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**B65, Jeff and Michelle classwork of the five-tall Towers problem (work view), Grade 4, Feb 6, 1992, raw footage [video].**Retrieved from https://doi.org/doi:10.7282/T3D50RK8

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TitleB65, Jeff and Michelle classwork of the five-tall Towers problem (work view), Grade 4, Feb 6, 1992, raw footage

PublisherNew Brunswick, NJ: Robert B. Davis Institute for Learning, 1992-02-06, c1992-02-06

Persistent URLhttps://doi.org/doi:10.7282/T3D50RK8

DescriptionThe fourth grade class was divided into pairs to work on a Towers problem on February 6, 1992. At the beginning of the session, there are two sheets of paper posted on the board with the following statement:

“Building Towers

Your group has two colors of Unifix cubes for building towers. Work together and make as many different towers as you can that are five cubes high. See if you and your partner can plan a good way to find all the towers that are five cubes high and decide a way to record what you find.”

Researcher Maher introduces the problem to the group, who are seated in pairs and have bags of Unifix cubes at their desks. The camera in this video follows Michelle and Jeff as they work on the problem and then after some problem solving, there is a full class discussion about the solutions of different groups moderated by Researcher Maher.

Before beginning to build, Jeff states that they should begin with basic towers that consist of only one color and Michelle describes switching the colors in each tower (Brookes, 2015). Jeff built the towers with only one color and michelle created towers with exactly one yellow cube. Jeff proceeded to create towers with exactly two red. They discussed their towers and checked for duplicates. Michelle used a color opposite strategy to build new towers. Jeff reorganized the towers in a staircase pattern. They also build the towers with alternating colors and two with exactly three yellows (RRYYY and YYYRR). Michelle suggested that the towers arranged in a staircase were not checked against the other towers, but Jeff confirmed that he did. The pair agreed that each tower should have an opposite. Eventually they obtain 32 towers. They get questioned by the researchers during their session and explain that they checked their towers for duplicates. At times Jeff does not let Michelle handle the cubes as he constructs new towers. Researcher Maher suggested to them to separate the towers into easy to explain sets similar to the staircase pattern. They run out of time to do this when the class discussion began.

Researcher Maher asks the students to think about how they will explain their findings to the rest of the class and gives them a few minutes to prepare. She asks groups one by one how many towers they found, at which point there are answers of 32 and 34. Robert and Sebastian say they have found 35. She asks the class if it is possible to have an odd number of towers. Some students say no. One student, Michael, explains that once you build a tower, it must have a color opposite. Another student says it makes sense to have an odd number of towers because a person has the choice to make or not make the color opposite. Milin explains that you must “duplicate” each tower, which meant one must create the color opposite.

The researcher invites students who think they have 32 towers to check for duplicates in the towers of the group that got 35. The class finds the three duplicates. Students get time to work on their own tower collections to check for duplicates and many groups find 32.

The students then are invited to listen to Ankur’s method for finding all of the patterns. Ankur and Joe had a different method for generating towers with awareness of potential duplicates. They describe a “staircase” pattern, beginning with all red, then having one yellow, two yellows, three yellows, four yellows, all five yellow, and then one red, two reds, three reds, four red. They did not include an all red tower because it was in the beginning of the line for this group. Researcher Maher discusses how many groups, including Jeff’s and Ankur’s, made towers with exactly one red on each “floor” (or each position) but in the case of Ankur’s group, their group of towers with exactly one red did not include the red on the top or bottom floor because they were already in their “staircase” pattern. Maher compares how Jeff’s group also did this and noticed duplication; therefore, both groups continued their new patterns to the extent of no duplication from the first patterns.

She then asked the students to study their towers with exactly two reds or two yellows, what they look like, and how many there are in total.

Researcher Maher poses the problem to the class where she wants them to find a way to convince her that they have all of the towers by using similar schemes for towers with exactly two, three, and four of the same color cubes. She gives them an idea to think about the different possibilities of exactly two reds together and how they might know that there cannot be any more ways, imagining the cubes moving up one floor starting from the bottom. They find 4 towers with two reds together. The researcher points out that these are not the only towers with exactly two red cubes. Together as a class, the researcher asks if there can be towers separated by one, two, three, and four floors, at which point Stephanie says no to the latter case. They discuss why there are no more cases. Stephanie and others suggest it is not possible to have four or more yellow cubes separating the red cubes because there are only five cubes in total. As a class they do a case based organization exploration for exactly two reds together, one yellow, two yellow, and three yellow cubes apart.

Researcher Maher pointed out to the whole class that they got ten total towers with exactly two reds. She asked if there is a set of towers that they can imagine in their heads and know the total immediately. Some students say they can make the opposites with exactly two yellows. Researcher Maher asks how many they have in total so far? The class says 20 towers total. She asks them to imagine the towers with exactly one red and how many there are. Students realized there is 5 and 5 more with exactly one yellow, the opposites. She asks what the remaining 2 towers are? Robert points out that it is the tower with all red or no yellow and its opposite. It ends with the class determining there are 32 towers in total when selecting from two colors.

