

Class: 7th grade through Geometry.

Materials: Data sheet, measuring devices are optional.

Goals: To introduce the students to mathematical modeling problems, write-ups, and presentations through a simple distance problem.

Time Required: About two days. The problem could first be described the last 10 minutes of class on Thursday. The entire class time on Friday could be used to measure. On Monday, the write-ups would be due, and the presentations would be given. Of course, the project could also be assigned as an out of class assignment, provided the students are able to meet before or after school.

Background: Prior knowledge of measuring distances.

Setting: Finding the shortest path between two points or the shortest route through many points is a problem faced by all sorts of people and companies. Some diverse examples include taxi drivers, mail carriers, and companies which process information through computer networks.

Problem: Find the shortest route from the classroom to the pay phone. Students may use any desired method of measurement, but the final result must be converted to feet. The handout may be used to record the data for two strategies, making sure to note any assumptions. Each group will submit a written report of their findings, and will present their results to the class.

Evaluation: One grade will be given for the entire group. Point values will be assigned for the data sheet, written report, and presentation. Example: data sheet/report, 10 points and presentation, five points. Bonus points may be awarded to the group finding the shortest route.

Extensions: The problem may be adapted to various school settings. It may be more feasible for the students to find the shortest distance to the restrooms, drinking fountain, Coke machine, nurse's office, candy machine, etc. Restrictions may be included, such as not allowing right turns. Another extension is to consider a more complex problem such as finding the shortest route through an entire day's schedule. Another twist to the problem is to find the quickest path between two points or through a route. The shortest and quickest are sometimes the same, but not always, as experience with crowded hallways and roadways attests.

Teacher Notes: In groups of three, one student could record the data while the other two measure the distances. Encourage students to be creative in their measurements by letting them devise their own method (using their feet, floor tiles, rulers, cinder blocks). If time permits, more than one strategy may be used. Be sure the expectations for the report and presentation are clear to the students before beginning the project.

! Write-up - must include a description of the problem and how it was approached, a list of assumptions, calculations, a conclusion, and the data sheet.

! Presentation - should discuss everything in the report.

Students may have difficulty recognizing the assumptions they make while measuring. Overlooked assumptions might include stating the exact starting point (teacher's desk, doorway?) and final destination (which pay phone?), consistent measurement (all floor tiles are the same size, a student's shoe measures one foot), route traveled (center of hallway, side of hallway?), not allowed to use Coke machines in teachers' lounge, and so forth.

Shortest Distance Problem

Sample Report

Project Description: Our problem was to find the shortest route from our classroom to the pay phone. Since our group had no available measuring devices, we decided to count the number of floor tiles along our routes. Our first path was to turn left outside the room and walk to the end of the hall, and then turn right and walk to the end of that hall. We then turned left and stopped at the phone closest to us. Our second strategy was to turn left outside the room, turn right down the first hallway and go to the end, turn left and follow that hall to the end, turn right and follow that hall to the end, and then finally turn left and stop at the closest phone. We recorded our data on the data sheet. Finally, we measured the length of one tile and converted the number of tiles to feet in order to see which path was the shortest.

Assumptions: The following is a list of assumptions we made during the project:

1. The starting point was the doorway of our classroom.

2. The route would be traveled down the middle of the hallways.

3. Each floor tile has the same length of one foot.

4. The final destination was in front of the phone closest to our classroom.

Calculations: We assumed that each floor tile represents one foot.

Strategy 1 - 376 floor tiles = 376 feet.

Strategy 2 - 402 floor tiles = 402 feet.

Conclusion: Based on our measurements, we have concluded that our first strategy was the best, and that this route is 376 feet.

Shortest Distance Problem

Sample Solution

Problem: Find the shortest route from the origin to the final destination. You may use any desired method of measurement, but the final result must be converted to feet. This handout may be used to record the data for each strategy. Be sure to note any assumptions you made while measuring. Each group will submit a written report of their findings, and will present their results to the class.

Origin Room 2011 Final Destination Phone

 Team Members: Kathy P, Chris B, Ray S.

 Method of Measurement Brief Path Description Distance (feet) Strategy 1 We counted the floor tiles from the room to the pay phone, then measured one of the tiles with a ruler. Turn left outside the room and walk to the end of the hall. Turn right and walk to the end of that hall. Turn left and use the phone closest to you. 376 floor tiles 1 tile = 1 foot 376 feet Strategy 2 Same as Strategy 1 Turn left outside the room and walk until you may turn right down a hallway. Turn right and follow the hall to the end. Turn left and follow the hall to the end. Turn right and follow the hall to the end. Turn left and use the first phone. 402 floor tiles 1 tile = 1 foot 402 feet

 Assumptions: We always walked down the middle of the hall and assumed that the length of the floor tiles was the same throughout the building. We started from the doorway of the classroom and ended in front of the first phone that we came to.

Shortest Distance Problem

Student Lab Sheet

Problem: Find the shortest route from the origin to the final destination. You may use any desired method of measurement, but the final result must be converted to feet. This handout may be used to record the data for each strategy. Be sure to note any assumptions you made while measuring. Each group will submit a written report of their findings, and will present their results to the class.

Origin____________________________ Final Destination___________________________

 Team Members:

 Method of Measurement Brief Path Description Distance (feet) Strategy 1 Strategy 2

 Assumptions:

Funded in part by the National Science Foundation and Indiana University 1995