Jsarmi - User contributions [en]

Ideas/abalorios

Jsarmi — Tue, 12 Jan 2010 21:26:47 GMT

Jsarmi:

"Narrative" Design como en la obra de Stuart [ ]
the suggestive power of objects, evocative, misterious
iconicidad pictórica, posible en el lenguaje poético
La autoreferencialidad libera los sistemas intertextuales de significado.
de acuerdo con Henri Bergson descubrimos el fluir de nuestra personalidad a través del tiempo

Juego de los Abalorios
Playing the game well requires years of hard study of music, mathematics, and cultural history. Essentially the game is an abstract synthesis of all arts and scholarship. It proceeds by players making deep connections between seemingly unrelated topics.
http://en.wikipedia.org/wiki/Glass_bead_game

Moritz Waldemeyer

No es una tarjeta es una cajita/grid con objetos
Los objetos son para que el que las recibe los pondere, sus caracterIsticas significado, su combinaciOn (rotaciOn), su sequencia, su procedencia

Los dos componentes a explorar son los diferentes grids y los objetos

El grid mas pequeNo tiene 1 objeto pero no tiene narrativa? [x]

Luego viene el grid de 2 que es la dualidad pero se puede orientar de muchas maneras:
[ [x] [x] ]

[X]
[X]

[X]
?
[X]

ESPIRAL? (a lo vigotsky... cada "vuelta" es lo mismo pero diferente! expandido, enriquecido)

En tres dimensiones uno enfrente del otro

El grid de tres tiene otras posibilidades una de las mAs comunes siendo el triAngulo que aunque tenga porciones iguales tiene una en el Apice. El rectAngulo puede ser un grid de 3

El grid de cuatro es el tipico rectAngulo pero no tiene que ser solo esto.

Grid/cajita Fen-Shuei tiene X divisiones y puede contenter muchos objetos

You could pitch this to hallmark but then you will loose it. You will have to construct a system to create many combinations and then see if that is a lot to own it. You could do it with poems, with phrases, with stories, with so many things ann emptiness counts.

You could make a grid that is growable so as people get more pieces they grow their grid or they grow it themselves. They could pass them around a group of friends or family every member adding one. Could we do this only... es esta una versiOn del juego de los abalorios?

[ Un verso de Vallejo ]
al lado o debajo, o sin conectarse pero esperando que alguien la conecte
[Sofa by Frank Zappa tocado por Michael Hedges]

Made out of wire so that you could hang them

Mom could help me find so many old things in Colombia

LOS OBJETOS

los objetos pueden evocar pero pueden tambien sugerir ciertas direcciones. Por ejemplo, los objetos pueden tener caracteristicas o posibilidades. Un galgo con un pata rota, un reloj detenido en una hora, un objeto que se puede rotar pero solo en una direcciOn. Un objeto que cabe dentro de otro, etc. etc.

WHAT TO DO?
Take 3-position-grids (which are the minimally interesting), 4 position grids, and 9 position grids and experiment with them. First creae a platform so that it would be extremely easy to create many (e.g. an html table where pics can be aeasily asigned)

Fill 20 of each and document them. Try some "anonimous" objects (white, not discrete) and empty positions (or missing positions) o siluetas o "archetipos' de objetos pero con caracterIsticas definidas (alto, ancho, roto/degradado). Estas figuras, a propOsito, se podrIan usar estos objetos para enseNar a los ninos la operaciOn cognitiva de "identificar" caracterIsticas de objetos.

EXPLORE THE PROBLEM SPACE

Makit it a "meaning-making" game for kids not a competition game

[x] [?] [?]

Make it so that people can write different "interpretations" online and have the interpretations play within their own grids :)

Is this like TAROT or ORACLE CARDS: "when X comes after Y then..." kind of but the language is much more varied and the interpretation more open, It has numerology because if you have n number of one object in one cell then you need to think about that

ABOUT NUMEROLOGY... what is the meaning of each one of the grid sizes: 1, 3, 4, 9?

Literature
Tarot was used as early as the 16th century to compose poems, called "tarocchi appropriati", describing ladies of the court or famous personages. In modern literature, two exceptional examples of novels centered on the Tarot are The Greater Trumps (1932) by Charles Williams and Il Castello dei destini incrociati (1969) (English translation: The Castle of Crossed Destinies [1979]) by Italo Calvino. In the former, the Tarot is used by the main characters to move through space and time, create matter, and raise powerful natural storms. In the latter, Mediaeval travellers meeting at a castle are inexplicably unable to speak, and use a Tarot deck to describe their stories, which are reconstructed by the narrator, calling forth implications of the nature of communication, fate, and the presence of the transcendent in daily life. Tarots also appear in T.S. Eliot's modernist poem The Waste Land (1922), in connection with the figure of Madame Sosostris, one of the characters which appear in the first part, "The Burial of the Dead". Some of the cards mentioned in the poem really exist in the tarot deck (the Hanged Man, the Wheel), some have been invented by Eliot. The 2007 novel Sepulchre by British author Kate Mosse features a fictional Tarot deck.

A GRID THAT HAS 10 "lines" for ten key words of a "dEcima"
Piedra de sol estA escrito en 584 endecasIlabos! (11 sIlabas)

[edit] Psychological
Carl Jung was the first psychologist to attach importance to tarot symbolism.[21] He may have regarded the tarot cards as representing archetypes: fundamental types of persons or situations embedded in the subconscious of all human beings. The theory of archetypes gives rise to several psychological uses. Since the cards represent these different archetypes within each individual, ideas of the subject's self-perception can be gained by asking them to select a card that they 'identify with'. Equally, the subject can try and clarify the situation by imagining it in terms of the archetypal ideas associated with each card. For instance, someone rushing in heedlessly like the Knight of Swords, or blindly keeping the world at bay like the Rider-Waite-Smith Two of Swords.

More recently Timothy Leary has suggested that the Tarot Trump cards are a pictorial representation of human development from infant to adult, with the Fool symbolizing the newborn infant, the Magician symbolizing the stage at which an infant begins to play with artifacts, etc. In Leary's view the Tarot Trumps may be viewed as a blueprint for the human race as it matures.

Immaterial Culture
folklore, beliefs, values, ideologies, philosophies, customs, traditions, legends, idioms, humour, methodologies, music, science, the sublime & the ridiculous.
http://typhon.autonomous.org/

Piedra de Sol tiene 584 lineas, como dias tiene el periodo synoidea de Venus.
Pregunta... ese periodo es un perido medido en dias de la tierra o en dias de Venus. Los dias en Venus duran 24 horas de la tierra? Lo dudo... pero por lo menos la definiciOn de "dia" es la misma" salida y puesta del sol?

Como era otra vez esos shatzkys en el peru? los nuditos aquellos usados para grabar y reconstruir memorias? QuE eran las filas y las columnas?

Que lindo que serIa hacer una piedra solar con "celdas" en las que en cada una se pone un elemento de cada una de las lIneas del poema de Octavio Paz... no toda la frase pero un objeto que evoque la frase... sauce chopo surtidor viento o

un sauce de cristal, un chopo de agua,
un alto surtidor que el viento arquea, U NAL TO SUR TI DOR QUE EL VIEN TO AR
un árbol bien plantado mas danzante, U NAR BOL BIEN PLAN TA DO MAS DAN ZAN TE
un caminar de río que se curva,
avanza, retrocede, da un rodeo A VAN ZA RE TRO CE DE DA UN RO DEO
y llega siempre: Y LLE GA SIEM PRE = 5

un caminar tranquilo UN CA MI NAR TRAN QUI LO = 7
de estrella o primavera sin premura,
agua que con los párpados cerrados
mana toda la noche profecías,
unánime presencia en oleaje,
ola tras ola hasta cubrirlo todo,
verde soberanía sin ocaso
como el deslumbramiento de las alas
cuando se abren en mitad del cielo,

un caminar entre las espesuras
de los días futuros y el aciago
fulgor de la desdicha como un ave
petrificando el bosque con su canto
y las felicidades inminentes
entre las ramas que se desvanecen,
horas de luz que pican ya los pájaros,
presagios que se escapan de la mano,

una presencia como un canto súbito,
como el viento cantando en el incendio,
una mirada que sostiene en vilo
al mundo con sus mares y sus montes,
cuerpo de luz filtrado por un ágata,
piernas de luz, vientre de luz, bahías,
roca solar, cuerpo color de nube,
color de día rápido que salta,
la hora centellea y tiene cuerpo,
el mundo ya es visible por tu cuerpo,
es transparente por tu transparencia,

Ideas/abalorios/ejemplos

Jsarmi — Tue, 12 Jan 2010 21:02:36 GMT

Jsarmi: Created page with '::4 pepas de cristal :: 1 cuerda de guitarra :: Soneto de Cortazar ::un sauce de cristal :: un chopo de agua :: un alto surtidor que el viento arquea'

::4 pepas de cristal :: 1 cuerda de guitarra :: Soneto de Cortazar
::un sauce de cristal :: un chopo de agua :: un alto surtidor que el viento arquea

Ideas/abalorios

Jsarmi — Tue, 12 Jan 2010 20:04:31 GMT

Jsarmi: Created page with '"Narrative" Design como en la obra de Stuart [ ] the suggestive power of objects, evocative, misterious iconicidad pictórica, posible en el lenguaje poético La autoreferencial…'

"Narrative" Design como en la obra de Stuart [ ]
the suggestive power of objects, evocative, misterious
iconicidad pictórica, posible en el lenguaje poético
La autoreferencialidad libera los sistemas intertextuales de significado.
de acuerdo con Henri Bergson descubrimos el fluir de nuestra personalidad a través del tiempo

Moritz Waldemeyer

No es una tarjeta es una cajita/grid con objetos
Los objetos son para que el que las recibe los pondere, sus caracterIsticas significado, su combinaciOn (rotaciOn), su sequencia, su procedencia

Los dos componentes a explorar son los diferentes grids y los objetos

El grid mas pequeNo tiene 1 objeto pero no tiene narrativa? [x]

Luego viene el grid de 2 que es la dualidad pero se puede orientar de muchas maneras:
[ [x] [x] ]

[X]
[X]

[X]
?
[X]

ESPIRAL? (a lo vigotsky... cada "vuelta" es lo mismo pero diferente! expandido, enriquecido)

En tres dimensiones uno enfrente del otro

El grid de tres tiene otras posibilidades una de las mAs comunes siendo el triAngulo que aunque tenga porciones iguales tiene una en el Apice. El rectAngulo puede ser un grid de 3

El grid de cuatro es el tipico rectAngulo pero no tiene que ser solo esto.