“Building Towers

Your group has two colors of Unifix cubes for building towers. Work together and make as many different towers as you can that are five cubes high. See if you and your partner can plan a good way to find all the towers that are five cubes high and decide a way to record what you find.”

Researcher Maher introduces the problem to the group, who are seated in pairs and have bags of Unifix cubes at their desks. The camera in this video follows Michelle and Jeff as they work on the problem and then after some problem solving, there is a full class discussion about the solutions of different groups moderated by Researcher Maher.

Before beginning to build, Jeff states that they should begin with basic towers that consist of only one color and Michelle describes switching the colors in each tower (Brookes, 2015). Jeff built the towers with only one color and michelle created towers with exactly one yellow cube. Jeff proceeded to create towers with exactly two red. They discussed their towers and checked for duplicates. Michelle used a color opposite strategy to build new towers. Jeff reorganized the towers in a staircase pattern. They also build the towers with alternating colors and two with exactly three yellows (RRYYY and YYYRR). Michelle suggested that the towers arranged in a staircase were not checked against the other towers, but Jeff confirmed that he did. The pair agreed that each tower should have an opposite. Eventually they obtain 32 towers. They get questioned by the researchers during their session and explain that they checked their towers for duplicates. At times Jeff does not let Michelle handle the cubes as he constructs new towers. Researcher Maher suggested to them to separate the towers into easy to explain sets similar to the staircase pattern. They run out of time to do this when the class discussion began.

Researcher Maher asks the students to think about how they will explain their findings to the rest of the class and gives them a few minutes to prepare. She asks groups one by one how many towers they found, at which point there are answers of 32 and 34. Robert and Sebastian say they have found 35. She asks the class if it is possible to have an odd number of towers. Some students say no. One student, Michael, explains that once you build a tower, it must have a color opposite. Another student says it makes sense to have an odd number of towers because a person has the choice to make or not make the color opposite. Milin explains that you must “duplicate” each tower, which meant one must create the color opposite.

The researcher invites students who think they have 32 towers to check for duplicates in the towers of the group that got 35. The class finds the three duplicates. Students get time to work on their own tower collections to check for duplicates and many groups find 32.

The students then are invited to listen to Ankur’s method for finding all of the patterns. Ankur and Joe had a different method for generating towers with awareness of potential duplicates. They describe a “staircase” pattern, beginning with all red, then having one yellow, two yellows, three yellows, four yellows, all five yellow, and then one red, two reds, three reds, four red. They did not include an all red tower because it was in the beginning of the line for this group. Researcher Maher discusses how many groups, including Jeff’s and Ankur’s, made towers with exactly one red on each “floor” (or each position) but in the case of Ankur’s group, their group of towers with exactly one red did not include the red on the top or bottom floor because they were already in their “staircase” pattern. Maher compares how Jeff’s group also did this and noticed duplication; therefore, both groups continued their new patterns to the extent of no duplication from the first patterns.

She then asked the students to study their towers with exactly two reds or two yellows, what they look like, and how many there are in total.

Researcher Maher poses the problem to the class where she wants them to find a way to convince her that they have all of the towers by using similar schemes for towers with exactly two, three, and four of the same color cubes. She gives them an idea to think about the different possibilities of exactly two reds together and how they might know that there cannot be any more ways, imagining the cubes moving up one floor starting from the bottom. They find 4 towers with two reds together. The researcher points out that these are not the only towers with exactly two red cubes. Together as a class, the researcher asks if there can be towers separated by one, two, three, and four floors, at which point Stephanie says no to the latter case. They discuss why there are no more cases. Stephanie and others suggest it is not possible to have four or more yellow cubes separating the red cubes because there are only five cubes in total. As a class they do a case based organization exploration for exactly two reds together, one yellow, two yellow, and three yellow cubes apart.

Researcher Maher pointed out to the whole class that they got ten total towers with exactly two reds. She asked if there is a set of towers that they can imagine in their heads and know the total immediately. Some students say they can make the opposites with exactly two yellows. Researcher Maher asks how many they have in total so far? The class says 20 towers total. She asks them to imagine the towers with exactly one red and how many there are. Students realized there is 5 and 5 more with exactly one yellow, the opposites. She asks what the remaining 2 towers are? Robert points out that it is the tower with all red or no yellow and its opposite. It ends with the class determining there are 32 towers in total when selecting from two colors.

Math ToolUnifix cubes

Math StrandCounting, Combinatorics

Math ProblemTowers

NCTM Grade Range3-5

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

Forms of Reasoning, Strategies and HeuristicsDirect reasoning, Guessing and checking, Recognizing a pattern

Grade Level4

Student ParticipantsJeff (student), Michelle I. (student)

SettingClassroom

Student GenderMixed

Student EthnicityWhite

Camera ViewsStudent 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 Captured1992-02-06

Local IdentifierB65-19920206-KNWH-SV-CLASS-JEFFMICH-GR4-CMB-T5T-RAW

Source

__Title__: B65,Combinatorics (work view) Grade 4,February 6,1992,raw footage.__Identifier__: B65-19920206-KNWH-SV-CLASS-JEFFMICH-GR4-CMB-T5T-RAW