From Jsarmi

Group Trajectory

 Session 1: 
 Session 2: Feedback attended to
 Session 3: 
 Session 4:

Group composition: Stable

 Session 1:  js   dc   ot   ss(L)
 Session 2:  js        ot          qw          
 Session 3:  js(E)     ot          qw  
 Session 4:  js        ot   ss(!)  qw
 
 (L) Late  (E) Leave Early  (!) Very Little Participation
 (*) js knows ss:  e.g. Jason 5/11/06 7:07:52 PM EDT: ssjnish says that his client is still loading
     ot knows dc:  e.g. 137 5/11/06 7:21:25 PM EDT: I think David forgot today... Our teacher didn't remind us.

Session I

They join within seconds:

 Jason joins the room 5/9/06 6:24:03 PM EDT
 davidcyl joins the room 5/9/06 6:24:04 PM EDT
 137 joins the room 5/9/06 6:24:15 PM EDT

We did this in class

Notice how it is 137 who ends up posting the formula

 Jason 5/9/06 6:25:44 PM EDT: ooh we just did this in math class about a week ago! :-)
 p M M M
 azemel 5/9/06 6:25:54 PM EDT: if you have any questions, just ask
 Jason 5/9/06 6:25:55 PM EDT: well, not the exact thing, but sequences and series
 p M M
 Jason 5/9/06 6:26:03 PM EDT: anyhow
 p M M M M M M M
 Jason 5/9/06 6:26:21 PM EDT: so do we see how the number of sticks grows in a sequence?
 davidcyl 5/9/06 6:26:25 PM EDT: ok i've drawn n=4,5,6
 Jason 5/9/06 6:26:29 PM EDT: 4(+6) = 10
 Jason 5/9/06 6:26:36 PM EDT: 10(+8) = 18
 p M M
 Jason 5/9/06 6:26:48 PM EDT: i'm guessing 18(+10) = 28 for the next one, according to this pattern
 davidcyl 5/9/06 6:27:32 PM EDT: the nth pattern has n more squares than the (n-1)th pattern
 davidcyl 5/9/06 6:27:55 PM EDT: basically it's 1+2+..+(n-1)+n for the number of squares in the nth pattern
 137 5/9/06 6:28:16 PM EDT: so n(n+1)/2
 davidcyl 5/9/06 6:28:24 PM EDT: and we can use the gaussian sum to determine the sum: n(1+n)/2
 davidcyl 5/9/06 6:28:36 PM EDT: 137 got it

Recursive or Explicit?

Notice that Jason ASKS and david offers his opinion. See Session II

 davidcyl 5/9/06 6:29:31 PM EDT: well to find the number of sticks:
 davidcyl 5/9/06 6:29:39 PM EDT: let's look on the board
 p M M M
 Jason 5/9/06 6:29:54 PM EDT: should we use a recursive or explicit definition for it
 p M M
 davidcyl 5/9/06 6:30:20 PM EDT: i don't think we need recursion

Lost Proposal?

137 posts these formulas which seem to be for the number of sticks and mentions overlaps but there is no follow up. Davidcyl and Jason seem to be engaged in figuring it out step by step.

 137 5/9/06 6:28:43 PM EDT: and 2(1+2+3...n-1) overlaps
 137 5/9/06 6:29:05 PM EDT: so n(n+1)/2-n(n-1)/2?

Horizontal and Vertical

As in Team B

(david circles horizontal lines in a pattern diagram on the whiteboard)
davidcyl 5/9/06 6:32:21 PM EDT: 137: i'm separating the sticks into vertical and horizontal sticks
 davidcyl 5/9/06 6:30:33 PM EDT: it's simpler to express it as 1+2+...+n

Not Shared

Disoriented, then MY formula

 Jason 5/9/06 6:32:28 PM EDT: # sticks = N*(3+N)
 davidcyl 5/9/06 6:32:30 PM EDT: wait what are you working on?
 Jason 5/9/06 6:32:32 PM EDT: wait lemme check
 davidcyl 5/9/06 6:32:35 PM EDT: (to 137)
 137 5/9/06 6:32:46 PM EDT: Great. Confused.
 Jason 5/9/06 6:33:03 PM EDT: 137 are you talking about # sticks or squares
 137 5/9/06 6:33:09 PM EDT: Sticks.
 Jason 5/9/06 6:33:18 PM EDT: ok
 davidcyl 5/9/06 6:33:21 PM EDT: i would think it's 2(n(1+n)/2) + n + n
 Jason 5/9/06 6:33:23 PM EDT: well i think my formula works
 Jason 5/9/06 6:33:33 PM EDT: provided that you have a value for N

but later (after an interruption)

 davidcyl 5/9/06 6:35:03 PM EDT: this simplifies to n(1+n) + 2n, or n(3+n)
 davidcyl 5/9/06 6:35:09 PM EDT: so jason, you're right
 Jason 5/9/06 6:35:16 PM EDT: :-)
 Jason 5/9/06 6:35:36 PM EDT: so now onto a formula for the total number of squares

but where is 137?