Grid/cajita Fen-Shuei tiene X divisiones y puede contenter muchos objetos

You could pitch this to hallmark but then you will loose it. You will have to construct a system to create many combinations and then see if that is a lot to own it. You could do it with poems, with phrases, with stories, with so many things ann emptiness counts.

You could make a grid that is growable so as people get more pieces they grow their grid or they grow it themselves. They could pass them around a group of friends or family every member adding one. Could we do this only... es esta una versiOn del juego de los abalorios?

[ Un verso de Vallejo ]
al lado o debajo, o sin conectarse pero esperando que alguien la conecte
[Sofa by Frank Zappa tocado por Michael Hedges]

Made out of wire so that you could hang them

Mom could help me find so many old things in Colombia

LOS OBJETOS

los objetos pueden evocar pero pueden tambien sugerir ciertas direcciones. Por ejemplo, los objetos pueden tener caracteristicas o posibilidades. Un galgo con un pata rota, un reloj detenido en una hora, un objeto que se puede rotar pero solo en una direcciOn. Un objeto que cabe dentro de otro, etc. etc.

WHAT TO DO?
Take 3-position-grids (which are the minimally interesting), 4 position grids, and 9 position grids and experiment with them. First creae a platform so that it would be extremely easy to create many (e.g. an html table where pics can be aeasily asigned)

Fill 20 of each and document them. Try some "anonimous" objects (white, not discrete) and empty positions (or missing positions) o siluetas o "archetipos' de objetos pero con caracterIsticas definidas (alto, ancho, roto/degradado). Estas figuras, a propOsito, se podrIan usar estos objetos para enseNar a los ninos la operaciOn cognitiva de "identificar" caracterIsticas de objetos.

EXPLORE THE PROBLEM SPACE

Makit it a "meaning-making" game for kids not a competition game

[x] [?] [?]

Make it so that people can write different "interpretations" online and have the interpretations play within their own grids :)

Is this like TAROT or ORACLE CARDS: "when X comes after Y then..." kind of but the language is much more varied and the interpretation more open, It has numerology because if you have n number of one object in one cell then you need to think about that

ABOUT NUMEROLOGY... what is the meaning of each one of the grid sizes: 1, 3, 4, 9?

Literature
Tarot was used as early as the 16th century to compose poems, called "tarocchi appropriati", describing ladies of the court or famous personages. In modern literature, two exceptional examples of novels centered on the Tarot are The Greater Trumps (1932) by Charles Williams and Il Castello dei destini incrociati (1969) (English translation: The Castle of Crossed Destinies [1979]) by Italo Calvino. In the former, the Tarot is used by the main characters to move through space and time, create matter, and raise powerful natural storms. In the latter, Mediaeval travellers meeting at a castle are inexplicably unable to speak, and use a Tarot deck to describe their stories, which are reconstructed by the narrator, calling forth implications of the nature of communication, fate, and the presence of the transcendent in daily life. Tarots also appear in T.S. Eliot's modernist poem The Waste Land (1922), in connection with the figure of Madame Sosostris, one of the characters which appear in the first part, "The Burial of the Dead". Some of the cards mentioned in the poem really exist in the tarot deck (the Hanged Man, the Wheel), some have been invented by Eliot. The 2007 novel Sepulchre by British author Kate Mosse features a fictional Tarot deck.

A GRID THAT HAS 10 "lines" for ten key words of a "dEcima"
Piedra de sol estA escrito en 584 endecasIlabos! (11 sIlabas)

[edit] Psychological
Carl Jung was the first psychologist to attach importance to tarot symbolism.[21] He may have regarded the tarot cards as representing archetypes: fundamental types of persons or situations embedded in the subconscious of all human beings. The theory of archetypes gives rise to several psychological uses. Since the cards represent these different archetypes within each individual, ideas of the subject's self-perception can be gained by asking them to select a card that they 'identify with'. Equally, the subject can try and clarify the situation by imagining it in terms of the archetypal ideas associated with each card. For instance, someone rushing in heedlessly like the Knight of Swords, or blindly keeping the world at bay like the Rider-Waite-Smith Two of Swords.

More recently Timothy Leary has suggested that the Tarot Trump cards are a pictorial representation of human development from infant to adult, with the Fool symbolizing the newborn infant, the Magician symbolizing the stage at which an infant begins to play with artifacts, etc. In Leary's view the Tarot Trumps may be viewed as a blueprint for the human race as it matures.

Immaterial Culture
folklore, beliefs, values, ideologies, philosophies, customs, traditions, legends, idioms, humour, methodologies, music, science, the sublime & the ridiculous.
http://typhon.autonomous.org/

Piedra de Sol tiene 584 lineas, como dias tiene el periodo synoidea de Venus.
Pregunta... ese periodo es un perido medido en dias de la tierra o en dias de Venus. Los dias en Venus duran 24 horas de la tierra? Lo dudo... pero por lo menos la definiciOn de "dia" es la misma" salida y puesta del sol?

Como era otra vez esos shatzkys en el peru? los nuditos aquellos usados para grabar y reconstruir memorias? QuE eran las filas y las columnas?

Que lindo que serIa hacer una piedra solar con "celdas" en las que en cada una se pone un elemento de cada una de las lIneas del poema de Octavio Paz... no toda la frase pero un objeto que evoque la frase... sauce chopo surtidor viento o

un sauce de cristal, un chopo de agua,
un alto surtidor que el viento arquea, U NAL TO SUR TI DOR QUE EL VIEN TO AR
un árbol bien plantado mas danzante, U NAR BOL BIEN PLAN TA DO MAS DAN ZAN TE
un caminar de río que se curva,
avanza, retrocede, da un rodeo A VAN ZA RE TRO CE DE DA UN RO DEO
y llega siempre: Y LLE GA SIEM PRE = 5

un caminar tranquilo UN CA MI NAR TRAN QUI LO = 7
de estrella o primavera sin premura,
agua que con los párpados cerrados
mana toda la noche profecías,
unánime presencia en oleaje,
ola tras ola hasta cubrirlo todo,
verde soberanía sin ocaso
como el deslumbramiento de las alas
cuando se abren en mitad del cielo,

un caminar entre las espesuras
de los días futuros y el aciago
fulgor de la desdicha como un ave
petrificando el bosque con su canto
y las felicidades inminentes
entre las ramas que se desvanecen,
horas de luz que pican ya los pájaros,
presagios que se escapan de la mano,

una presencia como un canto súbito,
como el viento cantando en el incendio,
una mirada que sostiene en vilo
al mundo con sus mares y sus montes,
cuerpo de luz filtrado por un ágata,
piernas de luz, vientre de luz, bahías,
roca solar, cuerpo color de nube,
color de día rápido que salta,
la hora centellea y tiene cuerpo,
el mundo ya es visible por tu cuerpo,
es transparente por tu transparencia,

Writing/list

Jsarmi — Thu, 07 Jan 2010 21:38:44 GMT

Jsarmi: Created page with ':From Mental Models to Experienced Situations ::Introduce Situated Cognition to the UX world ::Include Phenomenology & Pragmatism. Dewey's --> Experience'

:From Mental Models to Experienced Situations
::Introduce Situated Cognition to the UX world
::Include Phenomenology & Pragmatism. Dewey's --> Experience

Dharma/ideas

Jsarmi — Thu, 07 Jan 2010 21:37:43 GMT

Jsarmi:

Una iniciativa cercana a mi pasiOn en la que pueda participar mi mamA desde Colombia y que sea una fuente de ingresos (adicional o principal) y que sea una fuente de realizaciOn y alegria. Vamos a coleccionar ideas y a madurarlas.

:CafeLectura para PapAs e Hijos
::Vienes dejas tu nino en 2 o 3 pods de lectura compartida (e interacciOn) te tomas un cafE, hablas con una amigo/a y compras libros. Entras con un carnet anual que te da descuentos. El sitio solo vende 10 libros nuevos al mes pero son buenIsimos y los miembros pueden sugerir. El sitio es mAs un respiro para errands o para un pequeNo time off pero no es una guarderIa. Te dan un dispositivo para que te puedan ubicar en caso de llanto, problemas, etc. El diseNo del sitio le permite al padre tener una visual directa con el sitio donde estA el niNo. Los cuentos son cortos y rotan relativamente rApido.

