Dataset2/D2TBSS
From Jsarmi
Group Trajectory
Session 1: Bwang divided the figure into horizontal and vertical lines, Formulas for both sticks and squares were produced,
The values for N=4, 5, and 6 were agreed upon and posted to the Wiki. Quicksilver offers an explanation of how
the pattern grows but it is not discussed
Session 2: Feedback attended to
Session 3:
Session 4:
Group composition: Stable
Session 1: bw qs az Session 2: bw qs az Session 3: Session 4: (L) (N)
Session I
bwang8 5/9/06 6:23:18 PM EDT: hi Aznx 5/9/06 6:23:23 PM EDT: Hi Quicksilver 5/9/06 6:23:28 PM EDT: hey Aznx 5/9/06 6:23:35 PM EDT: So we can't have our own friends? bwang8 5/9/06 6:23:40 PM EDT: nope bwang8 5/9/06 6:23:49 PM EDT: lol Quicksilver 5/9/06 6:23:52 PM EDT: hey three of us are from miller Gerry 5/9/06 6:23:55 PM EDT: Hi everyone!
Gerry 5/9/06 6:24:59 PM EDT: Today's session is mainly to get to know the VMT system
Gerry 5/9/06 6:29:12 PM EDT: You can click on the button at the top that says "View Topic" to see the math problem bwang8 5/9/06 6:29:32 PM EDT: ok bwang8 5/9/06 6:29:50 PM EDT: are we suppose to solve it now? Gerry 5/9/06 6:30:13 PM EDT: Then you can click on the button in the little window that appears to open the topic in another big growser window Gerry 5/9/06 6:30:36 PM EDT: browser* Aznx 5/9/06 6:30:40 PM EDT: It didn't open. Aznx 5/9/06 6:30:52 PM EDT: Now it did. Aznx 5/9/06 6:31:32 PM EDT: So, are we supposed to work together? bwang8 5/9/06 6:31:49 PM EDT: yeah bwang8 5/9/06 6:31:50 PM EDT: ok Gerry 5/9/06 6:31:54 PM EDT: Exactly! Aznx 5/9/06 6:32:04 PM EDT: Aditya, you there? bwang8 5/9/06 6:32:05 PM EDT: you can divide the thing into two parts Aznx 5/9/06 6:32:10 PM EDT: Let's start this thing. (bwang draws the third iteration of the pattern splitted in 2 diagrams dividing the horizontal and vertical lines) ... Quicksilver 5/9/06 6:32:58 PM EDT: what are the lines for? Aznx 5/9/06 6:33:01 PM EDT: go to view topic bwang8 5/9/06 6:33:05 PM EDT: so you can see we only need to figur one out to get the total stick
bwang8 5/9/06 6:33:32 PM EDT: 1+2+3+........+N+N bwang8 5/9/06 6:33:38 PM EDT: times that by 2 Quicksilver 5/9/06 6:33:40 PM EDT: Never mind I figured it out.. Aznx 5/9/06 6:34:01 PM EDT: Can we collaborate this answer even more? Aznx 5/9/06 6:34:05 PM EDT: To make it even simpler? bwang8 5/9/06 6:34:15 PM EDT: ok Aznx 5/9/06 6:34:16 PM EDT: Because I think we can. bwang8 5/9/06 6:34:50 PM EDT: ((1+N)*N/2+N)*2 bwang8 5/9/06 6:34:58 PM EDT: that's the formula, right? Aznx 5/9/06 6:35:15 PM EDT: How did you come up with it? bwang8 5/9/06 6:35:16 PM EDT: for total sticks bwang8 5/9/06 6:35:34 PM EDT: is a common formual bwang8 5/9/06 6:35:40 PM EDT: formula Aznx 5/9/06 6:35:46 PM EDT: Yeah, I know. bwang8 5/9/06 6:35:59 PM EDT: and just slightly modify it to get this (whiteboard) Aznx 5/9/06 6:36:31 PM EDT: Aditya, you get this right? (bwang corrects his drawing so that the horizontal lines orient in the same way that the original problem drawing)
bwang8 5/9/06 6:38:38 PM EDT: The number of squares is just (1+N)*N/2 (points to his formula posted at 6:34:50 PM)
Gerry 5/9/06 6:38:52 PM EDT: I put BWang's formula on the whiteboard (whiteboard) Aznx 5/9/06 6:39:45 PM EDT: So how do we submit this? Quicksilver 5/9/06 6:40:26 PM EDT: We are still in the process Quicksilver 5/9/06 6:40:38 PM EDT: We are discussing Gerry 5/9/06 6:40:41 PM EDT: you can complete the table for different N and put that on the wiki
The practice of moving formulas to the whiteboard is not picked up, so the formula for the number of squares never makes it on to the whiteboard.