Ssjnish joins

ssjnish joins the room 5/9/06 6:34:25 PM EDT

Didn't we do that?

 davidcyl 5/9/06 6:35:03 PM EDT: this simplifies to n(1+n) + 2n, or n(3+n)
 davidcyl 5/9/06 6:35:09 PM EDT: so jason, you're right
 Jason 5/9/06 6:35:16 PM EDT: :-)
 Jason 5/9/06 6:35:36 PM EDT: so now onto a formula for the total number of squares
 davidcyl 5/9/06 6:35:42 PM EDT: ok let's complete the table
 Jason 5/9/06 6:35:46 PM EDT: if you take the change in the change of the number of squares, it's constant
 137 5/9/06 6:36:10 PM EDT: Didn't we do that?
 (points to Jason's message on 6:35:36 PM)
 davidcyl 5/9/06 6:36:19 PM EDT: yes
 Jason 5/9/06 6:36:27 PM EDT: oh, sorry i guess i must've not caught that
 davidcyl 5/9/06 6:36:33 PM EDT: look up
 (points to davidcyl 5/9/06 6:28:24 PM EDT: and we can use the gaussian sum to determine the sum: n(1+n)/2)
 Jason 5/9/06 6:36:33 PM EDT: could someone post it in a text box on the whiteboard
 davidcyl 5/9/06 6:36:38 PM EDT: sure
 azemel 5/9/06 6:36:40 PM EDT: be sure that SSJNISH is up to speed folks
 (Textbox created)

A summary for Ssjnish

Notice the diffeence between We've figured out and I divided

 davidcyl 5/9/06 6:38:30 PM EDT: basically, we've figured out that the number of squares in the nth pattern is 1 + 2 + ... + n
 p M
 137 5/9/06 6:38:33 PM EDT: It was blinding.
 p M M M M
 davidcyl 5/9/06 6:39:26 PM EDT: then, to find the number of sticks, I divided the figure into "vertical sticks" (|) and "horizontal   sticks" (--)
 Jason 5/9/06 6:39:41 PM EDT: the formulas are on the Whiteboard
 azemel 5/9/06 6:39:56 PM EDT: don't forget to post your ideas to the wiki when you think it's time!
 davidcyl 5/9/06 6:40:15 PM EDT: the number of vertical sticks is (1 + 2 + 3 + ... + n)+ n, and the number of horizontal sticks is the same
 p M

Ssjnish asks for an explanation

This is interesting because it prompts a form of "bridging" in a sense

 ssjnish 5/9/06 6:45:11 PM EDT: just to clarify sumthing, i am not overwhelmingly good at math as u guys seem to be, 
                                so it may take me more time than u guys to understand sumthing..
 azemel 5/9/06 6:45:44 PM EDT: can you tell us what's puzzling you?
 Jason 5/9/06 6:46:07 PM EDT: are we allowed to post images on the wiki? I could just download TeX real quick and get the 
                              summation notation in a small graphic
 ssjnish 5/9/06 6:46:12 PM EDT: the derivation of the number of squares
 Jason 5/9/06 6:46:21 PM EDT: oh
 Jason 5/9/06 6:46:31 PM EDT: so you see in the list a column for "N"
 Jason 5/9/06 6:46:50 PM EDT: when n=1, we have 1 square; for n=2, 3; and for n=3, 6
 Jason 5/9/06 6:47:00 PM EDT: we came up with a formula to find the total number of squares for any number N
 Jason 5/9/06 6:47:16 PM EDT: the purpose of the formula is so that you don't have to draw out the squares and count them
 ssjnish 5/9/06 6:47:39 PM EDT: um yes
 ssjnish 5/9/06 6:47:41 PM EDT: i know
 ssjnish 5/9/06 6:47:51 PM EDT: but how did u get that formula
 Jason 5/9/06 6:48:00 PM EDT: oh
 azemel 5/9/06 6:48:11 PM EDT: i believe so
 Jason 5/9/06 6:48:12 PM EDT: uh, basically you try to find a pattern in the total number of squares first
 Jason 5/9/06 6:48:47 PM EDT: we found a formula for that which we'll post on the wiki
 p M
 davidcyl 5/9/06 6:49:00 PM EDT: if you look at the patterns row by row, it's 1 + 2 + 3 + 4 + however many rows there are
 davidcyl 5/9/06 6:49:24 PM EDT: so for the nth pattern, we can say there are 1 + ... + n squares
 Jason 5/9/06 6:49:27 PM EDT: if N rows: 1+2+3+...N
 Jason 5/9/06 6:49:57 PM EDT: so then we incorporated the formula for finding the sum of an arithmetic series
 davidcyl 5/9/06 6:50:12 PM EDT: there's a formula for finding the sum of consecutive integers, which (when starting from 1) is: n(n+1)/2
 137 5/9/06 6:50:17 PM EDT: so you use gaussian sum to get n(n+1)/2
 Jason 5/9/06 6:50:25 PM EDT: that's it
 davidcyl 5/9/06 6:50:35 PM EDT: and as Jason said, it works for arithmetic sequences in general
 ssjnish 5/9/06 6:50:51 PM EDT: hmm...

Davidcyl Announces the Wiki

In a sense saying, it is done, check it out.

 davidcyl 5/9/06 7:03:59 PM EDT: http : //mathforum.org/wiki/VMTStudents/?PatternsOfTheSticks

but notice that 137 says that "they did"

 azemel 5/9/06 7:09:41 PM EDT: I assume you posted your results to the wiki, right?
 azemel 5/9/06 7:09:53 PM EDT: I don't think you can
 137 5/9/06 7:09:54 PM EDT: They did.

interesting

Moderator's Closing

 azemel 5/9/06 7:08:04 PM EDT: well, if you're done with the sticks problem as it stands, 
                              then this might be a time to stop. there will be another problem on Thursday
 p M M
 azemel 5/9/06 7:08:24 PM EDT: remember, 7 pm, Thursday, same room

Feedback

Dear 137, davidcyl, Jason, and ssjnish, It seemed to us that you had a very productive first session exploring the given pattern of sticks and squares. We were especially interested in the variety of strategies you used, such as constructing the next steps of the pattern on the whiteboard, separating the pattern in horizontal and vertical lines (other teams did that as well!) and deriving a formula for that sum.