Jsarmi — Sat, 03 May 2008 21:33:30 GMT

Jsarmi: /* What do we do now? */

Dataset2/D2TCSS

Jsarmi — Sat, 03 May 2008 21:32:09 GMT

Jsarmi: /* The "each square with 2 sides" thing doesn't work as neatly here */

==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...
(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

===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

== 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.

===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

===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

Jsarmi — Sat, 03 May 2008 21:21:12 GMT

Jsarmi: /* The "each square with 2 sides" thing doesn't work as neatly here */

==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...
(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===

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

== 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.

===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

===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

Jsarmi — Sat, 03 May 2008 21:06:07 GMT

Jsarmi — Sat, 03 May 2008 15:19:11 GMT

Jsarmi: /* Final action: the rest on the Wiki */

=Group Trajectory=
Session 1: Dyad from same school, technical problems, f2f work produces some reported results
Session 2: Feedback attended to
Session 3:
Session 4:

==Group composition: Stable/ Technical Problems==

Session 1: tc ro (TP)
Session 2: tc ro
Session 3: tc ro (TP)
Session 4: tc ro

(TP) Technical Problems

==Session I==

Work done F2F is reported to the chat... how is this "reporting" deployed? Initially Tc attempts a direct dialog style:

tcmath 5/9/06 5:12:43 PM EDT: So, Rook, we have the fact that squares is defined as a function of N
by the partial sum quadratic for the sequence 1,2,3,4,5....

This whiteboard textbook was created by tcmath before the previous message:

sequence:1,2,3,4,5
partial sum sequence:1,3,6,10,15
Function is quadratic, so these eqautions are true:
a1^2+b1+c=1

but later, because of technical problems, the style changes to a plural narrative:

jsarmi 5/9/06 5:19:46 PM EDT: It seems as if the system is giving Rook a lot of trouble, isn't it, tcmath?
jsarmi 5/9/06 5:21:41 PM EDT: tcmath?
tcmath 5/9/06 5:24:36 PM EDT: I'm done for now
tcmath 5/9/06 5:24:48 PM EDT: I was working with Rook since his computer wasn't working
jsarmi 5/9/06 5:24:57 PM EDT: I see
tcmath 5/9/06 5:25:16 PM EDT: We figured out the equation for squares, and we should easily solve it for sticks as well in the same manner

This lat posting works like a "projection/plan" which includes a report of what was done. This posting is deceptively simple. It is a report of what was done framed as "figuring out" but such achievement (i.e. the equation for the squares) is not recap, nor placed on the whiteboard. The projection made of what can be done in the future with two qualifications: level of difficulty (easily) and manner (the same way). It is presented to an unspecified recipient, most likely, the moderator who has been giving out instruction on the task and asking for the status of Rook's participation.

tcmath 5/9/06 5:25:16 PM EDT: We figured out the equation for squares, and we should easily solve it for sticks as well in the same manner.

==Wiki posting for Session I?==
It isn't clear that results were posted between Sessions I and II. Most likely, in Session II the team posted results for both sessions. Some results are posted to the "rows of the table" page. Interestingly, their results differ with what Team B posted in the same page (almost next to it) by other teams. There is uptake of this triggered by the facilitator during Session II.
For example:

Row 3

Team name: B
Row 3: 28, 10

Team name: A
Row 3: 18, 6

Team name:
Row 3:

==Feedback==
(No Feedback in between sessions)

==Session II==

Technical problems seemed to be resolved. Dyad must be in same room or connected otherwise, no greetings. Tc starts with this message after which the board is CLEARED of everything that was posted in the previous session.

tcmath 5/11/06 5:04:13 PM EDT: Okay. You must agree that the squares is defined by the equation squares=1/2N^2+1/2N.

Rook attempts to paste a table of values into a textbox but the formatting does not work too well and he ends up with a long list instead. Interestingly, this list lacks values that may have been posted to the wiki already? E.g., The values for N=4 and N=5 which in the wiki read "Team name: A Row 4: 28, 10 / Team name: A Row 5: 40, 15", and in Rooks texbox: "4 ? ? / 5 ? ?". Perhaps they created these values in the second session since they map their equation: N^2 + 3*N?

===Are we doing Session 1 still? Lack of coordination?:===

tcmath 5/11/06 5:09:53 PM EDT: I'm setting up the problem on the whiteboard
p M M M
tcmath 5/11/06 5:10:08 PM EDT:
p M M M M
Rook 5/11/06 5:11:18 PM EDT: Did you solve the sticks as function of N yet?
Rook 5/11/06 5:11:35 PM EDT: I think the equation is x^2+3x
Rook 5/11/06 5:11:50 PM EDT: Found the equation...
tcmath 5/11/06 5:11:54 PM EDT: The seperate lines here represent the sticks you would "add" when N increases by 1.
p M M
Rook 5/11/06 5:12:11 PM EDT: What are you doing?
p M M
Rook 5/11/06 5:12:17 PM EDT: !!
Rook 5/11/06 5:12:40 PM EDT: Are we doing Session 1 still?
tcmath 5/11/06 5:12:43 PM EDT: Okay. The top 4 sticks are there by default; you have to have them.

===Linking Sequences: Squares then Sticks===
How do they constitute the two to be related?

tcmath 5/11/06 5:26:20 PM EDT: So, for squares we found the partial sum equation for the sequence 1,2,3,4,5... Now we find it for the sequence 2n+2 (after distribution)

===Then we will be done===
Definitively in the same room but conversing/coordinationg on the chat. Notice long gaps between postings

tcmath 5/11/06 5:28:30 PM EDT: Here, you can use my TI-84 to do the whole regression thing and I'll do it algebraically on paper. Tell me your results.
Rook 5/11/06 5:29:34 PM EDT: oh...
Rook 5/11/06 5:30:36 PM EDT: x^2+3X
tcmath 5/11/06 5:31:51 PM EDT: Then we're done with the pattern:

===Typing on the Wiki===

tcmath 5/11/06 5:42:32 PM EDT: we're here--we're just typing up our patterns observed on VMT Wiki

===Reading the Wiki, Finding the "next" topic===

jsarmi 5/11/06 5:47:28 PM EDT: Well.. any ideas suggested by the notes from other groups in the Wiki?
tcmath 5/11/06 5:48:02 PM EDT: Well, they figured out the same thing for squares, but their approach was unique for the sticks.
tcmath 5/11/06 5:48:33 PM EDT: We used the same technique for both teh sticks and the squares (partial sums on a sequence), while they used two different ones.
jsarmi 5/11/06 5:48:41 PM EDT: how about the values of the table?
Rook 5/11/06 5:49:02 PM EDT: they got the same
Rook 5/11/06 5:49:33 PM EDT: and we checked with the other groups--same answers...
tcmath 5/11/06 5:49:38 PM EDT: They were ahead one row, though, starting with row 4 since it was the first row without a given answer.
Rook 5/11/06 5:49:53 PM EDT: on the 1st website (VMT Wiki)
jsarmi 5/11/06 5:50:12 PM EDT: Oh... I see
tcmath 5/11/06 5:50:31 PM EDT: For our discussion: I think that mathematiciancs would usually generalize a lot, finding a method that fit the problem even if initial values were changed.
Rook 5/11/06 5:50:46 PM EDT: Yeah
Rook 5/11/06 5:50:54 PM EDT: that's the next topic

===Our method would still work===

tcmath 5/11/06 5:51:39 PM EDT: For example, if every column in the diagram increased by 2 instead of 1.
Rook 5/11/06 5:52:20 PM EDT: I think that the polygons/figures have to be regular to have a chance at this problem, of course--or else there's really no pattern
Rook 5/11/06 5:52:39 PM EDT: Assume that the rest of the answers would just be doubled?
tcmath 5/11/06 5:52:57 PM EDT: If this happened, then our method would still work becuase we would determine the original sequence and then generalize it in a closed form, quadratic equation, for the partial sums.
Rook 5/11/06 5:53:02 PM EDT: The 2 wouldn't make a difference really
Rook 5/11/06 5:53:22 PM EDT: Yeah, or solve it as written, jumping by 2...
Rook 5/11/06 5:53:37 PM EDT: so, it should also work for n-gons that are regular
tcmath 5/11/06 5:54:18 PM EDT: Probably not for three since the sticks really depends on the number of squares we're increasing by. In ours, we had 4 in our equation, while if we increased by three, that number would be 10, I think.
Rook 5/11/06 5:54:34 PM EDT: Yeah, the equation, such as the initial starting point, would have to change to fit more or less starting sticks

===Projecting===

tcmath 5/11/06 5:58:42 PM EDT: We should try looking at the specific case of, say pentagons.
p M
Rook 5/11/06 5:59:11 PM EDT: Yeah
Rook 5/11/06 5:59:18 PM EDT: though do you have to leave?
tcmath 5/11/06 5:59:28 PM EDT: That'll have to wait till next time. I have to leave.
I wonder if we could even replicate the problem with regular pentagons?
Rook 5/11/06 5:59:45 PM EDT: mb in our spare time we could look at it and talk next session

=Feedback=

Dear tcmath and Rook,

We were very interested in the way you combined several approaches to find the formula
for the pattern of growth of sticks and squares. Using a regression method and thinking about
the difference between one step of the pattern and the next seem to work very nicely.
You also posted very complete descriptions of your ideas on the Wiki.

In your previous session you started to explore what the patterns would look like with other polygons and in 3-D,
and you made a couple of conjectures about them (e.g. “the methods for solving general situations like this would be the same”).
For the next step we will encourage you to continue thinking about the different approaches and the problems
that you can discover of your own and that are interesting to pursue.