Quicksilver 5/9/06 6:46:50 PM EDT: Are we Team B? Aznx 5/9/06 6:46:55 PM EDT: TEAM B bwang8 5/9/06 6:47:02 PM EDT: i am done Aznx 5/9/06 6:47:06 PM EDT: I think's its case sensitive. Aznx 5/9/06 6:47:11 PM EDT: Alright, my turn. Gerry 5/9/06 6:47:25 PM EDT: Nice! bwang8 5/9/06 6:47:27 PM EDT: is mine in the correct format
Aznx 5/9/06 6:49:34 PM EDT: The number of squares is 15. Gerry 5/9/06 6:49:47 PM EDT: You might want to draw the table on the whiteboard to make sure you all agree (table being constructed on the whiteboard by bwang) Quicksilver 5/9/06 6:51:00 PM EDT: There are 15 squares and 44 sticks for n=5? (correction 4 instead of 3 and the values for n=5 just mentioned by Quicksilver added by bwang) bwang8 5/9/06 6:52:03 PM EDT: isn't it 40 bwang8 5/9/06 6:52:18 PM EDT: (15+5)*2 (correction 40 instead of 44 by bwang)
Interesting... This an application of ((1+N)*N/2+N)*2 -> ((1+5)*5/2+5)*2) -> (15+5)*2 = 40
Later they have on the whiteboard two tables and one textbox which contain overlap but disagree in one number. In two, number of sticks for N=5 is 44 in the rest it is 40, so they try to coordinate this. Quicksilver does not seem responsive and later he posts this long chat message:
Quicksilver 5/9/06 7:00:39 PM EDT: Well, anyway, you can see a pattern that the amount of squares increases by the n. For the
sticks, The bottom row's square on the right has 2 new sticks. All the squares in the new row
to the left of it have 3 new sticks. So, If te row has 5 squares, 4 of the squares have 3 sticks,
the last on only has two. For the enitre Figure, you would add the amount to the previous ammount
After that technical problem the moderator sort of closes down the session:
Gerry 5/9/06 7:04:43 PM EDT: Maybe we should stop for today
Aznx 5/9/06 7:04:48 PM EDT: Alright.
jsarmi 5/9/06 7:04:50 PM EDT: Quicksilver... why don't you go ahead and close this window and try to enter again
Gerry 5/9/06 7:05:10 PM EDT: and let jsarmi help Quicksilver get ready to Thursday
jsarmi 5/9/06 7:05:27 PM EDT: (BTW, this room will now show up in the "My Rooms" tab not on the "Limited Access" one)
Quicksilver leaves the room 5/9/06 7:05:43 PM EDT
Gerry 5/9/06 7:05:48 PM EDT: You made a lot of progress already.
Aznx 5/9/06 7:05:53 PM EDT: So do we leave?
Aznx 5/9/06 7:06:01 PM EDT: And when do we come back?
...
Gerry 5/9/06 7:06:51 PM EDT: I think we will have the next session at the same time on Thursday.
You will come into the same room and start where you left off today
jsarmi 5/9/06 7:07:06 PM EDT: that's correct... same time, same team, same room
bwang8 5/9/06 7:07:12 PM EDT: ok
Quicksilver 5/9/06 7:07:15 PM EDT: I have a dentis't appointment...I don't think I can come
Aznx 5/9/06 7:07:16 PM EDT: Same time on Thursay...got it!
bwang8 5/9/06 7:07:37 PM EDT: but aren't we done now with this problem
Aznx 5/9/06 7:07:47 PM EDT: Yeah
Aznx 5/9/06 7:07:56 PM EDT: So I guess we would start on the second problem?
bwang8 5/9/06 7:08:05 PM EDT: ok
Quicksilver 5/9/06 7:08:16 PM EDT: Did you guys discuss the problem like it said to?
Aznx 5/9/06 7:08:21 PM EDT: Yeah.
Quicksilver 5/9/06 7:08:22 PM EDT: i didn't get half the messages'
Quicksilver 5/9/06 7:08:24 PM EDT: so i dont know
bwang8 5/9/06 7:08:25 PM EDT: yeah
Because Quicksilver can't make it on Thursday , they reschedule their session for the next day:
Quicksilver 5/9/06 7:09:45 PM EDT: Can we just reschedule for wednesday or something?
jsarmi 5/9/06 7:11:03 PM EDT: If it works out for everyone, you could re-schedule... is that an option?
Aznx 5/9/06 7:11:23 PM EDT: Wednesday should work for me.
Quicksilver 5/9/06 7:11:31 PM EDT: it works for me
Aznx 5/9/06 7:11:43 PM EDT: bwang?
bwang8 5/9/06 7:11:50 PM EDT: yes
Quicksilver 5/9/06 7:11:59 PM EDT: So this time tomorrow then
Quicksilver 5/9/06 7:12:11 PM EDT: in place of thursdays
bwang8 5/9/06 7:12:14 PM EDT: wednesday, 6-7 central?
Aznx 5/9/06 7:12:34 PM EDT: Same time as today.
Quicksilver 5/9/06 7:12:38 PM EDT: Yeah
bwang8 5/9/06 7:12:44 PM EDT: ok
Aznx 5/9/06 7:12:45 PM EDT: Except it's Wednesday instead of Thursday.