As far as working as a math team, you built on each other’s ideas and tried to work with them in interesting ways. We find it very important that ssjnish felt comfortable asking the team to explain in detail the reasoning for the work completed (e.g. ssjnish 5/9/06 6:47:51 PM EDT: but how did u get that formula?), and that as a team you provided that explanation. It looked useful to us when your group tested together the formula you found. One question that was left unexplored was whether a recursive function shows better how the number of sticks and square grow. Someone offered that as a possibility but you opted for using a summation notation. We notice when ideas or questions are stated in a group but not discussed. What do you think about that situation and how groups deal with it?

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 which you find interesting to pursue.

The VMT team. (Feel free to delete this note once everyone has read it)

Session II

Catching up Qw

Notice how Jason uses the formulas left on the whiteboard in his report to QW. Notice how Qw gest involved and asks a question "how do you get n(1+n)/2. Jason explanation is "that is the formula" basically.

 Jason 5/11/06 7:18:07 PM EDT: ok, so with this aside-- i guess we should discuss our feedback from the last session
 jsarmi 5/11/06 7:18:34 PM EDT: make sure you bring qwertyuiop up to speed
 Jason 5/11/06 7:18:41 PM EDT: ok
 Jason 5/11/06 7:19:35 PM EDT: for the problems last session, we came up with formulas to find the values for the columns
 qwertyuiop 5/11/06 7:20:02 PM EDT: in the view topic thing?
 Jason 5/11/06 7:20:03 PM EDT: You can see them to the left of this text; our formula for the total number 
                               of sticks or squares for any number N is given
 Jason 5/11/06 7:20:09 PM EDT: yes
 qwertyuiop 5/11/06 7:20:12 PM EDT: ok
 Jason 5/11/06 7:20:17 PM EDT: that was the problem we were given
 Jason 5/11/06 7:20:39 PM EDT: remains of our discussion is on the whiteboard and online wiki
 137 5/11/06 7:21:25 PM EDT: I think David forgot today... Our teacher didn't remind us.
 p M
 jsarmi 5/11/06 7:22:35 PM EDT: I see... hopefully he will join you next Tuesday
 qwertyuiop 5/11/06 7:23:35 PM EDT: n=3 is 3+2+1 squares, n=4 is 4+3+2+1 squares... how did you get n(1+n)/2
 Jason 5/11/06 7:23:42 PM EDT: oh
 Jason 5/11/06 7:23:53 PM EDT: that's the formula for finding a series of consecutive numbers
 Jason 5/11/06 7:24:08 PM EDT: 1+2+3+4+...n = ((n)(n+1))/2

Recursive Function (Again?) and the Feedback

Noticed how 137 uses "again" but qwerty, who is new, just asks plainly what is it (earlier he got a very different response with "how did you get it" )

 Jason 5/11/06 7:26:32 PM EDT: so apparently there's something with a recursive sequence that we should discuss
 137 5/11/06 7:26:55 PM EDT: What was a recursice sequence again?
 qwertyuiop 5/11/06 7:27:03 PM EDT: recursive sequence?
 
 ...
  
 Jason 5/11/06 7:27:18 PM EDT: i think that an explicit formula is better, but a recursive one would show how the number of squares/sticks increases as N increases
 Jason 5/11/06 7:27:35 PM EDT: it's something like this:
 Jason 5/11/06 7:27:45 PM EDT: a(n) = 5+ a(n-1)
 Jason 5/11/06 7:27:53 PM EDT: where the things in parentheses are supposed to be subscripts
 Jason 5/11/06 7:28:07 PM EDT: so a recursive formula relies on the value of a previous function
 137 5/11/06 7:28:09 PM EDT: Ah, I see.
 Jason 5/11/06 7:28:19 PM EDT: thus, you must specify something first, like a(1) = 4
 qwertyuiop 5/11/06 7:28:29 PM EDT: i get it
 Jason 5/11/06 7:28:54 PM EDT: great :-)
 qwertyuiop 5/11/06 7:30:07 PM EDT: for the number of squares, would that be: a(n)=n2-1
 137 5/11/06 7:30:15 PM EDT: so a(1)=1, a(n)=n+a(n-1)...
 
 ... 
 
 137 5/11/06 7:33:06 PM EDT: b(1)=4, b(n)=b(n-1)+4(n)-(n-1)-(n-1), b is the number ofr sticks...
 137 5/11/06 7:33:30 PM EDT: So b(n)=b(n-1)+2n+2?
 Jason 5/11/06 7:33:51 PM EDT: assuming only (n-1) is a subscript?
  
  ...
      
 Jason 5/11/06 7:35:13 PM EDT: did u check that
 Jason 5/11/06 7:35:39 PM EDT: looks correct
 Jason 5/11/06 7:35:45 PM EDT: how did you get it?

Noticed that this "how did you get it" is in essence the same question that QW asked at the beginning but the answer does not bring the "narrative" of the team (we did this last time and then this, and you can find it in the wiki and in the textboxes)

 Jason 5/11/06 7:35:45 PM EDT: how did you get it?
 137 5/11/06 7:36:23 PM EDT: There are n-1 overlaps here...
 137 5/11/06 7:36:24 PM EDT: Wait.
 p M M M
 137 5/11/06 7:36:36 PM EDT: Those.
 137 5/11/06 7:36:43 PM EDT: And n-1 here:
 qwertyuiop 5/11/06 7:36:45 PM EDT: those?
 p M M M M
 Jason 5/11/06 7:36:57 PM EDT: what do you mean by :"overlaps"
 137 5/11/06 7:37:17 PM EDT: They're counted twise; they belong to two boxes.
 p M M M M M
 Jason 5/11/06 7:38:13 PM EDT: are you guys still talking about that formula?

