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B60, Milin and Michael 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/T34M985H
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TitleB60, Milin and Michael 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/T34M985H
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 Milin and Michael 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.
During the group work, Milin and Michael generated towers using guess and check. Both students used local organization by first making a tower and then generating its partner tower by switching the colors of the corresponding cubes or by flipping the first tower upside down (Sran, 2010). Throughout their work on this task, Milin and Michael monitored their work by visually checking each new tower against the already made towers to eliminate any duplicates (Sran, 2010). Using the guess and check strategy and local organization of pairs the towers, both students were able to find all thirty-two towers using red and yellow cubes (Sran, 2010).
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. 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. 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 Milin and Michael 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.
During the group work, Milin and Michael generated towers using guess and check. Both students used local organization by first making a tower and then generating its partner tower by switching the colors of the corresponding cubes or by flipping the first tower upside down (Sran, 2010). Throughout their work on this task, Milin and Michael monitored their work by visually checking each new tower against the already made towers to eliminate any duplicates (Sran, 2010). Using the guess and check strategy and local organization of pairs the towers, both students were able to find all thirty-two towers using red and yellow cubes (Sran, 2010).
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. 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. 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 ParticipantsMilin (student), Michael M. (Kenilworth, student)
SettingClassroom
Student GenderMale
Student EthnicityWhite
Camera ViewsStudent view
Date Captured1992-02-06
Local IdentifierB60-19920206-KNWH-SV-CLASS-MILINMIKE-GR4-CMB-T5T-RAW