....
Aznx 5/9/06 7:13:37 PM EDT: So will we start on a new problem?
jsarmi 5/9/06 7:13:43 PM EDT: Is it clear to everyone???
Aznx 5/9/06 7:13:46 PM EDT: Tomorrow?
Quicksilver 5/9/06 7:13:48 PM EDT: Yes.
bwang8 5/9/06 7:13:59 PM EDT: ok
jsarmi 5/9/06 7:14:11 PM EDT: Some of you are on Central time and some of you on Pacific time, so do not get confused...
same time as today
bwang8 5/9/06 7:14:17 PM EDT: ok
Quicksilver 5/9/06 7:14:25 PM EDT: Ok. See you tomorrow then.
Feedback
VMT Feedback We were very interested in the approach that divided the figure into the horizontal lines and the vertical lines and the quickness with which formulas fell out of that approach. It seemed as though you also were paying attention to each other's work and quickly reached agreement. You handled the technology of the chat environment and the wiki easily. We also noticed two places in the chat where some kinds of conversation did not happen. There was a point where 44 was posted as the number of sticks and 40 was offered as a correction. There was no discussion of how 44 was calculated. At another moment, Quicksilver posted an explanation of the pattern of growth that was not discussed. There was a sense in which you indicated that your work was done when you had at least one answer for the questions in the problem. For the next step we will encourage you to think more about the different approaches and the problems that you can discover on your own and that are interesting to pursue.
Session II
Gerry 5/10/06 7:02:03 PM EDT: You have some feedback and a new Topic
Aznx 5/10/06 7:02:13 PM EDT: Has everyone read the feedback?
Quicksilver 5/10/06 7:02:17 PM EDT: yup
Quicksilver 5/10/06 7:02:25 PM EDT: NOw we have to discuss
bwang8 5/10/06 7:02:28 PM EDT: yes
Quicksilver 5/10/06 7:03:01 PM EDT: Well,, the part about converstaion not happening is because of me
Quicksilver 5/10/06 7:03:06 PM EDT: my computer was lagging....
Quicksilver 5/10/06 7:03:12 PM EDT: but that's out of our hands
Aznx 5/10/06 7:03:17 PM EDT: So, I think we should focus on discussing on each step more.
Quicksilver 5/10/06 7:03:30 PM EDT: and explain every answer thoroughly
Aznx 5/10/06 7:03:40 PM EDT: Even if the answer was "obvious."
bwang8 5/10/06 7:03:48 PM EDT: ok
Quicksilver 5/10/06 7:03:49 PM EDT: like i gave a wrong answer, but my explanations didn't come up
on the computer because of the lag
Quicksilver 5/10/06 7:03:58 PM EDT: so thats one thing
Quicksilver 5/10/06 7:04:12 PM EDT: maybe think of more ways of doing the same problem?
Aznx 5/10/06 7:04:21 PM EDT: Yeah.
They try to start an activity
Two proposals, looking at today's topic and talking about Quicksilvers wrong idea:
Quicksilver 5/10/06 7:04:50 PM EDT: Now did you two read today's topic? bwang8 5/10/06 7:05:08 PM EDT: what was your pattern of growth, quicksiler Quicksilver 5/10/06 7:06:20 PM EDT: i think it was something about the amount of squares that increased with each row....and how one of the new squares had 3 new sticks while the other new ones had 2 new sticks p M bwang8 5/10/06 7:06:41 PM EDT: oh, ok (Quicksilver starts to draw squares with sticks to illustrate his talk to which he refers to in his next chat message) Quicksilver 5/10/06 7:07:07 PM EDT: i drew some sqaures Quicksilver 5/10/06 7:07:19 PM EDT: the left one had three new sticks Quicksilver 5/10/06 7:07:33 PM EDT: the right one has a new stick on the bottom and on the right Quicksilver 5/10/06 7:07:39 PM EDT: the top one is from an old square Gerry 5/10/06 7:07:40 PM EDT: It was at 7:00:39 -- to get the old messages, click on the icon above here with the two circular arrows (this messages points to a message from the previous day's session, unclear if they followed the link?) (the feedback textbox is deleted from the whiteboard) Quicksilver 5/10/06 7:09:25 PM EDT: yea that's wrong Aznx 5/10/06 7:09:33 PM EDT: So let's brainstorm through some problems that we think are challenging.