-The VMT Team

=Session III=

The feedback talks about 3D so Rook orients to it:

tcmath 5/16/06 5:04:24 PM EDT: The 3-d seems like a good place to start.

Noticed that Rook had made some "projected" observations about 3D in the previous session:

Rook 5/11/06 5:58:40 PM EDT: but 3-D might include solving it in "2-D", such as seeing the patterns
on one "dimension" of the 3-D figures, and combing the formulats into one
equation for the 3-D problem

Tcmath reports on work they did in between. Because of technical problems they had to "talk" in person and report in the chat:

tcmath 5/16/06 5:06:04 PM EDT: We started talking about 3d after the last session and decided
that the method would just be double: The original sequence would be the 2-d problem
tcmath 5/16/06 5:08:51 PM EDT: This is true becuase if the problem is a pyramid, expanding n cubes in each direction,
then the number of squares starts with this pattern, then is the same pattern for n-1,
then n-2, all the way to one cube.
jsarmi 5/16/06 5:10:12 PM EDT: tcmath... you may want to wait until Rook gets here so that you can both discuss
the ideas you are posting
tcmath 5/16/06 5:11:36 PM EDT: Okay. we'll just talk (in person) and then type up what we said.

==A projectable (pentagons) recovered but discarded?==

They decide that Rook will type in parenthesis his ideas

jsarmi 5/16/06 5:22:41 PM EDT: (BTW, Once you read the feedback you can delete it so that you can work on the whiteboard freely)
tcmath 5/16/06 5:25:08 PM EDT: The feedback seems to indicate that we should work on 3d., since pentagons tesselate in a weird way (in 5 directions instead)
jsarmi 5/16/06 5:25:42 PM EDT: This is for you to decide... you can go in any direction that seems promising to you.
tcmath 5/16/06 5:25:59 PM EDT: Rook: if it's 2-D; assume that you add 2, then 3...figures every time...
tcmath 5/16/06 5:26:39 PM EDT: 3D seems like a good place to continue, since cubes are a lot like squares.
tcmath 5/16/06 5:27:42 PM EDT: [every side/plane is similar to the squares problem]
tcmath 5/16/06 5:28:06 PM EDT: [since we already figured out the equations for squares]
tcmath 5/16/06 5:30:00 PM EDT: 3d goes like this: the base has 1/2N^2+1/2N cubes, as we figured out in the squares problem, and each level above it has 1/2(N-1)^2+1/2(N-1), and so on.

==Reportable: Same method as before, Algebra==

tcmath 5/16/06 5:41:10 PM EDT: ANYhoo, the 3d partial sum equation is probably a cubic (I think the patter n is, the partial sum equation has a degree one higher than the sequence it is partial-summing)
jsarmi 5/16/06 5:41:30 PM EDT: ok
tcmath 5/16/06 5:42:22 PM EDT: We'll use the same method as we did last time, with algebra

==Wiki writing: a table==

Rook 5/16/06 5:51:34 PM EDT: I'm going to make a table for the data value for a the cube sequences
jsarmi 5/16/06 5:51:40 PM EDT: sure... probably better to concentrate on the math than fighting the computers... sorry about that
jsarmi 5/16/06 5:54:51 PM EDT: a table on the whitebaord?
Rook 5/16/06 5:57:47 PM EDT: on Wiki
jsarmi 5/16/06 6:00:56 PM EDT: I see
Rook leaves the room 5/16/06 6:04:27 PM EDT

Then, abrupt ending

==Feedback==

Dear tcmath and Rook:
We are sorry that you ran into technical problems last time. You tried to work with one computer but
that must have been awkward. Despite that, it seemed to us that you were able to start exploring a
3-D pattern based on your notes posted on the Wiki. Do you need to specify which 3-D pattern you are investigating?
Is it similar to the ones that other teams are exploring?

We hope that this time you will be able to collaborate via the system and continue
this work or explore other interesting math question you create. -The VMT team

This feedback message gets deleted very early on Session IV by tcmath. Rook claims he knows how to get it from the history

==Session IV==
The whitebaord stays clear for the most of the session after Tcmath erases everything, until a critical point when tcmath wants to "take a look" at something on the whitebaord

====Lets continue + Wiki====

Rook 5/18/06 5:25:41 PM EDT: o.k., let's continue the cubes version of the problem
Rook 5/18/06 5:25:44 PM EDT: tcmath
jsarmi 5/18/06 5:26:15 PM EDT: hopefully tcmath is just distracted with the Wiki and not having computer problems
Rook 5/18/06 5:26:31 PM EDT: I'll make a chart of the faces as function of N...
tcmath 5/18/06 5:27:40 PM EDT: Okay. I'm posting all of our ideas, since we came up with a lot of them. When I'm done, you can post your ideas on the wiki as you come up with them.

Rook 5/18/06 5:31:52 PM EDT: R you done posting?
Rook 5/18/06 5:31:54 PM EDT: tcmath
tcmath 5/18/06 5:32:09 PM EDT: Yeah. Its long, but its our work.

===I figured it out... everything in 3D has a strong correlation to the original squares problem===
Rook 5/18/06 5:34:50 PM EDT: so far, for N={1,2} I've gotten Faces={6,14}
Rook 5/18/06 5:34:54 PM EDT: it looks similar...
tcmath 5/18/06 5:35:19 PM EDT: I figured it out! (the faces). Its just 4*(1/2n^2+1/2n)
Rook 5/18/06 5:35:39 PM EDT: that makes sense!
Rook 5/18/06 5:35:55 PM EDT: the faces should be similar to the squares problem
Rook 5/18/06 5:36:05 PM EDT: it's just "magnified" times 4!
tcmath 5/18/06 5:36:41 PM EDT: I'll explain. There are two flat faces which are the full pyramid,
and one which is 3d from one angle, and that same side has another 1/2n^2+1/2n squares viewable
from a different abgle.
tcmath 5/18/06 5:36:51 PM EDT: Does this make any sense?
Rook 5/18/06 5:37:22 PM EDT: you mean the "opposite" side of the cube pyramid?
tcmath 5/18/06 5:37:34 PM EDT: Exactly. Its intuitive, not mathematical as one might expect.
However, we do notice that everything in 3D has a strong orrelation to the original squares problem

===Wait a minute...I'll look at it on the whiteboard===

tcmath 5/18/06 5:43:20 PM EDT: Wait a minute...I'll look at it on the whiteboard;
I think the actual formula is 2.5N^2+2.5N+1, or 5(1/2N^2+1/2N)+1, the extra 1 for the top face.
p M M M M M M M
Rook 5/18/06 5:44:08 PM EDT: (i get the four pyramids now: I was counting wrong)
p M M M M
tcmath 5/18/06 5:44:46 PM EDT: So the whiteboard now shows one view, a flat view of the 3d figure
p M M
Rook 5/18/06 5:45:14 PM EDT: where?
Rook 5/18/06 5:45:16 PM EDT: I don't see it
tcmath 5/18/06 5:45:43 PM EDT: You need to slide the slider on the left down to the bottom.
p M M
Rook 5/18/06 5:45:57 PM EDT: I think it just took some time...

===Something correlates===

Rook 5/18/06 5:48:30 PM EDT: something with the sticks in the original problem correlate to the number of faces in the cubes
tcmath 5/18/06 5:48:33 PM EDT: Yes, you're right. I think the ofrmula I came up with needs some adjustments.
Rook 5/18/06 5:48:50 PM EDT: the faces is similar to the sticks in the original
Rook 5/18/06 5:49:02 PM EDT: however, there are 6 faces compared to 4 sticks
Rook 5/18/06 5:49:16 PM EDT: and the faces increase by 8, then 10, then 12...
Rook 5/18/06 5:49:25 PM EDT: while the sticks are 6,8,10...
tcmath 5/18/06 5:50:09 PM EDT: We probably should use that method, with partial sums
tcmath 5/18/06 5:50:09 PM EDT: We probably should use that method, with partial sums.
Rook 5/18/06 5:50:41 PM EDT: yeah...
Rook 5/18/06 5:50:59 PM EDT: I think it's really the same equation, with slight addition modifications...
p M
Rook 5/18/06 5:51:21 PM EDT: what's the sticks original equation again?
Rook 5/18/06 5:51:42 PM EDT: let's see

==Final action: the rest on the Wiki==

tcmath 5/18/06 5:58:58 PM EDT: I'm leaving. Rook will put the rest of our ideas on the wiki
...
Rook 5/18/06 6:01:51 PM EDT: I think I'll quickly post what we found out just as tcmath was leaving (I checked with him in "life") on Wiki, I think I also have to leave in a minute

===Squares, Sticks, Cubes and Faces===
Interestingly, this group did not see a problem with working on a problem that was too similar to the original one, as other groups did. It almost seems that it was exciting for them to figure out that the same results applied in some parts of it. For other groups this was a reason not to approach the problem at all. In addition, this group broke the cube into "faces" which led them to reuse prior findings but did not break the "faces" into sticks. In other words, in the original pattern 2D (squares) was divided into 1D (sticks) but for their pyramid, 3D (cube) was only divided into 2D (faces).

Jsarmi — Thu, 01 May 2008 02:17:26 GMT

Jsarmi: /* Group composition: Stable */

Dataset2/D2TCSS

Jsarmi — Thu, 01 May 2008 02:14:40 GMT

Jsarmi: /* Session III */

==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

== 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.

