Dataset2/D2TASS
From Jsarmi
Contents
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Group Trajectory
Session 1: Dyad from same school, technical problems, f2f work produces some reported results Session 2: Feedback attended to Session 3: Session 4:
Group composition: Stable/ Technical Problems
Session 1: tc ro (TP) Session 2: tc ro Session 3: tc ro (TP) Session 4: tc ro (TP) Technical Problems
Session I
Work done F2F is reported to the chat... how is this "reporting" deployed? Initially Tc attempts a direct dialog style:
tcmath 5/9/06 5:12:43 PM EDT: So, Rook, we have the fact that squares is defined as a function of N
by the partial sum quadratic for the sequence 1,2,3,4,5....
This whiteboard textbook was created by tcmath before the previous message:
sequence:1,2,3,4,5 partial sum sequence:1,3,6,10,15 Function is quadratic, so these eqautions are true: a1^2+b1+c=1
but later, because of technical problems, the style changes to a plural narrative:
jsarmi 5/9/06 5:19:46 PM EDT: It seems as if the system is giving Rook a lot of trouble, isn't it, tcmath? jsarmi 5/9/06 5:21:41 PM EDT: tcmath? tcmath 5/9/06 5:24:36 PM EDT: I'm done for now tcmath 5/9/06 5:24:48 PM EDT: I was working with Rook since his computer wasn't working jsarmi 5/9/06 5:24:57 PM EDT: I see tcmath 5/9/06 5:25:16 PM EDT: We figured out the equation for squares, and we should easily solve it for sticks as well in the same manner
This lat posting works like a "projection/plan" which includes a report of what was done. This posting is deceptively simple. It is a report of what was done framed as "figuring out" but such achievement (i.e. the equation for the squares) is not recap, nor placed on the whiteboard. The projection made of what can be done in the future with two qualifications: level of difficulty (easily) and manner (the same way). It is presented to an unspecified recipient, most likely, the moderator who has been giving out instruction on the task and asking for the status of Rook's participation.
tcmath 5/9/06 5:25:16 PM EDT: We figured out the equation for squares, and we should easily solve it for sticks as well in the same manner.
Wiki posting for Session I?
It isn't clear that results were posted between Sessions I and II. Most likely, in Session II the team posted results for both sessions. Some results are posted to the "rows of the table" page. Interestingly, their results differ with what Team B posted in the same page (almost next to it) by other teams. There is uptake of this triggered by the facilitator during Session II. For example:
Row 3
Team name: B
Row 3: 28, 10
Team name: A
Row 3: 18, 6
Team name:
Row 3:
Feedback
(No Feedback in between sessions)
Session II
Technical problems seemed to be resolved. Dyad must be in same room or connected otherwise, no greetings. Tc starts with this message after which the board is CLEARED of everything that was posted in the previous session.
tcmath 5/11/06 5:04:13 PM EDT: Okay. You must agree that the squares is defined by the equation squares=1/2N^2+1/2N.
Rook attempts to paste a table of values into a textbox but the formatting does not work too well and he ends up with a long list instead. Interestingly, this list lacks values that may have been posted to the wiki already? E.g., The values for N=4 and N=5 which in the wiki read "Team name: A Row 4: 28, 10 / Team name: A Row 5: 40, 15", and in Rooks texbox: "4 ? ? / 5 ? ?". Perhaps they created these values in the second session since they map their equation: N^2 + 3*N?
Are we doing Session 1 still? Lack of coordination?:
tcmath 5/11/06 5:09:53 PM EDT: I'm setting up the problem on the whiteboard p M M M tcmath 5/11/06 5:10:08 PM EDT: p M M M M Rook 5/11/06 5:11:18 PM EDT: Did you solve the sticks as function of N yet? Rook 5/11/06 5:11:35 PM EDT: I think the equation is x^2+3x Rook 5/11/06 5:11:50 PM EDT: Found the equation... tcmath 5/11/06 5:11:54 PM EDT: The seperate lines here represent the sticks you would "add" when N increases by 1. p M M Rook 5/11/06 5:12:11 PM EDT: What are you doing? p M M Rook 5/11/06 5:12:17 PM EDT: !! Rook 5/11/06 5:12:40 PM EDT: Are we doing Session 1 still? tcmath 5/11/06 5:12:43 PM EDT: Okay. The top 4 sticks are there by default; you have to have them.