I liked the "original" one

and back to why recursive function is better? (Jason)

 Jason 5/11/06 7:41:48 PM EDT: well this requires you to first calculate number of squares; 
                               i think the formulas for each of these should be seperate
 Jason 5/11/06 7:41:58 PM EDT: i liked the original formula
 Jason 5/11/06 7:42:05 PM EDT: in my quick checking it worked
 137 5/11/06 7:42:06 PM EDT: So did I...
 137 5/11/06 7:42:18 PM EDT: The first one seeemed simpler.
 Jason 5/11/06 7:42:39 PM EDT: but this one has a nice explanation :-)
 Jason 5/11/06 7:42:43 PM EDT: i mean
 qwertyuiop 5/11/06 7:42:54 PM EDT: we already have the square formula; just include it: n(1+n)+n2
 Jason 5/11/06 7:43:16 PM EDT: yup
 qwertyuiop 5/11/06 7:43:41 PM EDT: that looks like the same thing as n*(N+3) at a glance...
 Jason 5/11/06 7:43:51 PM EDT: so speaking of formulas, we got both explicit and recursive definitions for sticks/squares; 
                               explicit is easier while recursive shows how each step grows from the previous

Notice, by "we got" does he mean "in both sessions" "all who have worked here". At that point the recursive functions have not been placed on the whiteboard but qwertyop quickly adds one of them to it:

 ----------------------------------
|  Formula for total # of squares: |
|                                  |
|    n(1+n)/2                      |
|    a(n)=n+a(n-1)                 |
|                                  |
 ----------------------------------

The "each square with 2 sides" thing doesn't work as neatly here

This is something that QW had said earlier when speaking about "overlap" ( qwertyuiop 5/11/06 7:39:02 PM EDT: might be easier if you think of each square corisponding to 2 sides-the right and bottom sides, and then add the upper left border)

 Jason 5/11/06 7:53:52 PM EDT: ok
 Jason 5/11/06 7:53:57 PM EDT: sorry for the delay on my part
 137 5/11/06 7:54:03 PM EDT: So the number of squares is n^2 +4, where n is a side length
 qwertyuiop 5/11/06 7:54:12 PM EDT: the "each square with 2 sides" thing doesn't work as neatly here
 Jason 5/11/06 7:54:13 PM EDT: if we look at rows, its 1, 3, 5, 3, 1

How did you get that

This is the third "how did you get that" of the session. It required a second call for an uptake.

qwertyuiop 5/11/06 7:54:28 PM EDT: how did you get that?
 Jason 5/11/06 7:54:29 PM EDT: by side length you mean...
 p M M M
 137 5/11/06 7:54:50 PM EDT: The orange.
 (...)
  Jason 5/11/06 7:55:52 PM EDT: can you explain how you got that formula?

137 attempts a number of diagrams but seems to have trouble with coloring. Later QW posts a formula for Number of sides. Jason rembers that 137 had posted a formula earlier and points to it (this one)

qwertyuiop 5/11/06 8:00:57 PM EDT: i think    NumberOfSides=NumberOfSquares*2+SideLength*3-2
 p M M
 Jason 5/11/06 8:01:29 PM EDT: wait-- you gave a formula earlier 
 p M
 Jason 5/11/06 8:01:34 PM EDT: lemme see if i can find it
 p M
 qwertyuiop 5/11/06 8:01:42 PM EDT: because it's like the first shape, but with 3 other edges
 p M
 Jason 5/11/06 8:01:55 PM EDT: this one
 p M M M M M M
 Jason 5/11/06 8:02:12 PM EDT: oh but you're talking about # of sides

later with more diagrams, 137 posts that "he screwed up somewhere" and Jason makes another observation:

 Jason 5/11/06 8:08:18 PM EDT: the fact that, when the squares are arranged in a diamond shape result in 1+3+5+3+1 
                               total squares in the diagram to the left, remind me of pascal's triangle
 Jason 5/11/06 8:08:35 PM EDT: just a thought... not sure if it'll work

Then QW makes a proposal "using your previous method" which they agree works:

 qwertyuiop 5/11/06 8:08:35 PM EDT: using your previous method: SideLenght^2 + (SideLength-1)^2
 qwertyuiop 5/11/06 8:08:51 PM EDT: does that work?
 qwertyuiop 5/11/06 8:10:06 PM EDT: the part in front of the plus is the large, rotatewd 45 square and 
                                    the part after it is the smaller, not rotated squared
 qwertyuiop 5/11/06 8:10:59 PM EDT: um... hello?
 137 5/11/06 8:11:03 PM EDT: Hi.
 Jason 5/11/06 8:11:17 PM EDT: well, looking at the center diagram with the orange boxed
 qwertyuiop 5/11/06 8:11:18 PM EDT: but does that work?
 Jason 5/11/06 8:11:20 PM EDT: boxes
 Jason 5/11/06 8:11:24 PM EDT: you'd be 1 off
 Jason 5/11/06 8:11:41 PM EDT: nvm, my arithmetic skills faltered ther
 Jason 5/11/06 8:11:42 PM EDT: e
 p M M
 Jason 5/11/06 8:11:59 PM EDT: there
 qwertyuiop 5/11/06 8:12:03 PM EDT: checking for SideLenght=3...
 Jason 5/11/06 8:12:35 PM EDT: works
 qwertyuiop 5/11/06 8:12:36 PM EDT: yes, it works
 Jason 5/11/06 8:12:52 PM EDT: cool, so should we call this a formula

And the forth "how do you get it" of the session:

 137 5/11/06 8:12:57 PM EDT: I don't get why though...
 Jason 5/11/06 8:13:29 PM EDT: i dont either
 p M M M M
 qwertyuiop 5/11/06 8:14:10 PM EDT: I used your previous method: take the (orange) side; that^2 gives part of the shape...
 qwertyuiop 5/11/06 8:15:03 PM EDT: the rest happens to be a square with a side length of one less, therfore th "(SideLength-1)^2"
 qwertyuiop 5/11/06 8:15:10 PM EDT: does that make sense?
 Jason 5/11/06 8:15:42 PM EDT: i get it!
 Jason 5/11/06 8:15:43 PM EDT: clever :)
 qwertyuiop 5/11/06 8:15:48 PM EDT: 137?
 137 5/11/06 8:15:56 PM EDT: Now I do.