===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

===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

Jsarmi — Thu, 01 May 2008 02:04:58 GMT

Jsarmi: /* Session III */

==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

== Session III ==
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.

===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

Dataset2/D2TCSS

Jsarmi — Thu, 01 May 2008 01:55:20 GMT

Jsarmi: /* What do we do now? */

Dataset2/D2TBSS

Jsarmi — Wed, 16 Apr 2008 15:50:56 GMT

Jsarmi: /* Sharedness */

=Group Trajectory=
Session 1: Bwang divided the figure into horizontal and vertical lines, Formulas for both sticks and squares were produced,
The values for N=4, 5, and 6 were agreed upon and posted to the Wiki. Quicksilver offers an explanation of how
the pattern grows but it is not discussed
Session 2: Feedback attended to
Session 3:
Session 4:

==Group composition: Stable==

Session 1: bw qs az
Session 2: bw qs az
Session 3:
Session 4:

(L) (N)

==Session I==

bwang8 5/9/06 6:23:18 PM EDT: hi
Aznx 5/9/06 6:23:23 PM EDT: Hi
Quicksilver 5/9/06 6:23:28 PM EDT: hey
Aznx 5/9/06 6:23:35 PM EDT: So we can't have our own friends?
bwang8 5/9/06 6:23:40 PM EDT: nope
bwang8 5/9/06 6:23:49 PM EDT: lol
Quicksilver 5/9/06 6:23:52 PM EDT: hey three of us are from miller
Gerry 5/9/06 6:23:55 PM EDT: Hi everyone!

Gerry 5/9/06 6:24:59 PM EDT: Today's session is mainly to get to know the VMT system

Gerry 5/9/06 6:29:12 PM EDT: You can click on the button at the top that says "View Topic" to see the math problem
bwang8 5/9/06 6:29:32 PM EDT: ok
bwang8 5/9/06 6:29:50 PM EDT: are we suppose to solve it now?
Gerry 5/9/06 6:30:13 PM EDT: Then you can click on the button in the little window that appears to open the topic in another big growser window
Gerry 5/9/06 6:30:36 PM EDT: browser*
Aznx 5/9/06 6:30:40 PM EDT: It didn't open.
Aznx 5/9/06 6:30:52 PM EDT: Now it did.
Aznx 5/9/06 6:31:32 PM EDT: So, are we supposed to work together?
bwang8 5/9/06 6:31:49 PM EDT: yeah
bwang8 5/9/06 6:31:50 PM EDT: ok
Gerry 5/9/06 6:31:54 PM EDT: Exactly!
Aznx 5/9/06 6:32:04 PM EDT: Aditya, you there?
bwang8 5/9/06 6:32:05 PM EDT: you can divide the thing into two parts
Aznx 5/9/06 6:32:10 PM EDT: Let's start this thing.
(bwang draws the third iteration of the pattern splitted in 2 diagrams dividing the horizontal and vertical lines)
...
Quicksilver 5/9/06 6:32:58 PM EDT: what are the lines for?
Aznx 5/9/06 6:33:01 PM EDT: go to view topic
bwang8 5/9/06 6:33:05 PM EDT: so you can see we only need to figur one out to get the total stick

===intersubjectivity===
good example of how whiteboar actions and chat are mutually informative,
but also how the sequential production of the conversation is used as a resource for making sense of the conversation.
This is all tentative, and the group works it out. What are the lines for?
Instructability: There is always a content and a procedural side to everything we do:
Saying "you can divide the thing into two parts" informs the others about how his subsequent actions are to be read.
Those whiteboard actions inform the meaning of that posting as well.

bwang8 5/9/06 6:33:32 PM EDT: 1+2+3+........+N+N
bwang8 5/9/06 6:33:38 PM EDT: times that by 2
Quicksilver 5/9/06 6:33:40 PM EDT: Never mind I figured it out..
Aznx 5/9/06 6:34:01 PM EDT: Can we collaborate this answer even more?
Aznx 5/9/06 6:34:05 PM EDT: To make it even simpler?
bwang8 5/9/06 6:34:15 PM EDT: ok
Aznx 5/9/06 6:34:16 PM EDT: Because I think we can.
bwang8 5/9/06 6:34:50 PM EDT: ((1+N)*N/2+N)*2
bwang8 5/9/06 6:34:58 PM EDT: that's the formula, right?
Aznx 5/9/06 6:35:15 PM EDT: How did you come up with it?
bwang8 5/9/06 6:35:16 PM EDT: for total sticks
bwang8 5/9/06 6:35:34 PM EDT: is a common formual
bwang8 5/9/06 6:35:40 PM EDT: formula
Aznx 5/9/06 6:35:46 PM EDT: Yeah, I know.
bwang8 5/9/06 6:35:59 PM EDT: and just slightly modify it to get this
(whiteboard)
Aznx 5/9/06 6:36:31 PM EDT: Aditya, you get this right?
(bwang corrects his drawing so that the horizontal lines orient in the same way that the original problem drawing)

bwang8 5/9/06 6:38:38 PM EDT: The number of squares is just (1+N)*N/2 (points to his formula posted at 6:34:50 PM)

Gerry 5/9/06 6:38:52 PM EDT: I put BWang's formula on the whiteboard
(whiteboard)
Aznx 5/9/06 6:39:45 PM EDT: So how do we submit this?
Quicksilver 5/9/06 6:40:26 PM EDT: We are still in the process
Quicksilver 5/9/06 6:40:38 PM EDT: We are discussing
Gerry 5/9/06 6:40:41 PM EDT: you can complete the table for different N and put that on the wiki

The practice of moving formulas to the whiteboard is not picked up, so the formula for the number of squares never makes it on to the whiteboard.

Quicksilver 5/9/06 6:46:50 PM EDT: Are we Team B?
Aznx 5/9/06 6:46:55 PM EDT: TEAM B
bwang8 5/9/06 6:47:02 PM EDT: i am done
Aznx 5/9/06 6:47:06 PM EDT: I think's its case sensitive.
Aznx 5/9/06 6:47:11 PM EDT: Alright, my turn.
Gerry 5/9/06 6:47:25 PM EDT: Nice!
bwang8 5/9/06 6:47:27 PM EDT: is mine in the correct format

Aznx 5/9/06 6:49:34 PM EDT: The number of squares is 15.
Gerry 5/9/06 6:49:47 PM EDT: You might want to draw the table on the whiteboard to make sure you all agree
(table being constructed on the whiteboard by bwang)
Quicksilver 5/9/06 6:51:00 PM EDT: There are 15 squares and 44 sticks for n=5?
(correction 4 instead of 3 and the values for n=5 just mentioned by Quicksilver added by bwang)
bwang8 5/9/06 6:52:03 PM EDT: isn't it 40
bwang8 5/9/06 6:52:18 PM EDT: (15+5)*2
(correction 40 instead of 44 by bwang)

Interesting... This an application of ((1+N)*N/2+N)*2 -> ((1+5)*5/2+5)*2) -> (15+5)*2 = 40

Later they have on the whiteboard two tables and one textbox which contain overlap but disagree in one number. In two, number of sticks for N=5 is 44 in the rest it is 40, so they try to coordinate this. Quicksilver does not seem responsive and later he posts this long chat message:

Quicksilver 5/9/06 7:00:39 PM EDT: Well, anyway, you can see a pattern that the amount of squares increases by the n. For the
sticks, The bottom row's square on the right has 2 new sticks. All the squares in the new row
to the left of it have 3 new sticks. So, If te row has 5 squares, 4 of the squares have 3 sticks,
the last on only has two. For the enitre Figure, you would add the amount to the previous ammount

After that technical problem the moderator sort of closes down the session:

Gerry 5/9/06 7:04:43 PM EDT: Maybe we should stop for today
Aznx 5/9/06 7:04:48 PM EDT: Alright.
jsarmi 5/9/06 7:04:50 PM EDT: Quicksilver... why don't you go ahead and close this window and try to enter again
Gerry 5/9/06 7:05:10 PM EDT: and let jsarmi help Quicksilver get ready to Thursday
jsarmi 5/9/06 7:05:27 PM EDT: (BTW, this room will now show up in the "My Rooms" tab not on the "Limited Access" one)
Quicksilver leaves the room 5/9/06 7:05:43 PM EDT
Gerry 5/9/06 7:05:48 PM EDT: You made a lot of progress already.
Aznx 5/9/06 7:05:53 PM EDT: So do we leave?
Aznx 5/9/06 7:06:01 PM EDT: And when do we come back?
...
Gerry 5/9/06 7:06:51 PM EDT: I think we will have the next session at the same time on Thursday.
You will come into the same room and start where you left off today
jsarmi 5/9/06 7:07:06 PM EDT: that's correct... same time, same team, same room
bwang8 5/9/06 7:07:12 PM EDT: ok
Quicksilver 5/9/06 7:07:15 PM EDT: I have a dentis't appointment...I don't think I can come
Aznx 5/9/06 7:07:16 PM EDT: Same time on Thursay...got it!
bwang8 5/9/06 7:07:37 PM EDT: but aren't we done now with this problem
Aznx 5/9/06 7:07:47 PM EDT: Yeah
Aznx 5/9/06 7:07:56 PM EDT: So I guess we would start on the second problem?
bwang8 5/9/06 7:08:05 PM EDT: ok
Quicksilver 5/9/06 7:08:16 PM EDT: Did you guys discuss the problem like it said to?
Aznx 5/9/06 7:08:21 PM EDT: Yeah.
Quicksilver 5/9/06 7:08:22 PM EDT: i didn't get half the messages'
Quicksilver 5/9/06 7:08:24 PM EDT: so i dont know
bwang8 5/9/06 7:08:25 PM EDT: yeah

