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TitleEarly algebra, investigating linear functions, series 5 of 7, ladder problem, Clip 5 of 7: Ariel revisits his solution for the 120 step ladder

PublisherNew Brunswick, N.J.: Robert B. Davis Institute for Learning, , c2005-12-15

Persistent URLhttps://doi.org/doi:10.7282/T31Z434M

DescriptionIn the fifth of seven clips from an after-school enrichment session in an urban middle school, Ariel, a 7th grade boy, continues his work on the Ladder Problem. Researcher John Francisco challenges Ariel to reexamine his solution for a ladder with 120 steps by comparing it with the procedure that he had written down as a general rule for finding the number of rods for ladders with even numbers of steps. The procedure that Ariel had articulated and followed for relatively small numbers had the following steps: (1) divide the number by two; (2) build a ladder with that number of steps with actual rods; (3) count the rods; then (4) double that number and subtract two from the product. Ariel insists that he is following his rule and explains that he divides 120 by 2 to find 60. He then refers to his earlier count of 20 for a ladder with 6 steps and multiplies that by 10 to get 200 for 60 steps. Ariel then refers to his rule and subtracts 2, for a result of 198. To find the number of rods for 120 steps, Ariel doubles 198, subtracts another 2, and insists that the result of 394 is the correct number. Despite the researcher' s attempts to get Ariel to follow his written procedure more closely, including an appeal to James, who briefly looks over Ariel's written solution, Ariel did not waver from this answer. Researcher Markus Hahkioniemi is interacting with James and researcher Prashant Baldev is observing.

The worksheet wording for the Ladder Problem:

A company makes ladders of different heights, from very short ones to very tall ones. The shortest ladder has only one rung, and looks like this (we could build a model of it with 5 light green Cuisenaire rods.) A two-rung ladder could be modeled using 8 light green rods, and looks like this. Build a rod model to represent a 3-rung ladder.

How many rods did you use? How could you represent the number of rods needed if you were to build a ladder with any number of rungs?

The questions as posed to Ariel and James:

Compare your solution for a ladder with 120 steps with your rule.

Are you following the steps that you outlined?

Explain how you used your rule to find your solution.

The worksheet wording for the Ladder Problem:

A company makes ladders of different heights, from very short ones to very tall ones. The shortest ladder has only one rung, and looks like this (we could build a model of it with 5 light green Cuisenaire rods.) A two-rung ladder could be modeled using 8 light green rods, and looks like this. Build a rod model to represent a 3-rung ladder.

How many rods did you use? How could you represent the number of rods needed if you were to build a ladder with any number of rungs?

The questions as posed to Ariel and James:

Compare your solution for a ladder with 120 steps with your rule.

Are you following the steps that you outlined?

Explain how you used your rule to find your solution.

Math ToolCuisenaire rods, Color markers

Math StrandAlgebra

Math ProblemLadder problem

NCTM Grade Range6-8

NCTM Content StandardAlgebra

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

Forms of Reasoning, Strategies and HeuristicsDirect reasoning, Considering a simpler problem, Recursive reasoning, Proportional reasoning

Grade Level7

Student ParticipantsAriel (student), James T. (student)

MediatorsFrancisco, John (Researcher), Baldev, Prashant V. (Researcher), Hahkioniemi, Markus (Researcher)

SettingInformal learning

Student GenderMale

Student EthnicityMixed

Camera ViewsWork view, Student 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 Captured2005-12-15

Local IdentifierB17B18-ALG-VAR-CLIP005

Related Publication

__Type__: Related publication

__Label__: Ed.D. dissertation references the video footage that includes Early algebra, investigating linear functions, series 5 of 7, ladder problem, Clip 5 of 7: Ariel revisits his solution for the 120 step ladder.

__Date__: 2009

__Author__: Baldev, Prashant V. (Rutgers, the State University of New Jersey)

__Name__: Urban, seventh-grade students building early algebra ideas in an informal after school program

__Reference__: QA.B175 2009

Source

__Title__: B17, Early algebra, investigating linear functions, Series 5 of 7, Ladder problem (student view), Grade 7, December 15, 2005, raw footage

__Identifier__: 17-20051215-PFLD-SV-IFML-GR7-ALG-VAR-RAW

Source

__Title__: B18, Early algebra, investigating linear functions, Series 5 of 7, Ladder problem (student view), Grade 7, December 15, 2005, raw footage

__Identifier__: B18-20051215-PFLD-SV-IFML-GR7-ALG-VAR-RAW