Dataset2/D2TCSS
From Jsarmi
Contents
|
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:
Session 4:
(L) Late
(*) 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
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
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
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
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
Noticed how Qw gest involved and asks a question:
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 noticed later how "how did you get it" is also a request for prior activity to be reported... how is that different?
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 7:35:13 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?
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 OUT OF SEQUENCE
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
What do we do now?
137 5/11/06 7:48:12 PM EDT: Er... Am I lagging or is nobody typing? qwertyuiop 5/11/06 7:48:21 PM EDT: nobody is typing Jason 5/11/06 7:48:21 PM EDT: typing now :) 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? 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.. p M Jason 5/11/06 7:50:17 PM EDT: ok p M M Jason 5/11/06 7:50:27 PM EDT: if the squares were arranged in a diamond-like shape... p M M M M Jason 5/11/06 7:50:37 PM EDT: wait -- lemme make sure i read the directions carefully
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
Triangular Numbers
Something other teams discovered but for which there is no group-to-group interaction
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