3D but time runs out

and the moderator suggests working on the Wiki

qwertyuiop 5/11/06 8:22:40 PM EDT: to make it more 3D, you could continue the pattern along the new dimension

 qwertyuiop 5/11/06 8:22:57 PM EDT: not sure how you would draw that...
 Jason 5/11/06 8:23:38 PM EDT: well going back to our session problems
 Jason 5/11/06 8:23:38 PM EDT:  What are the different methods (induction, series, recursion, graphing, tables, etc.) you can use to analye these different patterns?
 Jason 5/11/06 8:23:57 PM EDT: i'd say series (recursive or explicit) for many of them
 Jason 5/11/06 8:24:49 PM EDT: anyone else?
 qwertyuiop 5/11/06 8:24:59 PM EDT: induction?
 Jason 5/11/06 8:25:58 PM EDT: what's that
 ...

Notice "last time" but no incidence in problem solving

Cynthia 5/11/06 8:26:13 PM EDT: so i'm wondering if this is a good time to figure out the information that you are going to post to the wiki?
 Jason 5/11/06 8:26:30 PM EDT: hmm
 qwertyuiop 5/11/06 8:26:30 PM EDT: the equations...
 Jason 5/11/06 8:26:55 PM EDT: well im not sure if we formulated too many questions to post, but i guess yeah we can write up our formulas and how we got them
 qwertyuiop 5/11/06 8:28:25 PM EDT: there are 4, I think: number of squares and number of sides for both diamonds and the first problem
 Jason 5/11/06 8:28:47 PM EDT: alright
 Jason 5/11/06 8:29:05 PM EDT: last time we designated one person to write up the wiki... 
                                do you guys want to do that again or should we all assign each other some part to write
 qwertyuiop 5/11/06 8:29:35 PM EDT: we could just do it here
 Jason 5/11/06 8:29:47 PM EDT: do what here
 qwertyuiop 5/11/06 8:29:59 PM EDT: write it

Feedback II

Dear Jason, 137, davidcyl, ssjnish, and qwertyuiop

Last time you had a very creative session where you explored a number of new ideas 
related to the sticks and squares problem.  We found it interesting how you worked 
as a team to clarify what a recursive formula is and how to find one for the sticks and squares.  
We also noticed that you concluded that "an explicit formula is better, but a recursive 
one would show how the number of squares/sticks increases as N increases".  
We wondered what other teams working on the problem might think about this idea 
so maybe you want to post it to the Wiki?

Your exploration of the diamond shape was also very interesting to us, and your posting to the Wiki 
should be helpful to other teams thinking about similar cases.  For the next step 
we will encourage you to continue thinking about the problems that you can discover on your own 
and that are interesting to pursue, and also to explore the different approaches to solve them.  
BTW, remember that 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.)

-The VMT team.

Session III

Noticed that this session's feedback does not really contain any statements about things "you should have done different". A couple of suggestions for things to post on the wiki are followed up eventually but there is really no other direct discussion about the feedback. One could say that there is uptake because the suggestion of working on problems the group finds interesting is, indeed, what they do

we do what we did last time again?

 137 5/16/06 7:04:41 PM EDT: So we do what we did last time again?
 nan 5/16/06 7:04:47 PM EDT: yes

Triangular Numbers

137 creates a hexagonal grid on the whiteboard but it doesn't look too good so he asks somebody else to try it. QW does it:

 qwertyuiop 5/16/06 7:14:51 PM EDT: triangles are done
 137 5/16/06 7:15:08 PM EDT: So do you want to first calculate the number of triangles in a hexagonal array?

They work on this grid for a while and come to the idea of triangular numbers. This is something that Team B also worked withbut for which there is no group-to-group exchange. First both QW and 137 make references to things they had contributed in the previous session: "each polygon corrisponds to 2 sides" and the 1-3-5 pattern. Only QW does it explicitly with a reference to LAST TIME and quotes:

 Jason 5/16/06 7:20:02 PM EDT: so... should we try to find a formula i guess
 Jason 5/16/06 7:20:22 PM EDT: input: side length; output: # triangles
 qwertyuiop 5/16/06 7:20:39 PM EDT: It might be easier to see it as the 6 smaller triangles.
 137 5/16/06 7:20:48 PM EDT: Like this?
 p M M M
 qwertyuiop 5/16/06 7:21:02 PM EDT: yes
 Jason 5/16/06 7:21:03 PM EDT: yup
 p M M
 qwertyuiop 5/16/06 7:21:29 PM EDT: side length is the same...
 Jason 5/16/06 7:22:06 PM EDT: yeah
 Jason 5/16/06 7:22:13 PM EDT: so it'll just be x6 for # triangles in the hexagon
 137 5/16/06 7:22:19 PM EDT: Each one has 1+3+5 triangles.
 Jason 5/16/06 7:22:23 PM EDT: but then we're assuming just regular hexagons
 qwertyuiop 5/16/06 7:22:29 PM EDT: the "each polygon corrisponds to 2 sides" thing we did last time doesn't work for triangles

later 137 proposes a formula and JS offers support with an objection:

