From Jsarmi

Revision as of 12:57, 3 May 2008

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

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) but, as will be seen, there is no uptake of this (by any of the 2 teams): 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.

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