Linking Sequences: Squares then Sticks
How do they constitute the two to be related?
tcmath 5/11/06 5:26:20 PM EDT: So, for squares we found the partial sum equation for the sequence 1,2,3,4,5... Now we find it for the sequence 2n+2 (after distribution)
Then we will be done
Definitively in the same room but conversing/coordinationg on the chat. Notice long gaps between postings
tcmath 5/11/06 5:28:30 PM EDT: Here, you can use my TI-84 to do the whole regression thing and I'll do it algebraically on paper. Tell me your results. Rook 5/11/06 5:29:34 PM EDT: oh... Rook 5/11/06 5:30:36 PM EDT: x^2+3X tcmath 5/11/06 5:31:51 PM EDT: Then we're done with the pattern:
Typing on the Wiki
tcmath 5/11/06 5:42:32 PM EDT: we're here--we're just typing up our patterns observed on VMT Wiki
Reading the Wiki, Finding the "next" topic
jsarmi 5/11/06 5:47:28 PM EDT: Well.. any ideas suggested by the notes from other groups in the Wiki? tcmath 5/11/06 5:48:02 PM EDT: Well, they figured out the same thing for squares, but their approach was unique for the sticks. tcmath 5/11/06 5:48:33 PM EDT: We used the same technique for both teh sticks and the squares (partial sums on a sequence), while they used two different ones. jsarmi 5/11/06 5:48:41 PM EDT: how about the values of the table? Rook 5/11/06 5:49:02 PM EDT: they got the same Rook 5/11/06 5:49:33 PM EDT: and we checked with the other groups--same answers... tcmath 5/11/06 5:49:38 PM EDT: They were ahead one row, though, starting with row 4 since it was the first row without a given answer. Rook 5/11/06 5:49:53 PM EDT: on the 1st website (VMT Wiki) jsarmi 5/11/06 5:50:12 PM EDT: Oh... I see tcmath 5/11/06 5:50:31 PM EDT: For our discussion: I think that mathematiciancs would usually generalize a lot, finding a method that fit the problem even if initial values were changed. Rook 5/11/06 5:50:46 PM EDT: Yeah Rook 5/11/06 5:50:54 PM EDT: that's the next topic
Our method would still work
tcmath 5/11/06 5:51:39 PM EDT: For example, if every column in the diagram increased by 2 instead of 1. Rook 5/11/06 5:52:20 PM EDT: I think that the polygons/figures have to be regular to have a chance at this problem, of course--or else there's really no pattern Rook 5/11/06 5:52:39 PM EDT: Assume that the rest of the answers would just be doubled? tcmath 5/11/06 5:52:57 PM EDT: If this happened, then our method would still work becuase we would determine the original sequence and then generalize it in a closed form, quadratic equation, for the partial sums. Rook 5/11/06 5:53:02 PM EDT: The 2 wouldn't make a difference really Rook 5/11/06 5:53:22 PM EDT: Yeah, or solve it as written, jumping by 2... Rook 5/11/06 5:53:37 PM EDT: so, it should also work for n-gons that are regular tcmath 5/11/06 5:54:18 PM EDT: Probably not for three since the sticks really depends on the number of squares we're increasing by. In ours, we had 4 in our equation, while if we increased by three, that number would be 10, I think. Rook 5/11/06 5:54:34 PM EDT: Yeah, the equation, such as the initial starting point, would have to change to fit more or less starting sticks
Projecting
tcmath 5/11/06 5:58:42 PM EDT: We should try looking at the specific case of, say pentagons.
p M
Rook 5/11/06 5:59:11 PM EDT: Yeah
Rook 5/11/06 5:59:18 PM EDT: though do you have to leave?
tcmath 5/11/06 5:59:28 PM EDT: That'll have to wait till next time. I have to leave.
I wonder if we could even replicate the problem with regular pentagons?
Rook 5/11/06 5:59:45 PM EDT: mb in our spare time we could look at it and talk next session
Feedback
Dear tcmath and Rook, We were very interested in the way you combined several approaches to find the formula for the pattern of growth of sticks and squares. Using a regression method and thinking about the difference between one step of the pattern and the next seem to work very nicely. You also posted very complete descriptions of your ideas on the Wiki. In your previous session you started to explore what the patterns would look like with other polygons and in 3-D, and you made a couple of conjectures about them (e.g. “the methods for solving general situations like this would be the same”). For the next step we will encourage you to continue thinking about the different approaches and the problems that you can discover of your own and that are interesting to pursue. -The VMT Team
Session III
The feedback talks about 3D so Rook orients to it:
tcmath 5/16/06 5:04:24 PM EDT: The 3-d seems like a good place to start.