 137 5/16/06 7:24:49 PM EDT: And there are n terms so... n(2n/2)
 137 5/16/06 7:25:07 PM EDT: or n^2
 Jason 5/16/06 7:25:17 PM EDT: yeah
 Jason 5/16/06 7:25:21 PM EDT: then multiply by 6
 137 5/16/06 7:25:31 PM EDT: To get 6n^2
 Jason 5/16/06 7:25:39 PM EDT: but this is only with regular hexagons... is it possible to have one 
                               definite formula for irregular hexagons as well

So they do some more work and triangular numbers emerge

 qwertyuiop 5/16/06 7:33:54 PM EDT: I don't see the pattern yet...
 137 5/16/06 7:34:01 PM EDT: We're ignoring the bottom one?
 p M M M
 qwertyuiop 5/16/06 7:34:29 PM EDT: no, 3 is only for side length 2.
 137 5/16/06 7:34:52 PM EDT: And I think the'y;re all triangular numbers.
 p M M M
 qwertyuiop 5/16/06 7:35:17 PM EDT: "triangular numbers"?
 p M M M M M M
 Jason 5/16/06 7:35:37 PM EDT: you mean like 1, 3, 7, ...
 Jason 5/16/06 7:35:39 PM EDT: ?
 137 5/16/06 7:35:59 PM EDT: Like 1,3,6,10,15,21,28.
 qwertyuiop 5/16/06 7:36:02 PM EDT: the sequence is 1, 3, 6...
 137 5/16/06 7:36:30 PM EDT: Numbers that can be expressed as n(n+1)/2, where n is an integer.
 qwertyuiop 5/16/06 7:36:45 PM EDT: ah

Team B Asks a Question

 nan 5/16/06 7:48:49 PM EDT: (we got a question for you from another team, which was posted in the lobby:
 nan 5/16/06 7:48:53 PM EDT:  Quicksilver 7:44:50 PM EDT: Hey anyone from team c, our team needs to know what n was in your equations last week
 Jason 5/16/06 7:49:04 PM EDT: oh
 137 5/16/06 7:49:15 PM EDT: The length of a side.
 qwertyuiop 5/16/06 7:49:16 PM EDT: was n side length?
 Jason 5/16/06 7:49:33 PM EDT: are you talking about the original problem with the squares
 137 5/16/06 7:49:48 PM EDT: I think nan is.
 qwertyuiop 5/16/06 7:49:58 PM EDT: i think it's squares and diamonds
 Jason 5/16/06 7:49:58 PM EDT: oh
 Jason 5/16/06 7:50:12 PM EDT: then if you look in the topic description, theres a column for N; 
 Jason 5/16/06 7:50:14 PM EDT: thats what it is
 nan 5/16/06 7:50:17 PM EDT: ok, quicksilver said they got it
 Jason 5/16/06 7:50:25 PM EDT: so yes it is # sides
 nan 5/16/06 7:50:26 PM EDT: thanks guys

Collecting Formulas on Textbox

Jason leaves first, but Qw and 137 continue to work:

 ...
 qwertyuiop 5/16/06 8:06:20 PM EDT: it's x3 for the 3 colinear sets, then x6 for 6 triangles in a hexagon... where's the 9 and 2?
 qwertyuiop 5/16/06 8:06:28 PM EDT: oh
 137 5/16/06 8:06:38 PM EDT: So 18/2.
 137 5/16/06 8:06:50 PM EDT: A.K.A. 9
 qwertyuiop 5/16/06 8:07:08 PM EDT: (n(n+1)/2)x3x6
 137 5/16/06 8:07:15 PM EDT: Yeah.
 qwertyuiop 5/16/06 8:07:27 PM EDT: which can be simplified...
 137 5/16/06 8:07:46 PM EDT: To 9n(n+1)
 qwertyuiop 5/16/06 8:08:04 PM EDT: that's it?
 137 5/16/06 8:08:12 PM EDT: -6n.
 137 5/16/06 8:08:24 PM EDT: So 9n(n+1)-6n
 qwertyuiop 5/16/06 8:08:34 PM EDT: i'll put it with the other formulas...

There is a bit of collision while editing the texbox but at the end it looks like this:

 SIDES:
 N(N+3)
 diamond:
 (n^2+(n-1)^2)*2+n*3-2
 hexagon w/ triangles:
 9n(n+1)-6n
 
 
 POLYHEDRA:
 square
 n(n-1)/2
 diamond:
 n^2+(n-1)^2
 hexagon w/ triangles
 6n^2

The hypercube: Imagine our first problem with a grid of squares.... Resume here next time