Aznx goes back to his idea of orienting to a new task
Instead of quicksilver's error to which bwang and quicksilver oriented, others agree:
Quicksilver 5/10/06 7:09:40 PM EDT: yes...new topic bwang8 5/10/06 7:09:42 PM EDT: ok Quicksilver 5/10/06 7:10:20 PM EDT: 3-d figures? (quicksilver adds sticks to the 2 squares he drew earlier to make them 3-D)
Projecting
Later on, Aznx adds a third 3D square and calls them a "row of blocks". Aznx makes some "projections" of what they could do:
Aznx 5/10/06 7:11:06 PM EDT: I think we should discuss on the different methods. Aznx 5/10/06 7:11:24 PM EDT: So that we can easily apply our thoughts quickly when seeing a problem. Quicksilver 5/10/06 7:11:30 PM EDT: Yes....but we must find a question or problem to investigate Aznx 5/10/06 7:11:37 PM EDT: Yeah. p M M Aznx 5/10/06 7:11:50 PM EDT: I think we should start off with a conjecture, that we need to prove. p M M M Aznx 5/10/06 7:12:03 PM EDT: Not a hard one, but one that can be challenging. Quicksilver 5/10/06 7:12:17 PM EDT: Maybe a row of blocks Quicksilver 5/10/06 7:12:27 PM EDT: likethis Aznx 5/10/06 7:12:44 PM EDT: What about them? Quicksilver 5/10/06 7:12:59 PM EDT: The amount of sticks may increase in a pattern? bwang8 5/10/06 7:13:01 PM EDT: The problem from yesterday, but only 3-d
The problem from yesterday! BUT different.
Notice no reference to members! (They are stable!) and this is taken by Quicksilver to be a negative assessment, a complaint! BUT ONLY?
Quicksilver 5/10/06 7:13:07 PM EDT: yea i guess Quicksilver 5/10/06 7:13:09 PM EDT: not that good
He repairs the offer, bwang accepts, he mitigates it, bwant escalates, aznx endorses it, quicksilver mitigates put down:
Quicksilver 5/10/06 7:13:18 PM EDT: maybe a pyramind bwang8 5/10/06 7:13:24 PM EDT: yeah Quicksilver 5/10/06 7:13:30 PM EDT: although that's hard to draw bwang8 5/10/06 7:13:35 PM EDT: pryamind is good Aznx 5/10/06 7:13:36 PM EDT: Yeah, I liked that. Quicksilver 5/10/06 7:13:36 PM EDT: but we shoudl be able to managt Quicksilver 5/10/06 7:13:36 PM EDT: e (Quicksilver draws a 2d version of a stack of cubes in pyramid shape and marks it as "side view")
Yesterday is still a problem:
bwang8 5/10/06 7:14:56 PM EDT: isn't this the same as yesterday problem Quicksilver 5/10/06 7:15:03 PM EDT: Really? Aznx 5/10/06 7:15:10 PM EDT: Except it's 3-D. Quicksilver 5/10/06 7:15:12 PM EDT: no it's three d
What can we use that we ALREADY KNOW:
Interesting:
Aznx 5/10/06 7:16:45 PM EDT: So, how should we approach this? Aznx 5/10/06 7:16:54 PM EDT: What can we use that we already know? Quicksilver 5/10/06 7:16:57 PM EDT: Layer by layer shown in a chart? bwang8 5/10/06 7:17:01 PM EDT: well we can divide it into a front and a back Aznx 5/10/06 7:17:02 PM EDT: I'd suggest yesterday's problem. bwang8 5/10/06 7:17:10 PM EDT: yeah bwang8 5/10/06 7:17:22 PM EDT: using the formula from yesterday's problem bwang8 5/10/06 7:17:32 PM EDT: we can figure the front and back easily Quicksilver 5/10/06 7:17:36 PM EDT: this (Points to the formula from Session I which is still on the whiteboard)
Notice here Projections again, in this case very collaboratively constructed: "break it down" is something they used in Session I.
