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TitleTaxicab problem, clip 3 of 5: It's Pascal's triangle! But Why?

PublisherNew Brunswick, NJ: Robert B. Davis Institute for learning, , c2000-05-05

Persistent URLhttps://doi.org/doi:10.7282/T3K937CF

DescriptionIn the third of five clips, the four twelfth grade students attempt to justify for themselves and then demonstrate the relationship that they have conjectured between the Taxicab problem and Pascal's Triangle. After Romina is convinced that points on her chart fit the pattern of Pascal's Triangle for the first 4 rows, the students work to confirm that there are twenty possible paths to the point (3, -3) on the problem grid, which would correctly fit the pattern of Pascal's Triangle. Jeff suggests that, if they are able to answer the question of "why" the number of shortest routes corresponds to the numbers in Pascal's Triangle, they will no longer have to check each position on the grid. The students make further conjectures about a possible relationship between the Taxicab problem and building unifix towers when selecting from two colors..

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 ToolGraph paper, Color markers

Math StrandCounting, Combinatorics

Math ProblemTaxicab

NCTM Grade Range9-12

NCTM Content StandardNumber and operations, Algebra, Geometry, Data analysis and probability

NCTM Process StandardProblem solving, Reasoning and proof, Communication, Connections, Representation

Forms of Reasoning, Strategies and HeuristicsDirect reasoning, Considering a simpler problem, Guessing and checking, Recognizing an isomorphism, Recognizing equivalence

Grade Level12

Student ParticipantsBrian (Kenilworth, student), Romina (student), Jeff (student), Michael A. (Kenilworth, student)

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-CLIP003

Related Publication

__Type__: Dissertation

__Label__: Ph.D. dissertation references the video footage that includes Taxicab problem, clip 3 of 5.

__Date__: 2003-05-01

__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

Source

__Title__: A26, Taxicab problem: full session, grade 12, May 5, 2000, raw footage

__Identifier__: A26-20000505-KNWH-WV-AFTRS-GR12-GMY-TAXI-RAW