Because Quicksilver can't make it on Thursday , they reschedule their session for the next day:

Quicksilver 5/9/06 7:09:45 PM EDT: Can we just reschedule for wednesday or something?
jsarmi 5/9/06 7:11:03 PM EDT: If it works out for everyone, you could re-schedule... is that an option?
Aznx 5/9/06 7:11:23 PM EDT: Wednesday should work for me.
Quicksilver 5/9/06 7:11:31 PM EDT: it works for me
Aznx 5/9/06 7:11:43 PM EDT: bwang?
bwang8 5/9/06 7:11:50 PM EDT: yes
Quicksilver 5/9/06 7:11:59 PM EDT: So this time tomorrow then
Quicksilver 5/9/06 7:12:11 PM EDT: in place of thursdays
bwang8 5/9/06 7:12:14 PM EDT: wednesday, 6-7 central?
Aznx 5/9/06 7:12:34 PM EDT: Same time as today.
Quicksilver 5/9/06 7:12:38 PM EDT: Yeah
bwang8 5/9/06 7:12:44 PM EDT: ok
Aznx 5/9/06 7:12:45 PM EDT: Except it's Wednesday instead of Thursday.
....
Aznx 5/9/06 7:13:37 PM EDT: So will we start on a new problem?
jsarmi 5/9/06 7:13:43 PM EDT: Is it clear to everyone???
Aznx 5/9/06 7:13:46 PM EDT: Tomorrow?
Quicksilver 5/9/06 7:13:48 PM EDT: Yes.
bwang8 5/9/06 7:13:59 PM EDT: ok
jsarmi 5/9/06 7:14:11 PM EDT: Some of you are on Central time and some of you on Pacific time, so do not get confused...
same time as today
bwang8 5/9/06 7:14:17 PM EDT: ok
Quicksilver 5/9/06 7:14:25 PM EDT: Ok. See you tomorrow then.

==Feedback==

VMT Feedback

We were very interested in the approach that divided the figure into
the horizontal lines and the vertical lines and the quickness with
which formulas fell out of that approach. It seemed as though you
also were paying attention to each other's work and quickly reached
agreement. You handled the technology of the chat environment and the
wiki easily.

We also noticed two places in the chat where some kinds of
conversation did not happen. There was a point where 44 was posted as
the number of sticks and 40 was offered as a correction. There was no
discussion of how 44 was calculated. At another moment, Quicksilver
posted an explanation of the pattern of growth that was not discussed.

There was a sense in which you indicated that your work was done when
you had at least one answer for the questions in the problem. For the
next step we will encourage you to think more about the different
approaches and the problems that you can discover on your own and
that are interesting to pursue.

==Session II==

Gerry 5/10/06 7:02:03 PM EDT: You have some feedback and a new Topic
Aznx 5/10/06 7:02:13 PM EDT: Has everyone read the feedback?
Quicksilver 5/10/06 7:02:17 PM EDT: yup
Quicksilver 5/10/06 7:02:25 PM EDT: NOw we have to discuss
bwang8 5/10/06 7:02:28 PM EDT: yes
Quicksilver 5/10/06 7:03:01 PM EDT: Well,, the part about converstaion not happening is because of me
Quicksilver 5/10/06 7:03:06 PM EDT: my computer was lagging....
Quicksilver 5/10/06 7:03:12 PM EDT: but that's out of our hands
Aznx 5/10/06 7:03:17 PM EDT: So, I think we should focus on discussing on each step more.
Quicksilver 5/10/06 7:03:30 PM EDT: and explain every answer thoroughly
Aznx 5/10/06 7:03:40 PM EDT: Even if the answer was "obvious."
bwang8 5/10/06 7:03:48 PM EDT: ok
Quicksilver 5/10/06 7:03:49 PM EDT: like i gave a wrong answer, but my explanations didn't come up
on the computer because of the lag
Quicksilver 5/10/06 7:03:58 PM EDT: so thats one thing
Quicksilver 5/10/06 7:04:12 PM EDT: maybe think of more ways of doing the same problem?
Aznx 5/10/06 7:04:21 PM EDT: Yeah.

===They try to start an activity===
Two proposals, looking at today's topic and talking about Quicksilvers wrong idea:

Quicksilver 5/10/06 7:04:50 PM EDT: Now did you two read today's topic?
bwang8 5/10/06 7:05:08 PM EDT: what was your pattern of growth, quicksiler
Quicksilver 5/10/06 7:06:20 PM EDT: i think it was something about the amount of squares that increased with each row....and how one of the new squares had 3 new sticks while the other new ones had 2 new sticks
p M
bwang8 5/10/06 7:06:41 PM EDT: oh, ok
(Quicksilver starts to draw squares with sticks to illustrate his talk to which he refers to in his next chat message)
Quicksilver 5/10/06 7:07:07 PM EDT: i drew some sqaures
Quicksilver 5/10/06 7:07:19 PM EDT: the left one had three new sticks
Quicksilver 5/10/06 7:07:33 PM EDT: the right one has a new stick on the bottom and on the right
Quicksilver 5/10/06 7:07:39 PM EDT: the top one is from an old square
Gerry 5/10/06 7:07:40 PM EDT: It was at 7:00:39 -- to get the old messages, click on the icon above here with the two circular arrows
(this messages points to a message from the previous day's session, unclear if they followed the link?)
(the feedback textbox is deleted from the whiteboard)
Quicksilver 5/10/06 7:09:25 PM EDT: yea that's wrong
Aznx 5/10/06 7:09:33 PM EDT: So let's brainstorm through some problems that we think are challenging.

===Aznx goes back to his idea of orienting to a new task===
Instead of quicksilver's error to which bwang and quicksilver oriented, others agree:

Quicksilver 5/10/06 7:09:40 PM EDT: yes...new topic
bwang8 5/10/06 7:09:42 PM EDT: ok
Quicksilver 5/10/06 7:10:20 PM EDT: 3-d figures?
(quicksilver adds sticks to the 2 squares he drew earlier to make them 3-D)

===Projecting===
Later on, Aznx adds a third 3D square and calls them a "row of blocks". Aznx makes some "projections" of what they could do:

Aznx 5/10/06 7:11:06 PM EDT: I think we should discuss on the different methods.
Aznx 5/10/06 7:11:24 PM EDT: So that we can easily apply our thoughts quickly when seeing a problem.
Quicksilver 5/10/06 7:11:30 PM EDT: Yes....but we must find a question or problem to investigate
Aznx 5/10/06 7:11:37 PM EDT: Yeah.
p M M
Aznx 5/10/06 7:11:50 PM EDT: I think we should start off with a conjecture, that we need to prove.
p M M M
Aznx 5/10/06 7:12:03 PM EDT: Not a hard one, but one that can be challenging.
Quicksilver 5/10/06 7:12:17 PM EDT: Maybe a row of blocks
Quicksilver 5/10/06 7:12:27 PM EDT: likethis
Aznx 5/10/06 7:12:44 PM EDT: What about them?
Quicksilver 5/10/06 7:12:59 PM EDT: The amount of sticks may increase in a pattern?
bwang8 5/10/06 7:13:01 PM EDT: The problem from yesterday, but only 3-d

===The problem from yesterday! BUT different.===
Notice no reference to members! (They are stable!) and this is taken by Quicksilver to be a negative assessment, a complaint! BUT ONLY?

Quicksilver 5/10/06 7:13:07 PM EDT: yea i guess
Quicksilver 5/10/06 7:13:09 PM EDT: not that good

He repairs the offer, bwang accepts, he mitigates it, bwant escalates, aznx endorses it, quicksilver mitigates put down:

Quicksilver 5/10/06 7:13:18 PM EDT: maybe a pyramind
bwang8 5/10/06 7:13:24 PM EDT: yeah
Quicksilver 5/10/06 7:13:30 PM EDT: although that's hard to draw
bwang8 5/10/06 7:13:35 PM EDT: pryamind is good
Aznx 5/10/06 7:13:36 PM EDT: Yeah, I liked that.
Quicksilver 5/10/06 7:13:36 PM EDT: but we shoudl be able to managt
Quicksilver 5/10/06 7:13:36 PM EDT: e
(Quicksilver draws a 2d version of a stack of cubes in pyramid shape and marks it as "side view")

Yesterday is still a problem:

bwang8 5/10/06 7:14:56 PM EDT: isn't this the same as yesterday problem
Quicksilver 5/10/06 7:15:03 PM EDT: Really?
Aznx 5/10/06 7:15:10 PM EDT: Except it's 3-D.
Quicksilver 5/10/06 7:15:12 PM EDT: no it's three d

===What can we use that we ALREADY KNOW:===
Interesting:

Aznx 5/10/06 7:16:45 PM EDT: So, how should we approach this?
Aznx 5/10/06 7:16:54 PM EDT: What can we use that we already know?
Quicksilver 5/10/06 7:16:57 PM EDT: Layer by layer shown in a chart?
bwang8 5/10/06 7:17:01 PM EDT: well we can divide it into a front and a back
Aznx 5/10/06 7:17:02 PM EDT: I'd suggest yesterday's problem.
bwang8 5/10/06 7:17:10 PM EDT: yeah
bwang8 5/10/06 7:17:22 PM EDT: using the formula from yesterday's problem
bwang8 5/10/06 7:17:32 PM EDT: we can figure the front and back easily
Quicksilver 5/10/06 7:17:36 PM EDT: this
(Points to the formula from Session I which is still on the whiteboard)

