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Taxicab problem, clip 2 of 5: investigating the number of shortest paths. [video] Retrieved from https://doi.org/doi:10.7282/T3FJ2GNQ
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TitleTaxicab problem, clip 2 of 5: investigating the number of shortest paths.
PublisherNew Brunswick, NJ: Robert B. Davis Institute for learning, , c2000-05-05
Persistent URLhttps://doi.org/doi:10.7282/T3FJ2GNQ
DescriptionIn the second of five clips, the four twelfth grade students employ various strategies to determine the number of shortest paths to the remaining two points, B and C, on the problem grid. Various patterns, invented notations, and color-coded diagrams are shared in an attempt to find a solution for the problem. As they calculate total numbers of routes for successively larger areas of the grid, the students conjecture about a possible relationship to the Towers Problem . When Romina calculates the numbers of shortest routes for the points where each shortest route has length 5, she makes the conjecture that the problem follows the pattern of Pascal's Triangle.
PROBLEM STATEMENT: The problem was presented to the students with an accompanying representation on a single (fourth) quadrant of a coordinate grid of squares with the “taxi stand” located at (0,0) and the three “pick-up” points A (blue), B(red) and C(green) at (1,-4), (4,-3) and (5,-5) respectively, implying that movement could only occur horizontally or vertically toward a point. The problem states that: A taxi driver is given a specific territory of a town, as represented by the grid. All trips originate at the taxi stand. One very slow night, the driver is dispatched only three times; each time, she picks up passengers at one of the intersections indicated on the map. To pass the time, she considers all the possible routes she could have taken to each pick-up point and wonders if she could have chosen a shorter route. What is the shortest route from the taxi stand to each point? How do you know it is the shortest? Is there more than one shortest route to each point? If not, why not? If so, how many? Justify your answers.
PROBLEM STATEMENT: The problem was presented to the students with an accompanying representation on a single (fourth) quadrant of a coordinate grid of squares with the “taxi stand” located at (0,0) and the three “pick-up” points A (blue), B(red) and C(green) at (1,-4), (4,-3) and (5,-5) respectively, implying that movement could only occur horizontally or vertically toward a point. The problem states that: A taxi driver is given a specific territory of a town, as represented by the grid. All trips originate at the taxi stand. One very slow night, the driver is dispatched only three times; each time, she picks up passengers at one of the intersections indicated on the map. To pass the time, she considers all the possible routes she could have taken to each pick-up point and wonders if she could have chosen a shorter route. What is the shortest route from the taxi stand to each point? How do you know it is the shortest? Is there more than one shortest route to each point? If not, why not? If so, how many? Justify your answers.
Math ToolColor markers, Graph paper
Math StrandCounting, Combinatorics
Math ProblemTaxicab
NCTM Grade Range9-12
NCTM Content StandardAlgebra, Data analysis and probability, Geometry, Number and operations
NCTM Process StandardProblem solving, Reasoning and proof, Communication, Connections, Representation
Forms of Reasoning, Strategies and HeuristicsDirect reasoning, Considering a simpler problem
Grade Level12
Student ParticipantsBrian (Kenilworth, student), Romina (student), Jeff (student), Michael A. (Kenilworth, student)
MediatorsKiczek, Regina (Researcher)
SettingInformal learning
Student GenderMixed
Student EthnicityMixed
RightsThe video is protected by copyright. It is available for reviewing and use within the Video Mosaic Collaborative (VMC) portal. Please contact the Robert B. Davis Institute for Learning (RBDIL) for further information about the use of this video.
Date Captured2000-05-05
Local IdentifierA02A26-GMY-TAXI-CLIP002
Related Publication
Type: Dissertation
Label: Ph.D. dissertation references the video footage that includes Taxicab problem, clip 2 of 5.
Detail: Dissertation available in digital and paper formats in the Rutgers University Libraries dissertation collection.
Author: Powell, Arthur B. (Rutgers Graduate School of Education)
Name: So let's prove it!: emergent and elaborated mathematical ideas and reasoning in the discourse and inscriptons of learners engaged in a combinatorial task .
Type: Dissertation
Label: Ph.D. dissertation references the video footage that includes Taxicab problem, clip 2 of 5.
Detail: Dissertation available in digital and paper formats in the Rutgers University Libraries dissertation collection.
Author: Powell, Arthur B. (Rutgers Graduate School of Education)
Name: So let's prove it!: emergent and elaborated mathematical ideas and reasoning in the discourse and inscriptons of learners engaged in a combinatorial task .
Source
Title: A02, Taxicab problem, full session, grade 12, May 5, 2000, raw footage
Identifier: A02-20000505-KNWH-SV-AFTRS-GR12-GMY-TAXI-RAW
Title: A02, Taxicab problem, full session, grade 12, May 5, 2000, raw footage
Identifier: A02-20000505-KNWH-SV-AFTRS-GR12-GMY-TAXI-RAW
Source
Title: A26,Taxicab problem: full session, grade 12, May 5, 2000, raw footage
Identifier: A26-20000505-KNWH-WV-AFTRS-GR12-GMY-TAXI-RAW
Title: A26,Taxicab problem: full session, grade 12, May 5, 2000, raw footage
Identifier: A26-20000505-KNWH-WV-AFTRS-GR12-GMY-TAXI-RAW