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TitleA28, Night Session, Pascal's Identity (presentation view), Grade 11, May 12, 1999, raw footage

PublisherNew Brunswick, NJ: Robert B. Davis Institute for Learning, , c1999-05-12

Persistent URLhttps://doi.org/doi:10.7282/T3319TTG

DescriptionIn this full-session, raw footage video, students have come to school in the evening for a night session. The group, made up of Jeff, Michael and Romina begin discussing the coefficients of the binomial expansion, specifically (a+b) to the 10th power. In attempting to explain why 45 is the coefficient of the third term in this expansion, the students refer to counting how many 10-tall towers have exactly two cubes of a specific color. As they are joined by another member, Ankur, they discuss the formula for “n choose x” using factorial notation and what the factorial symbol means. When asked to explain “why you multiply,” Ankur responds by making use of an analogy of counting the number of ways to arrange three different colors. They then investigate the reason for dividing n! by (n-x)! and x! when calculating “n choose x.” In explaining the specific example of “5 choose 2,” they use the analogies: of arranging five people on a line when you are concerned about the positions of only two of the people and counting the number of 5-tall towers having exactly two cubes of one color. They then discuss the notation for combinations, which they called “choose” notation, and how it relates to Pascal’s Identity (the addition rule for Pascal’s Triangle). Michael describes how the third row of Pascal’s Triangle can be written in “choose notation” (1 3 3 1 becomes “3 choose 0, “3 choose 1,” “3 choose 2,” and “3 choose 3”). He and the other students explain how the answers to the 3-topping pizza problem are related to row 3 of Pascal’s Triangle, and how a specific instance of Pascal’s Identity can be understood as generating specific 4-topping pizzas from 3-topping pizzas. The researcher then asks the students to write Pascal’s Triangle in this notation, including a general row (row n). The students then explain to Brian, a late-comer, the meaning of Pascal’s Identity (the addition rule for Pascal’s Triangle) in terms of operations on the pizzas that are represented by specific entries in Pascal’s Triangle. They write Pascal’s Identity in general form using standard notation.

Notes:

“Choose” notation is the notation for counting the number of combinations; “n choose r” gives the number of ways of selecting subsets containing r objects from a set containing n objects. When counting combinations, the order of selection is irrelevant. “N choose r” is equal to n!/[(n – r)!r!].

The n-tall towers problem is: How many towers n cubes tall is it possible to make when there are two colors of cubes to choose from?

The n-topping pizza problem is: How many pizzas can be made when there are n different pizza toppings to choose from?

Notes:

“Choose” notation is the notation for counting the number of combinations; “n choose r” gives the number of ways of selecting subsets containing r objects from a set containing n objects. When counting combinations, the order of selection is irrelevant. “N choose r” is equal to n!/[(n – r)!r!].

The n-tall towers problem is: How many towers n cubes tall is it possible to make when there are two colors of cubes to choose from?

The n-topping pizza problem is: How many pizzas can be made when there are n different pizza toppings to choose from?

Math ToolCalculator

Math StrandCombinatorics, Counting

Math ProblemPascal's identity

NCTM Grade Range9-12

NCTM Content StandardNumber and operations

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

Forms of Reasoning, Strategies and HeuristicsDirect reasoning, Inductive reasoning, Referencing a previous problem, Recognizing a pattern, Recursive reasoning

Grade Level11

Student ParticipantsJeff (student), Romina (student), Michael A. (Kenilworth, student), Ankur (student), Brian (Kenilworth, student)

MediatorsMaher, Carolyn Alexander (Researcher), Pantozzi, Ralph S. (Researcher), Kiczek, Regina D. (Researcher), Steencken, Elena Perrone (Researcher)

SettingInformal learning

Student GenderMixed

Student EthnicityMixed

Camera ViewsClassroom view

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 Captured1999-05-12

Local IdentifierA28-19990512-KNWH-PV-CLASS-GR11-CMB-TW10T-RAW

Related Publication

__Type__: Related publication

__Label__: Ed.D. dissertation references video footage that includes A28, Night Session, Pascal's Identity (presentation view), Grade 11. May 12, 1999, raw footage.

__Date__: 2005

__Author__: Uptegrove, Elizabeth B. (Rutgers, the State University of New Jersey)

__Name__: To symbols from meaning: Students' investigations in counting

__Reference__: http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000055263

Related Publication

__Type__: Excerpt or clip creation

__Label__: Video clips created from the video footage A28, Night Session, Pascal's Identity (presentation view), Grade 11, May 12, 1999, raw footage

__Name__: Night session, Pascal's identity, clip 1 of 7: thinking about the meaning of combinatorics notation

__Reference__: http://hdl.rutgers.edu/1782.1/rucore00000001201.Video.000055279