 qwertyuiop 5/16/06 8:15:02 PM EDT: If you have a square, it extends to make a grid that fills a plane. A cube fills a space. A simaller pattern of hypercubes fills a "hyperspace". 
 p M M
 137 5/16/06 8:15:19 PM EDT: The heck?
 137 5/16/06 8:15:29 PM EDT: That's kinda confusing.
 p M M M M M
 qwertyuiop 5/16/06 8:15:43 PM EDT: So, how many planes in a hyper cube latice of space n?
 p M M M M M M M
 137 5/16/06 8:16:05 PM EDT: Er...
 qwertyuiop 5/16/06 8:16:07 PM EDT: instead of "how many lines in a grid of length n"
 qwertyuiop 5/16/06 8:16:17 PM EDT: does that make any sense?
 137 5/16/06 8:16:30 PM EDT: No. No offense, of course.
 qwertyuiop 5/16/06 8:16:43 PM EDT: ok... let me think...
 p M M M M M M
 qwertyuiop 5/16/06 8:17:19 PM EDT: Imagine our first problem with a grid of squares. 
 p M M
 137 5/16/06 8:17:31 PM EDT: Right.
 p M M
 qwertyuiop 5/16/06 8:18:07 PM EDT: The squares are 2 dimensional and they can be arranged in a grid to tessalate over a plane. The plane is also 2 dimensional.
 p M
 137 5/16/06 8:18:41 PM EDT: Right.
 qwertyuiop 5/16/06 8:18:54 PM EDT: If you use 3 dimensional cubes, they can be arranged to fill a 3 dimensional space.
 137 5/16/06 8:19:17 PM EDT: And that structure's 4 dimensional?
 qwertyuiop 5/16/06 8:19:25 PM EDT: If you have hypercubes, they can be arranged to fill a 4 dimensional "hyperspace"
 qwertyuiop 5/16/06 8:19:36 PM EDT: what's 4D?
 137 5/16/06 8:19:46 PM EDT: ?
 nan 5/16/06 8:20:04 PM EDT: you may want to make your ideas available on the wiki before you go
 nan 5/16/06 8:20:09 PM EDT: which may take some time
 137 5/16/06 8:20:24 PM EDT: Actually, I only have around 10 minutes left.
 ...
 (Moderator prompts to stop and work on the wiki)
 ...
 137 5/16/06 8:21:33 PM EDT: So how the heck are we supposed to calculate the number of four-dimentional figures? 
 nan 5/16/06 8:21:42 PM EDT: do you want to stop here and start putting ideas on wiki?
 qwertyuiop 5/16/06 8:21:47 PM EDT: ok
 137 5/16/06 8:21:52 PM EDT: Sure.
 qwertyuiop 5/16/06 8:22:09 PM EDT: resume from here next time?
 p M M
 nan 5/16/06 8:22:17 PM EDT: sure
 137 5/16/06 8:22:19 PM EDT: Ya.

Wiki

 qwertyuiop 5/16/06 8:22:48 PM EDT: We have the 2 hexagon equations to put on the wiki.
 p M M
 137 5/16/06 8:23:04 PM EDT: Right.
 p M M M
 qwertyuiop 5/16/06 8:23:18 PM EDT: Where's the wiki again?
 p M
 nan 5/16/06 8:23:30 PM EDT: open "view topic"
 137 5/16/06 8:23:31 PM EDT: Somewhere in the View topic button
 p M
 nan 5/16/06 8:23:41 PM EDT: there's link
 qwertyuiop 5/16/06 8:23:54 PM EDT: I see it.
 137 leaves the room 5/16/06 8:24:28 PM EDT
 qwertyuiop 5/16/06 8:25:02 PM EDT: i'll write it.

Feedback III

 Dear Team C:   We noticed that you continued working on your idea of a pattern of 
 “hexagons made of triangles” and you discussed some possible formulas for the number of sides and triangles.  
 There is a sense in your conversation that at the end you ran out of time to fully check and discuss 
 these formulas, so if you want to do that today, that would be ok.  If you feel that you understand that pattern 
 already and that your notes on the Wiki are good for others to understand your ideas, you can explore other patterns 
 that you find interesting.  You could also explore some of the ideas presented by other groups on the Wiki.  
 When revisiting the idea of a hypercube and a pattern in 4-dimensions wee also noticed that you were looking 
 for the relationship between similar patterns in different dimensions, and that made us think of a number 
 of interesting ideas.  Enjoy your fourth session!  -The VMT Team

Session IV

Jason missed the last part of Session III, so asks to be caught up. Differentiates between "keep thinking" and "start" ss is coming back after missing 2 sessions and participates very little in this final session.

Keep thinking... or start?

 qwertyuiop 5/18/06 7:06:22 PM EDT: do we want to keep thinking about the hexagon thing or start on the hypercube?
 p M M M M M M M
 Jason 5/18/06 7:06:43 PM EDT: well if we want to start on the hypercube i'll need to first fully 
 understand the concept of the fourth dimension
 p M M M M M M
 qwertyuiop 5/18/06 7:07:05 PM EDT: what needs clairifying?
 p M M M M M
 Jason 5/18/06 7:07:12 PM EDT: everything, pretty much
 ssjnish 5/18/06 7:07:19 PM EDT: same here
 qwertyuiop 5/18/06 7:07:30 PM EDT: ok... imagine a cube
 (Moves objects on the Whiteboard to make space)
 qwertyuiop 5/18/06 7:07:56 PM EDT: it tessalates in a grid to fill a 2 dimensional plane
 p M M M M M M M M M M
 qwertyuiop 5/18/06 7:08:39 PM EDT: it's made of 4 parts (sides) of one dimension lower than itself (1 dimensional lines)
 qwertyuiop 5/18/06 7:08:49 PM EDT: does that make sense so far?
 p M M M
 137 5/18/06 7:09:08 PM EDT: Yeah.
 p M M M M M M M M
 Jason 5/18/06 7:09:42 PM EDT: hmm
 p M
 qwertyuiop 5/18/06 7:10:01 PM EDT: a cube is made of 6 parts one dimension lower than itself (6 2 dimensional faces) and can be arranged to fill a 3 dimensional space
 Jason 5/18/06 7:10:03 PM EDT: well u guys can go ahead with the hypercube if you want, i probably wont be able to contribute much though
 137 5/18/06 7:10:20 PM EDT: Neither will I.
 qwertyuiop 5/18/06 7:10:36 PM EDT: we could just do a cube
 Jason 5/18/06 7:10:45 PM EDT: ok, that'll probably be a bit easier :-)
 qwertyuiop 5/18/06 7:11:01 PM EDT: i'll start drawing

The first problem and the hypercube + Like last time

 qwertyuiop 5/18/06 7:14:09 PM EDT: on the first problem, the pattern continued in 2 dimensions (up and left) so this should continue in 3 (along each axis)
 qwertyuiop 5/18/06 7:14:18 PM EDT: does that make sense?
 Jason 5/18/06 7:14:31 PM EDT: yea
 137 5/18/06 7:14:32 PM EDT: Yes.
  ...
 137 5/18/06 7:20:31 PM EDT: I think we should just look at it in 3 groups of parallel lines like last time.