Noticed that Rook had made some "projected" observations about 3D in the previous session:
Rook 5/11/06 5:58:40 PM EDT: but 3-D might include solving it in "2-D", such as seeing the patterns
on one "dimension" of the 3-D figures, and combing the formulats into one
equation for the 3-D problem
Tcmath reports on work they did in between. Because of technical problems they had to "talk" in person and report in the chat:
tcmath 5/16/06 5:06:04 PM EDT: We started talking about 3d after the last session and decided
that the method would just be double: The original sequence would be the 2-d problem
tcmath 5/16/06 5:08:51 PM EDT: This is true becuase if the problem is a pyramid, expanding n cubes in each direction,
then the number of squares starts with this pattern, then is the same pattern for n-1,
then n-2, all the way to one cube.
jsarmi 5/16/06 5:10:12 PM EDT: tcmath... you may want to wait until Rook gets here so that you can both discuss
the ideas you are posting
tcmath 5/16/06 5:11:36 PM EDT: Okay. we'll just talk (in person) and then type up what we said.
A projectable (pentagons) recovered but discarded?
They decide that Rook will type in parenthesis his ideas
jsarmi 5/16/06 5:22:41 PM EDT: (BTW, Once you read the feedback you can delete it so that you can work on the whiteboard freely) tcmath 5/16/06 5:25:08 PM EDT: The feedback seems to indicate that we should work on 3d., since pentagons tesselate in a weird way (in 5 directions instead) jsarmi 5/16/06 5:25:42 PM EDT: This is for you to decide... you can go in any direction that seems promising to you. tcmath 5/16/06 5:25:59 PM EDT: Rook: if it's 2-D; assume that you add 2, then 3...figures every time... tcmath 5/16/06 5:26:39 PM EDT: 3D seems like a good place to continue, since cubes are a lot like squares. tcmath 5/16/06 5:27:42 PM EDT: [every side/plane is similar to the squares problem] tcmath 5/16/06 5:28:06 PM EDT: [since we already figured out the equations for squares] tcmath 5/16/06 5:30:00 PM EDT: 3d goes like this: the base has 1/2N^2+1/2N cubes, as we figured out in the squares problem, and each level above it has 1/2(N-1)^2+1/2(N-1), and so on.
Reportable: Same method as before, Algebra
tcmath 5/16/06 5:41:10 PM EDT: ANYhoo, the 3d partial sum equation is probably a cubic (I think the patter n is, the partial sum equation has a degree one higher than the sequence it is partial-summing) jsarmi 5/16/06 5:41:30 PM EDT: ok tcmath 5/16/06 5:42:22 PM EDT: We'll use the same method as we did last time, with algebra
Wiki writing: a table
Rook 5/16/06 5:51:34 PM EDT: I'm going to make a table for the data value for a the cube sequences jsarmi 5/16/06 5:51:40 PM EDT: sure... probably better to concentrate on the math than fighting the computers... sorry about that jsarmi 5/16/06 5:54:51 PM EDT: a table on the whitebaord? Rook 5/16/06 5:57:47 PM EDT: on Wiki jsarmi 5/16/06 6:00:56 PM EDT: I see Rook leaves the room 5/16/06 6:04:27 PM EDT
Then, abrupt ending
Feedback
Dear tcmath and Rook: We are sorry that you ran into technical problems last time. You tried to work with one computer but that must have been awkward. Despite that, it seemed to us that you were able to start exploring a 3-D pattern based on your notes posted on the Wiki. Do you need to specify which 3-D pattern you are investigating? Is it similar to the ones that other teams are exploring? We hope that this time you will be able to collaborate via the system and continue this work or explore other interesting math question you create. -The VMT team
This feedback message gets deleted very early on Session IV by tcmath. Rook claims he knows how to get it from the history
Session IV
The whitebaord stays clear for the most of the session after Tcmath erases everything, until a critical point when tcmath wants to "take a look" at something on the whitebaord
Lets continue + Wiki
Rook 5/18/06 5:25:41 PM EDT: o.k., let's continue the cubes version of the problem Rook 5/18/06 5:25:44 PM EDT: tcmath jsarmi 5/18/06 5:26:15 PM EDT: hopefully tcmath is just distracted with the Wiki and not having computer problems Rook 5/18/06 5:26:31 PM EDT: I'll make a chart of the faces as function of N... tcmath 5/18/06 5:27:40 PM EDT: Okay. I'm posting all of our ideas, since we came up with a lot of them. When I'm done, you can post your ideas on the wiki as you come up with them.