bwang8 5/10/06 7:27:55 PM EDT: the last level have 9 Quicksilver 5/10/06 7:28:07 PM EDT: yeah p M M M M M bwang8 5/10/06 7:28:28 PM EDT: so we will just have to figure out how many sticks make up 3 by 3 blocks p M M M M M Aznx 5/10/06 7:29:06 PM EDT: Yes. p M Aznx 5/10/06 7:29:15 PM EDT: After that, we go up to Nth step. Quicksilver 5/10/06 7:29:20 PM EDT: Yes p M M M M M M M M M M M bwang8 5/10/06 7:30:07 PM EDT: ok, how do we figure that out p M bwang8 5/10/06 7:30:17 PM EDT: 3*3 blocks p M Quicksilver 5/10/06 7:30:26 PM EDT: Break it down Aznx 5/10/06 7:30:27 PM EDT: I'd say look for a pattern. p M Aznx 5/10/06 7:30:33 PM EDT: and yes, break it down. p M Aznx 5/10/06 7:30:40 PM EDT: What other possible ways are there? Aznx 5/10/06 7:30:44 PM EDT: That we know of? bwang8 5/10/06 7:30:52 PM EDT: top, middle and bottom p M M M M M M bwang8 5/10/06 7:31:29 PM EDT: top and bottom are 3 by 3 squares Quicksilver 5/10/06 7:31:33 PM EDT: whoops i drew it wrong
"Yesterday stuff:"
bwang8 5/10/06 7:32:27 PM EDT: do you mind if i erase some yesterday stuff? Quicksilver 5/10/06 7:32:34 PM EDT: yea u can do it Diagrams and other materials (e.g. table) are deleted, the formula is kept and then a "decomposed" diagram with horizontal and vertical sticks for a 3x3 grid is created by bwang (as he did in S1!) ... bwang8 5/10/06 7:33:57 PM EDT: when they combine bwang8 5/10/06 7:34:05 PM EDT: they make a 3 by 3 square Quicksilver 5/10/06 7:34:08 PM EDT: yes Quicksilver 5/10/06 7:34:14 PM EDT: so just count these bwang8 5/10/06 7:34:28 PM EDT: and the equation for it is 2N(N+1) bwang8 5/10/06 7:34:35 PM EDT: right? p M M bwang8 5/10/06 7:34:58 PM EDT: N is the level Quicksilver 5/10/06 7:34:59 PM EDT: I don't know Aznx 5/10/06 7:35:04 PM EDT: Prove it. Quicksilver 5/10/06 7:35:11 PM EDT: Where did you get it? Aznx 5/10/06 7:35:16 PM EDT: I'll help you as you go along. Aznx 5/10/06 7:35:27 PM EDT: I kind of get it, but not clearly. bwang8 5/10/06 7:35:27 PM EDT: i mean just from the top and bottom 3 by 3 squares Aznx 5/10/06 7:35:44 PM EDT: Where did the 2 come from? bwang8 5/10/06 7:35:54 PM EDT: this is 3(3+1) (reference to the whitebaord)
Sharedness
Interesting explanation of bwang8's formula. Is it shared now? Was it just in bwang's problem space before? Was it in everbody's problem space but positioned differently? With different meanings?
Bwang had sugested that their pyramid can be broken down into "top, middle and bottom" and that the "top and bottom are 3 by 3 squares" Then bwang draws a "broken down" version of a 3 by 3 square that looks like this on the whiteboard:
_ _ _
_ _ _ | | | |
_ _ _ | | | |
_ _ _ | | | |
Then he follows up with this explanation:
bwang8 5/10/06 7:33:57 PM EDT: when they combine bwang8 5/10/06 7:34:05 PM EDT: they make a 3 by 3 square Quicksilver 5/10/06 7:34:08 PM EDT: yes Quicksilver 5/10/06 7:34:14 PM EDT: so just count these bwang8 5/10/06 7:34:28 PM EDT: and the equation for it is 2N(N+1) bwang8 5/10/06 7:34:35 PM EDT: right? (bwang manipulates some objects on the whiteboard, invisible) bwang8 5/10/06 7:34:58 PM EDT: N is the level Quicksilver 5/10/06 7:34:59 PM EDT: I don't know Aznx 5/10/06 7:35:04 PM EDT: Prove it. Quicksilver 5/10/06 7:35:11 PM EDT: Where did you get it? Aznx 5/10/06 7:35:16 PM EDT: I'll help you as you go along. Aznx 5/10/06 7:35:27 PM EDT: I kind of get it, but not clearly. bwang8 5/10/06 7:35:27 PM EDT: i mean just from the top and bottom 3 by 3 squares Aznx 5/10/06 7:35:44 PM EDT: Where did the 2 come from? bwang8 5/10/06 7:35:54 PM EDT: this is 3(3+1) (Points to the left side of his diagram, the array of horizontal lines)
Claim Markings:
Quicksilver 5/10/06 7:36:05 PM EDT: ok Aznx 5/10/06 7:36:05 PM EDT: Ah. Quicksilver 5/10/06 7:36:07 PM EDT: i c Aznx 5/10/06 7:36:08 PM EDT: I get it.
