A28, Night Session, Pascal's Identity (presentation view), Grade 11, May 12, 1999, raw footage
Descriptive
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Research data
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Observational data
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Raw data
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Repurposed data
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Longitudinal data
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School
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Qualitative research
Genre
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Educational interventions (small group)
Subject
Name
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Subject
Name
(authority = RBDIL_personal)
NamePart
Romina (student)
Subject
Name
(authority = RBDIL_personal)
NamePart
Michael A. (Kenilworth, student)
Subject
Name
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Subject
Name
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NamePart
Brian (Kenilworth, student)
Subject
Name
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NamePart
David Brearley High School (Kenilworth, N.J.)
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Subject
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Topic
Mathematics education
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Topic
Learning, Psychology of--Case studies
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Topic
Critical thinking in children--New Jersey--Case studies
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Topic
Number and operations
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(authority = NCTM Process)
Topic
Reasoning and proof
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(authority = NCTM Process)
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(authority = NCTM Process)
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Subject
(authority = rbdil_gradeLevel)
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(authority = rbdil_forms of reasoning, strategies and heuristics)
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(authority = rbdil_forms of reasoning, strategies and heuristics)
Topic
Inductive reasoning
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(authority = rbdil_forms of reasoning, strategies and heuristics)
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Referencing a previous problem
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(authority = rbdil_forms of reasoning, strategies and heuristics)
Topic
Recognizing a pattern
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(authority = rbdil_forms of reasoning, strategies and heuristics)
Topic
Recursive reasoning
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(authority = rbdil_topic)
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TypeOfResource
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TitleInfo
Title
A28, Night Session, Pascal's Identity (presentation view), Grade 11, May 12, 1999, raw footage
Identifier
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A28-19990512-KNWH-PV-CLASS-GR11-CMB-TW10T-RAW
Identifier
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http://hdl.rutgers.edu/1782.1/rucore00000001201.Video.000065130
Subject
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Geographic
Kenilworth Public Schools
Classification
(authority = RUresearch);
(edition = Data)
Abstract
(type = summary)
In 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?
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1
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Social science
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Mathematics education
Note
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Transcript is also available.
Note
(type = APA citation)
Robert B. Davis Institute for Learning. (1999). A28, Night Session, Pascal's Identity (presentation view), Grade 11, May 12, 1999, raw footage [video]. Retrieved from http://hdl.rutgers.edu/1782.1/rucore00000001201.Video.000065130
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Maher
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Carolyn Alexander
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Rutgers, the State University of New Jersey
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Pantozzi
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Ralph S.
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Rutgers, the State University of New Jersey
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Kiczek
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Regina D.
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Rutgers, the State University of New Jersey
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Steencken
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Elena Perrone
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Rutgers, the State University of New Jersey
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New Brunswick, NJ
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Robert B. Davis Institute for Learning
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To symbols from meaning: Students' investigations in counting / Elizabeth B. Uptegrove
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Night session, Pascal's identity, clip 1 of 7: thinking about the meaning of combinatorics notation
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Night session, Pascal's identity, clip 2 of 7: making sense of factorial notation and "why you multiply”
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Night Session, Pascal’s Identity, Clip 5 of 7: Another way to write Pascal’s Triangle
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Night Session, Pascal’s Identity, Clip 6 of 7: Examples of Pascal’s Identity in the notation for combinations
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Night Session, Pascal’s Identity, Clip 7 of 7: Generating and generalizing Pascal’s Identity
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Ed.D. dissertation references video footage that includes A28, Night Session, Pascal's Identity (presentation view), Grade 11. May 12, 1999, raw footage.
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2005
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Dissertation available in digital and paper formats in the Rutgers University Libraries' dissertation collection
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