Triangular numbers

Notice the reapearance of the definition of triangular numbers

 Jason 5/18/06 7:26:29 PM EDT: can you get the next level for these cubes
 Jason 5/18/06 7:26:38 PM EDT: then we might be able to start seeing a pattern
 p M M M M M
 qwertyuiop 5/18/06 7:27:20 PM EDT: I think that's n=4.
 137 5/18/06 7:27:34 PM EDT: The number of cubes is the sum of n consecutive triangular numbers...
 qwertyuiop 5/18/06 7:27:49 PM EDT: triangular?
 qwertyuiop 5/18/06 7:27:49 PM EDT: 
 Jason 5/18/06 7:27:57 PM EDT: woah
 Jason 5/18/06 7:28:15 PM EDT: (reacting to the diagram i mean)
 137 5/18/06 7:28:22 PM EDT: Numbers that are the sum of counstecutibe numbers starting from 1...
 Jason 5/18/06 7:28:43 PM EDT: how many cubes are in that diagram that you just posted?
 qwertyuiop 5/18/06 7:28:45 PM EDT: counstecutible?
 137 5/18/06 7:28:57 PM EDT: consecutive.
 137 5/18/06 7:29:07 PM EDT: Sry.
 qwertyuiop 5/18/06 7:29:09 PM EDT: 1 min
 137 5/18/06 7:29:34 PM EDT: 10+6+3+1=20.

The evolution of Whiteboard Notes

It is interesting that this team keeps all its results in a single textbox on the whiteboard. In the wiki, there is more explanation but on the whiteboard it is a place to keep all the formulas organized and, as new formulas/results are added, some of the existing labels are changes to represent the "expanding" notions that they are working with. For example, in the original problem, there were sticks and squares as the "primary" elements but they labeled them "squares" and "sides" which worked for both the original pattern and their diamond pattern, however, when they work with the hexagon made up of triangles, they changed squares to "POLYHEDRA" and recorded that hexagons as "hexagon/w triangles". This is how the formula for the number of cubes in their piramid gets recorded in that textbox:

  SIDES:
  square:
  N(N+3)
  diamond:
  (n^2+(n-1)^2)*2+n*3-2
  hexagon w/ triangles:
  9n(n+1)-6n
  
  
  POLYHEDRA:
  square
  n(n-1)/2
  diamond:
  n^2+(n-1)^2
  hexagon w/ triangles
  6n^2
  cubes in pyramid shape:
  f(n)=f(n-1)+x(X+1)/2

Now that they are working with cubes, they can use, sticks, squares ("faces"), and cubes.

Like we said earlier

Earlier in Session II

 qwertyuiop 5/18/06 7:41:25 PM EDT: or do we want to put that function in a different form?
 Jason 5/18/06 7:41:35 PM EDT: i think that form is good
 Jason 5/18/06 7:41:49 PM EDT: like we said earlier, recursiveness=easy to track pattern  of growth

Wouldn't this work?

OT points to his earlier posting indicating a reuse of something they did in last session on the hexagon. Notice that "overlap" resurfaces as a relevant concern.

137 5/18/06 7:41:52 PM EDT: Wouldn't this work?  [Points to: I think we should just look at it in 3 groups of parallel lines like last time.]
 137 5/18/06 7:42:06 PM EDT: Each cube has 2 pointing each direction.
 137 5/18/06 7:42:11 PM EDT: Wait.
 137 5/18/06 7:42:13 PM EDT: 4.
 qwertyuiop 5/18/06 7:42:28 PM EDT: i think so   [Points to: Wouldn't this work?]
 qwertyuiop 5/18/06 7:43:38 PM EDT: the overlap is different depending on wether the cube is on the outside of the shape or on the inside

Later QW makes a series of very interesting graphs that form a proposal for how to calculate the number of edges and posts:

 qwertyuiop 5/18/06 7:48:21 PM EDT: an idea: Each cube is assigned to 3 edges. Then add the edges on the diagonalish face.
 qwertyuiop 5/18/06 7:48:44 PM EDT: does that make sense?
 137 5/18/06 7:48:58 PM EDT: Not really...
 qwertyuiop 5/18/06 7:49:13 PM EDT: let me think how to explain it...
 Jason 5/18/06 7:49:13 PM EDT: well some would be the same
 p M M M
 qwertyuiop 5/18/06 7:51:36 PM EDT: oops
 p M
 qwertyuiop 5/18/06 7:52:44 PM EDT: see those 3 highlighted edges?
 Jason 5/18/06 7:53:32 PM EDT: yeah
 Jason 5/18/06 7:53:37 PM EDT: theyre in red
 qwertyuiop 5/18/06 7:54:09 PM EDT: those would be the edges assigned to that cube (the cube at the bottom corner)
 qwertyuiop 5/18/06 7:54:15 PM EDT: still make sense?

 qwertyuiop 5/18/06 8:02:49 PM EDT: so the first part of the equation would be 3 times the number of cubes
 qwertyuiop 5/18/06 8:03:12 PM EDT: I cn't think how to get the other part, though.

but unfortunately time runs out and the moderator closes the session with some questions.

Usually....

qwertyuiop 5/18/06 8:09:15 PM EDT: Usually when I work in a group, I don't do much. Here, I had a lot more to say.

End

Dataset2/D2TCSS