Bwang aknowledges, moves on:
bwang8 5/10/06 7:36:10 PM EDT: ok bwang8 5/10/06 7:36:10 PM EDT: ok bwang8 5/10/06 7:36:32 PM EDT: so now we get the top and bottom bwang8 5/10/06 7:36:40 PM EDT: we need to find the middle
But this "sharedness" is contingent, not cumulative! For example... in this passage both Quicksilver and bwang create different graphics and at the end, they use it to understand each other's views:
Quicksilver 5/10/06 7:37:14 PM EDT: i don't understand something Quicksilver 5/10/06 7:37:15 PM EDT: Quicksilver 5/10/06 7:37:15 PM EDT: Quicksilver 5/10/06 7:37:15 PM EDT: Quicksilver 5/10/06 7:37:19 PM EDT: sorry\\ Quicksilver 5/10/06 7:37:28 PM EDT: um...what do you mean it is the top and bottom bwang8 5/10/06 7:37:37 PM EDT: oh Quicksilver 5/10/06 7:37:39 PM EDT: it is a pyramid with a flat face right p M bwang8 5/10/06 7:37:47 PM EDT: let me explain (Quicksilver and Bwang, each, create a drawing. Bwang's is 3D, Quicksilver's 2D with colors) Quicksilver 5/10/06 7:38:38 PM EDT: This face could go againts a wall(this is a top view) [Points to his drawing] bwang8 5/10/06 7:38:48 PM EDT: it is just a way to divide the problem (Quicksilver colors a part of his diagram in yellow) Quicksilver 5/10/06 7:39:05 PM EDT: so the yellow is the top level? (Quicksilver creates a yellow 3 by 3 grid of squares, Bwang marks lines in his original diagram with scribbles) bwang8 5/10/06 7:39:45 PM EDT: we are just focusing on the bottommost level Aznx 5/10/06 7:39:54 PM EDT: Yeah. bwang8 5/10/06 7:40:04 PM EDT: and all those edges with scribble on it are the top Quicksilver 5/10/06 7:40:18 PM EDT: So this is against the floor underneath the other two levels [Points to his yellow 3 by 3 grid] bwang8 5/10/06 7:40:35 PM EDT: yes Quicksilver 5/10/06 7:40:43 PM EDT: ok...continue then
Q: Is "coloring" (like "labeling") a a more "persistent" diectic? (Note: the use of whiteboard references is very sophisticated here) How do the 2 diagrams stand in relation to each other, their authors and other participants
bwang8 5/10/06 7:41:02 PM EDT: all the vertical lines are consider the middle [Points to his diagram]] Quicksilver 5/10/06 7:41:21 PM EDT: ok bwang8 5/10/06 7:41:32 PM EDT: and the rest are the bottom bwang8 5/10/06 7:41:48 PM EDT: which has the same number of sticks as the top
p M M
Explaining to others and to oneself
Also, bwang, who is "explaining" is also monitoring the intelegibility of his own explanation, and complains about it:
bwang8 5/10/06 7:42:51 PM EDT: my explanation is pretty bad bwang8 5/10/06 7:43:01 PM EDT: do you understand? Quicksilver 5/10/06 7:43:12 PM EDT: enough to go on with the problem Aznx 5/10/06 7:43:12 PM EDT: I understand it.
Bwang posts this equation on the whiteboard, not on the chat:
top/bottom: 2n(n+1) middle: (n+1)^2
"recursive equation/function" appears
bwang8 5/10/06 7:46:30 PM EDT: so we can use recursive equation to figure out n 3d pryamind bwang8 5/10/06 7:46:36 PM EDT: stick number bwang8 5/10/06 7:46:42 PM EDT: lol bwang8 5/10/06 7:46:48 PM EDT: bad grammar (bwang scribles on the whiteboard: Sum (n=1, n) = 4n(n+1) + (n+1)^2
Later he clarifies how is that this is a "recursive function"
Quicksilver 5/10/06 7:50:18 PM EDT: It goes up by one every time because n=1 rite? bwang8 5/10/06 7:50:38 PM EDT: no, n start at 1 Quicksilver 5/10/06 7:50:46 PM EDT: oh yeah!!! bwang8 5/10/06 7:50:46 PM EDT: this is a recursive function Quicksilver 5/10/06 7:50:46 PM EDT: bwang8 5/10/06 7:51:26 PM EDT: when n=1, plug 1 into the right equation\\ Quicksilver 5/10/06 7:51:40 PM EDT: 12? bwang8 5/10/06 7:51:58 PM EDT: when n=2, plug 2 into the right equation and add the equation when n=1 Quicksilver 5/10/06 7:52:13 PM EDT: Oh...so u add the thing that came before bwang8 5/10/06 7:52:17 PM EDT: yes bwang8 5/10/06 7:52:31 PM EDT: 12 is right (points to Quicksilver's 7:51:40 PM)
Transition to the Wiki
What to put, who/how to put it.
Quicksilver 5/10/06 7:57:45 PM EDT: so we have to go to the wiki now ... Quicksilver 5/10/06 8:01:03 PM EDT: so our equation Quicksilver 5/10/06 8:01:11 PM EDT: no first lets state our problem bwang8 5/10/06 8:01:49 PM EDT: The number of sticks to make a N level pryamind bwang8 5/10/06 8:02:04 PM EDT: ? Aznx 5/10/06 8:02:17 PM EDT: Yeah, that's our question. Quicksilver 5/10/06 8:02:56 PM EDT: Err... Quicksilver 5/10/06 8:03:03 PM EDT: I kind of put somethingt different Quicksilver 5/10/06 8:03:04 PM EDT: Quicksilver 5/10/06 8:03:09 PM EDT: But it's similar Aznx 5/10/06 8:03:24 PM EDT: Wait, do we all have to put it in? Aznx 5/10/06 8:03:30 PM EDT: Only one of us right?
