Dataset2/D2TCSS
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) | | | ----------------------------------
Other problems to do?
qwertyuiop 5/11/06 7:48:37 PM EDT: are there other problems to do? 137 5/11/06 7:48:51 PM EDT: I do not think so... Jason 5/11/06 7:48:58 PM EDT: well Jason 5/11/06 7:49:03 PM EDT: we are supposed to come up with some qwertyuiop 5/11/06 7:49:10 PM EDT: ok... Cynthia 5/11/06 7:49:13 PM EDT: did you view the topic for tonight, session 2 Jason 5/11/06 7:49:17 PM EDT: WHAT IF? Mathematicians do not just solve other people's problems - they also explore little worlds of patterns that they define and find interesting. Think about other mathematical problems related to the problem with the sticks. For instance, consider other arrangements of squares in addition to the triangle arrangement (diamond, cross, etc.). What if instead of squares you use other polygons like triangles, hexagons, etc.? Which polygons work well for building patterns like this? How about 3-D figures, like cubes with edges, sides and cubes? What are the different methods (induction, series, recursion, graphing, tables, etc.) you can use to analye these different patterns? --->(Taken from the topic) Cynthia 5/11/06 7:49:45 PM EDT: thanks, jason 137 5/11/06 7:50:07 PM EDT: Let's try diamonds first..
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 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