Notice here Projections again, in this case very collaboratively constructed:
"break it down" is something they used in Session I.

bwang8 5/10/06 7:27:55 PM EDT: the last level have 9
Quicksilver 5/10/06 7:28:07 PM EDT: yeah
p M M M M M
bwang8 5/10/06 7:28:28 PM EDT: so we will just have to figure out how many sticks make up 3 by 3 blocks
p M M M M M
Aznx 5/10/06 7:29:06 PM EDT: Yes.
p M
Aznx 5/10/06 7:29:15 PM EDT: After that, we go up to Nth step.
Quicksilver 5/10/06 7:29:20 PM EDT: Yes
p M M M M M M M M M M M
bwang8 5/10/06 7:30:07 PM EDT: ok, how do we figure that out
p M
bwang8 5/10/06 7:30:17 PM EDT: 3*3 blocks
p M
Quicksilver 5/10/06 7:30:26 PM EDT: Break it down
Aznx 5/10/06 7:30:27 PM EDT: I'd say look for a pattern.
p M
Aznx 5/10/06 7:30:33 PM EDT: and yes, break it down.
p M
Aznx 5/10/06 7:30:40 PM EDT: What other possible ways are there?
Aznx 5/10/06 7:30:44 PM EDT: That we know of?
bwang8 5/10/06 7:30:52 PM EDT: top, middle and bottom
p M M M M M M
bwang8 5/10/06 7:31:29 PM EDT: top and bottom are 3 by 3 squares
Quicksilver 5/10/06 7:31:33 PM EDT: whoops i drew it wrong

==="Yesterday stuff:"===

bwang8 5/10/06 7:32:27 PM EDT: do you mind if i erase some yesterday stuff?
Quicksilver 5/10/06 7:32:34 PM EDT: yea u can do it
Diagrams and other materials (e.g. table) are deleted, the formula is kept and then
a "decomposed" diagram with horizontal and vertical sticks for a 3x3 grid is created by bwang (as he did in S1!)
...
bwang8 5/10/06 7:33:57 PM EDT: when they combine
bwang8 5/10/06 7:34:05 PM EDT: they make a 3 by 3 square
Quicksilver 5/10/06 7:34:08 PM EDT: yes
Quicksilver 5/10/06 7:34:14 PM EDT: so just count these
bwang8 5/10/06 7:34:28 PM EDT: and the equation for it is 2N(N+1)
bwang8 5/10/06 7:34:35 PM EDT: right?
p M M
bwang8 5/10/06 7:34:58 PM EDT: N is the level
Quicksilver 5/10/06 7:34:59 PM EDT: I don't know
Aznx 5/10/06 7:35:04 PM EDT: Prove it.
Quicksilver 5/10/06 7:35:11 PM EDT: Where did you get it?
Aznx 5/10/06 7:35:16 PM EDT: I'll help you as you go along.
Aznx 5/10/06 7:35:27 PM EDT: I kind of get it, but not clearly.
bwang8 5/10/06 7:35:27 PM EDT: i mean just from the top and bottom 3 by 3 squares
Aznx 5/10/06 7:35:44 PM EDT: Where did the 2 come from?
bwang8 5/10/06 7:35:54 PM EDT: this is 3(3+1) (reference to the whitebaord)

===Sharedness===

Interesting explanation of bwang8's formula. Is it shared now?
Was it just in bwang's problem space before? Was it in everbody's problem space but positioned differently?
With different meanings?

Bwang had sugested that their pyramid can be broken down into "top, middle and bottom" and that the "top and bottom are 3 by 3 squares" Then bwang draws a "broken down" version of a 3 by 3 square that looks like this on the whiteboard:

_ _ _
_ _ _ | | | |
_ _ _ | | | |
_ _ _ | | | |

Then he follows up with this explanation:

bwang8 5/10/06 7:33:57 PM EDT: when they combine
bwang8 5/10/06 7:34:05 PM EDT: they make a 3 by 3 square
Quicksilver 5/10/06 7:34:08 PM EDT: yes
Quicksilver 5/10/06 7:34:14 PM EDT: so just count these
bwang8 5/10/06 7:34:28 PM EDT: and the equation for it is 2N(N+1)
bwang8 5/10/06 7:34:35 PM EDT: right?
(bwang manipulates some objects on the whiteboard, invisible)
bwang8 5/10/06 7:34:58 PM EDT: N is the level
Quicksilver 5/10/06 7:34:59 PM EDT: I don't know
Aznx 5/10/06 7:35:04 PM EDT: Prove it.
Quicksilver 5/10/06 7:35:11 PM EDT: Where did you get it?
Aznx 5/10/06 7:35:16 PM EDT: I'll help you as you go along.
Aznx 5/10/06 7:35:27 PM EDT: I kind of get it, but not clearly.
bwang8 5/10/06 7:35:27 PM EDT: i mean just from the top and bottom 3 by 3 squares
Aznx 5/10/06 7:35:44 PM EDT: Where did the 2 come from?
bwang8 5/10/06 7:35:54 PM EDT: this is 3(3+1) (Points to the left side of his diagram, the array of horizontal lines)

Claim Markings:

Quicksilver 5/10/06 7:36:05 PM EDT: ok
Aznx 5/10/06 7:36:05 PM EDT: Ah.
Quicksilver 5/10/06 7:36:07 PM EDT: i c
Aznx 5/10/06 7:36:08 PM EDT: I get it.

Bwang aknowledges, moves on:

bwang8 5/10/06 7:36:10 PM EDT: ok
bwang8 5/10/06 7:36:10 PM EDT: ok
bwang8 5/10/06 7:36:32 PM EDT: so now we get the top and bottom
bwang8 5/10/06 7:36:40 PM EDT: we need to find the middle

But this "sharedness" is contingent, not cumulative! For example... in this passage both Quicksilver and bwang create different graphics and at the end, they use it to understand each other's views:

Quicksilver 5/10/06 7:37:14 PM EDT: i don't understand something
Quicksilver 5/10/06 7:37:15 PM EDT:
Quicksilver 5/10/06 7:37:15 PM EDT:
Quicksilver 5/10/06 7:37:15 PM EDT:
Quicksilver 5/10/06 7:37:19 PM EDT: sorry\\
Quicksilver 5/10/06 7:37:28 PM EDT: um...what do you mean it is the top and bottom
bwang8 5/10/06 7:37:37 PM EDT: oh
Quicksilver 5/10/06 7:37:39 PM EDT: it is a pyramid with a flat face right
p M
bwang8 5/10/06 7:37:47 PM EDT: let me explain
(''Quicksilver and Bwang, each, create a drawing. Bwang's is 3D, Quicksilver's 2D with colors'')
Quicksilver 5/10/06 7:38:38 PM EDT: This face could go againts a wall(this is a top view) [''Points to his drawing'']
bwang8 5/10/06 7:38:48 PM EDT: it is just a way to divide the problem
(''Quicksilver colors a part of his diagram in yellow'')
Quicksilver 5/10/06 7:39:05 PM EDT: so the yellow is the top level?
(Quicksilver creates a yellow 3 by 3 grid of squares, Bwang marks lines in his original diagram with scribbles)
bwang8 5/10/06 7:39:45 PM EDT: we are just focusing on the bottommost level
Aznx 5/10/06 7:39:54 PM EDT: Yeah.
bwang8 5/10/06 7:40:04 PM EDT: and all those edges with scribble on it are the top
Quicksilver 5/10/06 7:40:18 PM EDT: So this is against the floor underneath the other two levels [''Points to his yellow 3 by 3 grid'']
bwang8 5/10/06 7:40:35 PM EDT: yes
Quicksilver 5/10/06 7:40:43 PM EDT: ok...continue then

Q: Is "coloring" (like "labeling") a a more "persistent" diectic?
(Note: the use of whiteboard references is very sophisticated here)
How do the 2 diagrams stand in relation to each other, their authors and other participants

bwang8 5/10/06 7:41:02 PM EDT: all the vertical lines are consider the middle [''Points to his diagram'']]
Quicksilver 5/10/06 7:41:21 PM EDT: ok
bwang8 5/10/06 7:41:32 PM EDT: and the rest are the bottom
bwang8 5/10/06 7:41:48 PM EDT: which has the same number of sticks as the top
p M M

===Explaining to others and to oneself===
Also, bwang, who is "explaining" is also monitoring the intelegibility of his own explanation, and complains about it:

bwang8 5/10/06 7:42:51 PM EDT: my explanation is pretty bad
bwang8 5/10/06 7:43:01 PM EDT: do you understand?
Quicksilver 5/10/06 7:43:12 PM EDT: enough to go on with the problem
Aznx 5/10/06 7:43:12 PM EDT: I understand it.