Today's result was "technically" the same as yesterday but today was "really" a discussion
Quicksilver 5/10/06 8:04:03 PM EDT: you guys can add on Quicksilver 5/10/06 8:04:08 PM EDT: i just put the basic Quicksilver 5/10/06 8:04:20 PM EDT: Maybe share our results? Aznx 5/10/06 8:04:53 PM EDT: We technically had the same result. Quicksilver 5/10/06 8:05:07 PM EDT: Whaddya mean? Quicksilver 5/10/06 8:05:21 PM EDT: oh as yesterday? Aznx 5/10/06 8:05:31 PM EDT: Yeah. Aznx 5/10/06 8:05:36 PM EDT: And today. Quicksilver 5/10/06 8:05:40 PM EDT: Still... Aznx 5/10/06 8:05:43 PM EDT: Well today was really a discussion. Quicksilver 5/10/06 8:05:46 PM EDT: we should say that' ... Quicksilver 5/10/06 8:06:39 PM EDT: That may mean these types of problems all are similar in one way
The "natural" person to write their findings on the Wiki might be bwang since he is the one who was driving the process and explaining things to others, but he says he is "bad with words". So Quicksilver suggests that they do it together. In reality, he does it by himself almost, which is a great display of his level of engagement with prior group activity:
Aznx 5/10/06 8:06:55 PM EDT: Wait, who is submitting? Aznx 5/10/06 8:06:57 PM EDT: bwang? bwang8 5/10/06 8:06:59 PM EDT: tell them the intervals between levels Aznx 5/10/06 8:07:06 PM EDT: or quicksilver, or me? bwang8 5/10/06 8:07:08 PM EDT: sorry, i am bad with words Quicksilver 5/10/06 8:07:14 PM EDT: So am i Aznx 5/10/06 8:07:15 PM EDT: Not to worry. Aznx 5/10/06 8:07:21 PM EDT: We should write it out Aznx 5/10/06 8:07:22 PM EDT: here Quicksilver 5/10/06 8:07:24 PM EDT: Aznx to the rescue lol Aznx 5/10/06 8:07:26 PM EDT: Together Quicksilver 5/10/06 8:07:30 PM EDT: sure Aznx 5/10/06 8:07:30 PM EDT: No not that Quicksilver 5/10/06 8:07:34 PM EDT: a conclusion Aznx 5/10/06 8:07:34 PM EDT: Yeah lol bwang8 5/10/06 8:07:36 PM EDT: yeah Aznx 5/10/06 8:07:51 PM EDT: So first, we started off with the basic problem. Aznx 5/10/06 8:07:57 PM EDT: and changed it to a 3-D problem Aznx 5/10/06 8:08:02 PM EDT: to make it more challenging Quicksilver 5/10/06 8:08:04 PM EDT: Yea Aznx 5/10/06 8:08:22 PM EDT: then we divided the top, bottom, and middle area down, to make the problem simpler for us Aznx 5/10/06 8:08:42 PM EDT: Then, we recognized the 2N(N+1) pattern in the middle area Quicksilver 5/10/06 8:08:44 PM EDT: That was one strategy\\ Aznx 5/10/06 8:08:54 PM EDT: that's our second: finding a pattern Quicksilver 5/10/06 8:08:55 PM EDT: dividing it Quicksilver 5/10/06 8:08:59 PM EDT: yes Aznx 5/10/06 8:09:07 PM EDT: Finally, we used the recursion method to figure out the obttom area, Quicksilver 5/10/06 8:09:15 PM EDT: obttom? bwang8 5/10/06 8:09:16 PM EDT: Aznx 5/10/06 8:09:19 PM EDT: So that's three strategies right there. Aznx 5/10/06 8:09:32 PM EDT: I meant bottom ...
retrospective-prospective
Later, Quicksilver still assigns the task to bwang. Also, here there is a retrospective-prospective move: The feedback said X, we can do that next time
Aznx 5/10/06 8:10:32 PM EDT: I think bwang should put it in, since he's more familiar with the recursion method and how to use it than we are, Aznx 5/10/06 8:10:36 PM EDT: Agreed? Quicksilver 5/10/06 8:10:51 PM EDT: Today's topic siad go to the wiki and share the most intersting math problems that your group chose to work on Quicksilver 5/10/06 8:10:59 PM EDT: agreed Quicksilver 5/10/06 8:11:15 PM EDT: but we do understand it now Quicksilver 5/10/06 8:11:18 PM EDT: that's important Aznx 5/10/06 8:11:28 PM EDT: Well, we should just say we wanted to explore yesterday's problem more. Quicksilver 5/10/06 8:11:29 PM EDT: maybe we can apply it next time...who knows? Aznx 5/10/06 8:11:32 PM EDT: Yes, we do. bwang8 5/10/06 8:11:32 PM EDT: ok bwang8 5/10/06 8:12:06 PM EDT: we can use the strategy we used to solve this problem to solve future problems bwang8 5/10/06 8:12:31 PM EDT: the method is important bwang8 5/10/06 8:12:36 PM EDT: not the answer Aznx 5/10/06 8:12:48 PM EDT: Yup. Quicksilver 5/10/06 8:12:49 PM EDT: definiteyly Aznx 5/10/06 8:12:56 PM EDT: Always learned that whereever I learned math. =P
Interestingly, here they find it relevant to talk about things that they "learned." How did they do that? For what purpose?