Rook 5/18/06 5:31:52 PM EDT: R you done posting? Rook 5/18/06 5:31:54 PM EDT: tcmath tcmath 5/18/06 5:32:09 PM EDT: Yeah. Its long, but its our work.
I figured it out... everything in 3D has a strong correlation to the original squares problem
Rook 5/18/06 5:34:50 PM EDT: so far, for N={1,2} I've gotten Faces={6,14}
Rook 5/18/06 5:34:54 PM EDT: it looks similar...
tcmath 5/18/06 5:35:19 PM EDT: I figured it out! (the faces). Its just 4*(1/2n^2+1/2n)
Rook 5/18/06 5:35:39 PM EDT: that makes sense!
Rook 5/18/06 5:35:55 PM EDT: the faces should be similar to the squares problem
Rook 5/18/06 5:36:05 PM EDT: it's just "magnified" times 4!
tcmath 5/18/06 5:36:41 PM EDT: I'll explain. There are two flat faces which are the full pyramid,
and one which is 3d from one angle, and that same side has another 1/2n^2+1/2n squares viewable
from a different abgle.
tcmath 5/18/06 5:36:51 PM EDT: Does this make any sense?
Rook 5/18/06 5:37:22 PM EDT: you mean the "opposite" side of the cube pyramid?
tcmath 5/18/06 5:37:34 PM EDT: Exactly. Its intuitive, not mathematical as one might expect.
However, we do notice that everything in 3D has a strong orrelation to the original squares problem
Wait a minute...I'll look at it on the whiteboard
tcmath 5/18/06 5:43:20 PM EDT: Wait a minute...I'll look at it on the whiteboard;
I think the actual formula is 2.5N^2+2.5N+1, or 5(1/2N^2+1/2N)+1, the extra 1 for the top face.
p M M M M M M M
Rook 5/18/06 5:44:08 PM EDT: (i get the four pyramids now: I was counting wrong)
p M M M M
tcmath 5/18/06 5:44:46 PM EDT: So the whiteboard now shows one view, a flat view of the 3d figure
p M M
Rook 5/18/06 5:45:14 PM EDT: where?
Rook 5/18/06 5:45:16 PM EDT: I don't see it
tcmath 5/18/06 5:45:43 PM EDT: You need to slide the slider on the left down to the bottom.
p M M
Rook 5/18/06 5:45:57 PM EDT: I think it just took some time...
Something correlates
Rook 5/18/06 5:48:30 PM EDT: something with the sticks in the original problem correlate to the number of faces in the cubes tcmath 5/18/06 5:48:33 PM EDT: Yes, you're right. I think the ofrmula I came up with needs some adjustments. Rook 5/18/06 5:48:50 PM EDT: the faces is similar to the sticks in the original Rook 5/18/06 5:49:02 PM EDT: however, there are 6 faces compared to 4 sticks Rook 5/18/06 5:49:16 PM EDT: and the faces increase by 8, then 10, then 12... Rook 5/18/06 5:49:25 PM EDT: while the sticks are 6,8,10... tcmath 5/18/06 5:50:09 PM EDT: We probably should use that method, with partial sums tcmath 5/18/06 5:50:09 PM EDT: We probably should use that method, with partial sums. Rook 5/18/06 5:50:41 PM EDT: yeah... Rook 5/18/06 5:50:59 PM EDT: I think it's really the same equation, with slight addition modifications... p M Rook 5/18/06 5:51:21 PM EDT: what's the sticks original equation again? Rook 5/18/06 5:51:42 PM EDT: let's see
Final action: the rest on the Wiki
tcmath 5/18/06 5:58:58 PM EDT: I'm leaving. Rook will put the rest of our ideas on the wiki ... Rook 5/18/06 6:01:51 PM EDT: I think I'll quickly post what we found out just as tcmath was leaving (I checked with him in "life") on Wiki, I think I also have to leave in a minute
Squares, Sticks, Cubes and Faces
Interestingly, this group did not see a problem with working on a problem that was too similar to the original one, as other groups did. It almost seems that it was exciting for them to figure out that the same results applied in some parts of it. For other groups this was a reason not to approach the problem at all. In addition, this group broke the cube into "faces" which led them to reuse prior findings but did not break the "faces" into sticks. In other words, in the original pattern 2D (squares) was divided into 1D (sticks) but for their pyramid, 3D (cube) was only divided into 2D (faces).