bwang8 5/10/06 8:13:05 PM EDT: we learn that divide the problem up can make it simpler and easier to solve Quicksilver 5/10/06 8:13:09 PM EDT: so bwang...are you updating the wiki? Quicksilver 5/10/06 8:13:14 PM EDT: yea Aznx 5/10/06 8:13:21 PM EDT: we also learned finding a pattern is a good step Quicksilver 5/10/06 8:13:39 PM EDT: yes and we could have also started with a simpler problem Quicksilver 5/10/06 8:13:42 PM EDT: in fact...we did Aznx 5/10/06 8:13:43 PM EDT: and recursion can be usually used when solving for a pattern, after finding the designated pattern of course Quicksilver 5/10/06 8:13:48 PM EDT: yesterday's problem was simpler Aznx 5/10/06 8:13:52 PM EDT: yes, we did! Aznx 5/10/06 8:14:00 PM EDT: so we actually used 4 strategies =D Quicksilver 5/10/06 8:14:12 PM EDT: yes Aznx 5/10/06 8:14:34 PM EDT: We also tried to look at the problem from different views, although it's not really a strategy.
The moderator kind of "closes" down the session by bringing up next session:
Gerry 5/10/06 8:15:46 PM EDT: So, can you all come back here at the same time next Tuesday?
What gets on the Wiki
An elaborate narrative of what they did (to which they will add after the other 2 sessions)
To investigate the number of sticks in a flat faced pyramid with n levels with 1 block increase in length and width per level. Also, to find as many approaches and put them to use. We eventually found 4 different strategies and applied them, such as divide the problem up, finding a basic pattern, and use recursion to solve problems. We also found a formula, its origins, and how to use it. f(n)=4n(n+1)+(n+1)^2+f(n-1) and f(0)=0. We first determine the number of squares in each level of the pyramid, 1 cubes in first level, 4 cubes in second level, 9 cubes in third level, and so on. Then we divide each level into 3 parts, the top, the bottom and the middle. The top is the same as the bottom part. They are just a bunch of squares in square format. When divided top or bottom into vertical or horizontal, the equation for # of sticks is n(n+1). Times that by 2 and you get a top or bottom, and times it by 2 again to get the total for top and bottom. The result is 4n(n+1). Then there is the middle which are straight columns, and they are n+1 by n+1 on the side, so (n+1)^2. The equation for each level is 4n(n+1)+(n+1)^2, and use a recursive fuction and you can get the total sticks for the pyramid.
Feedback II
VMT Feedback on Session #2 You can load the old chat messages by clicking on the double arrow icon above the chat scroll bar. You can look through the history of the whiteboard by using the scroll bar all the way on the left (be sure to scroll all the way down to the present in order to draw anything new.) You did a great job of defining a challenging problem and solving it by using a combination of methods that were well suited to the problem. And you shared what you did with the other groups in the wiki. You noticed that stating the problem and making it clear to everyone is a big part of working on a problem. In going to 3-D, you selected a particular kind of pyramid. How would your problem change if you had two flat sides, with each layer in a corner of the layer underneath, so that some cube faces and edges (sticks) were shared between layers? Can you explain your formula for the number of sticks so that someone in a different group can see how you got it by breaking each layer into its top surface, bottom and middle and then counting the horizontal and vertical sticks separately? Do you understand how team C got its formulae for the diamond pattern of squares? What if they had a diamond pattern of diamonds (just rotate the squares 45 degrees)? What shapes make mathematically interesting patterns in 2-D or in 3-D?
Session III
Bwang's message in between sessions III and IV
bwang8 joins the room 5/16/06 8:42:43 PM EDT p M M M M M M M M M M M M M M M M M bwang8 5/17/06 12:18:23 AM EDT: Sorry i have to leave early bwang8 5/17/06 12:18:56 AM EDT: i updated our wiki page to explain the equation from the last session p M M M bwang8 5/17/06 12:34:54 AM EDT: i also added some stuff, change anything you want bwang8 5/17/06 12:39:55 AM EDT: i hope you can read this
Feedback III
We started doing the problems you were doing and got very interested in them. We did not always arrive at the same answers and we weren't always able to decide if we were looking at it in the same way that you were, but we liked working "with you" in this sense. From our perspective, the goal has not been for you to answer “our” problems, but for you to figure out math questions that interest you and to pursue them. That is why we set the challenge for the iPod as having your group excel at collaboration, not at finding some answers. You seem to have pursued some interesting questions and are all contributing and also making use of each other’s ideas. It is sometimes hard for us to tell what you are writing and thinking. It seems that there are times when you say you are following each other, but it is not clear that you are really in agreement or completely understand each other. You might actually discover some more math if you state things in more detail – to be completely sure you are in agreement. For session four, you could revisit a problem you were working on before, in order to state more clearly for other groups in the wiki: (a) a definition of your problem, (b) a solution and (c) how you solved the problem. Or you could try a new variation of these pattern problems, like a 3-D version of group C’s diamond pattern. It is up to you to pursue whatever most interests you and what enables you to improve and enjoy your ability to work together. As you know, one hour goes by pretty quickly, so it’s easy to run out of time for a complicated problem. Be creative and enjoy the session.