Bwang posts this equation on the whiteboard, not on the chat:

top/bottom: 2n(n+1)
middle: (n+1)^2

==="recursive equation/function" appears===

bwang8 5/10/06 7:46:30 PM EDT: so we can use recursive equation to figure out n 3d pryamind
bwang8 5/10/06 7:46:36 PM EDT: stick number
bwang8 5/10/06 7:46:42 PM EDT: lol
bwang8 5/10/06 7:46:48 PM EDT: bad grammar
(bwang scribles on the whiteboard: Sum (n=1, n) = 4n(n+1) + (n+1)^2

Later he clarifies how is that this is a "recursive function"

Quicksilver 5/10/06 7:50:18 PM EDT: It goes up by one every time because n=1 rite?
bwang8 5/10/06 7:50:38 PM EDT: no, n start at 1
Quicksilver 5/10/06 7:50:46 PM EDT: oh yeah!!!
bwang8 5/10/06 7:50:46 PM EDT: this is a recursive function
Quicksilver 5/10/06 7:50:46 PM EDT:
bwang8 5/10/06 7:51:26 PM EDT: when n=1, plug 1 into the right equation\\
Quicksilver 5/10/06 7:51:40 PM EDT: 12?
bwang8 5/10/06 7:51:58 PM EDT: when n=2, plug 2 into the right equation and add the equation when n=1
Quicksilver 5/10/06 7:52:13 PM EDT: Oh...so u add the thing that came before
bwang8 5/10/06 7:52:17 PM EDT: yes
bwang8 5/10/06 7:52:31 PM EDT: 12 is right (points to Quicksilver's 7:51:40 PM)

===Transition to the Wiki===
What to put, who/how to put it.

Quicksilver 5/10/06 7:57:45 PM EDT: so we have to go to the wiki now
...
Quicksilver 5/10/06 8:01:03 PM EDT: so our equation
Quicksilver 5/10/06 8:01:11 PM EDT: no first lets state our problem
bwang8 5/10/06 8:01:49 PM EDT: The number of sticks to make a N level pryamind
bwang8 5/10/06 8:02:04 PM EDT: ?
Aznx 5/10/06 8:02:17 PM EDT: Yeah, that's our question.
Quicksilver 5/10/06 8:02:56 PM EDT: Err...
Quicksilver 5/10/06 8:03:03 PM EDT: I kind of put somethingt different
Quicksilver 5/10/06 8:03:04 PM EDT:
Quicksilver 5/10/06 8:03:09 PM EDT: But it's similar
Aznx 5/10/06 8:03:24 PM EDT: Wait, do we all have to put it in?
Aznx 5/10/06 8:03:30 PM EDT: Only one of us right?

Today's result was "technically" the same as yesterday but today was "really" a discussion

Quicksilver 5/10/06 8:04:03 PM EDT: you guys can add on
Quicksilver 5/10/06 8:04:08 PM EDT: i just put the basic
Quicksilver 5/10/06 8:04:20 PM EDT: Maybe share our results?
Aznx 5/10/06 8:04:53 PM EDT: We technically had the same result.
Quicksilver 5/10/06 8:05:07 PM EDT: Whaddya mean?
Quicksilver 5/10/06 8:05:21 PM EDT: oh as yesterday?
Aznx 5/10/06 8:05:31 PM EDT: Yeah.
Aznx 5/10/06 8:05:36 PM EDT: And today.
Quicksilver 5/10/06 8:05:40 PM EDT: Still...
Aznx 5/10/06 8:05:43 PM EDT: Well today was really a discussion.
Quicksilver 5/10/06 8:05:46 PM EDT: we should say that'
...
Quicksilver 5/10/06 8:06:39 PM EDT: That may mean these types of problems all are similar in one way

====We should write it out here====

The "natural" person to write their findings on the Wiki might be bwang since he is the one who was driving the process and explaining things to others, but he says he is "bad with words". So Quicksilver suggests that they do it together. In reality, he does it by himself almost, which is a great display of his level of engagement with prior group activity:

Aznx 5/10/06 8:06:55 PM EDT: Wait, who is submitting?
Aznx 5/10/06 8:06:57 PM EDT: bwang?
bwang8 5/10/06 8:06:59 PM EDT: tell them the intervals between levels
Aznx 5/10/06 8:07:06 PM EDT: or quicksilver, or me?
bwang8 5/10/06 8:07:08 PM EDT: sorry, i am bad with words
Quicksilver 5/10/06 8:07:14 PM EDT: So am i
Aznx 5/10/06 8:07:15 PM EDT: Not to worry.
Aznx 5/10/06 8:07:21 PM EDT: We should write it out
Aznx 5/10/06 8:07:22 PM EDT: here
Quicksilver 5/10/06 8:07:24 PM EDT: Aznx to the rescue lol
Aznx 5/10/06 8:07:26 PM EDT: Together
Quicksilver 5/10/06 8:07:30 PM EDT: sure
Aznx 5/10/06 8:07:30 PM EDT: No not that
Quicksilver 5/10/06 8:07:34 PM EDT: a conclusion
Aznx 5/10/06 8:07:34 PM EDT: Yeah lol
bwang8 5/10/06 8:07:36 PM EDT: yeah
Aznx 5/10/06 8:07:51 PM EDT: So first, we started off with the basic problem.
Aznx 5/10/06 8:07:57 PM EDT: and changed it to a 3-D problem
Aznx 5/10/06 8:08:02 PM EDT: to make it more challenging
Quicksilver 5/10/06 8:08:04 PM EDT: Yea
Aznx 5/10/06 8:08:22 PM EDT: then we divided the top, bottom, and middle area down, to make the problem simpler for us
Aznx 5/10/06 8:08:42 PM EDT: Then, we recognized the 2N(N+1) pattern in the middle area
Quicksilver 5/10/06 8:08:44 PM EDT: That was one strategy\\
Aznx 5/10/06 8:08:54 PM EDT: that's our second: finding a pattern
Quicksilver 5/10/06 8:08:55 PM EDT: dividing it
Quicksilver 5/10/06 8:08:59 PM EDT: yes
Aznx 5/10/06 8:09:07 PM EDT: Finally, we used the recursion method to figure out the obttom area,
Quicksilver 5/10/06 8:09:15 PM EDT: obttom?
bwang8 5/10/06 8:09:16 PM EDT:
Aznx 5/10/06 8:09:19 PM EDT: So that's three strategies right there.
Aznx 5/10/06 8:09:32 PM EDT: I meant bottom
...

===retrospective-prospective===
Later, Quicksilver still assigns the task to bwang. Also, here there is a '''retrospective-prospective''' move: The feedback said X, we can do that next time

Aznx 5/10/06 8:10:32 PM EDT: I think bwang should put it in, since he's more familiar with the recursion method and how to use it than we are,
Aznx 5/10/06 8:10:36 PM EDT: Agreed?
Quicksilver 5/10/06 8:10:51 PM EDT: Today's topic siad go to the wiki and share the most intersting math problems that your group chose to work on
Quicksilver 5/10/06 8:10:59 PM EDT: agreed
Quicksilver 5/10/06 8:11:15 PM EDT: but we do understand it now
Quicksilver 5/10/06 8:11:18 PM EDT: that's important
Aznx 5/10/06 8:11:28 PM EDT: Well, we should just say we wanted to explore yesterday's problem more.
Quicksilver 5/10/06 8:11:29 PM EDT: maybe we can apply it next time...who knows?
Aznx 5/10/06 8:11:32 PM EDT: Yes, we do.
bwang8 5/10/06 8:11:32 PM EDT: ok
bwang8 5/10/06 8:12:06 PM EDT: we can use the strategy we used to solve this problem to solve future problems
bwang8 5/10/06 8:12:31 PM EDT: the method is important
bwang8 5/10/06 8:12:36 PM EDT: not the answer
Aznx 5/10/06 8:12:48 PM EDT: Yup.
Quicksilver 5/10/06 8:12:49 PM EDT: definiteyly
Aznx 5/10/06 8:12:56 PM EDT: Always learned that whereever I learned math. =P

Interestingly, here they find it relevant to talk about things that they "learned." How did they do that? For what purpose?

bwang8 5/10/06 8:13:05 PM EDT: we learn that divide the problem up can make it simpler and easier to solve
Quicksilver 5/10/06 8:13:09 PM EDT: so bwang...are you updating the wiki?
Quicksilver 5/10/06 8:13:14 PM EDT: yea
Aznx 5/10/06 8:13:21 PM EDT: we also learned finding a pattern is a good step
Quicksilver 5/10/06 8:13:39 PM EDT: yes and we could have also started with a simpler problem
Quicksilver 5/10/06 8:13:42 PM EDT: in fact...we did
Aznx 5/10/06 8:13:43 PM EDT: and recursion can be usually used when solving for a pattern, after finding the designated pattern of course
Quicksilver 5/10/06 8:13:48 PM EDT: yesterday's problem was simpler
Aznx 5/10/06 8:13:52 PM EDT: yes, we did!
Aznx 5/10/06 8:14:00 PM EDT: so we actually used 4 strategies =D
Quicksilver 5/10/06 8:14:12 PM EDT: yes
Aznx 5/10/06 8:14:34 PM EDT: We also tried to look at the problem from different views, although it's not really a strategy.

The moderator kind of "closes" down the session by bringing up next session:

Gerry 5/10/06 8:15:46 PM EDT: So, can you all come back here at the same time next Tuesday?

===What gets on the Wiki===
An elaborate narrative of what they did (to which they will add after the other 2 sessions)

To investigate the number of sticks in a flat faced pyramid with n levels with 1 block increase in length and width per level.
Also, to find as many approaches and put them to use. We eventually found 4 different strategies and applied them, such as divide
the problem up, finding a basic pattern, and use recursion to solve problems. We also found a formula, its origins, and how to use it.

f(n)=4n(n+1)+(n+1)^2+f(n-1) and f(0)=0.
We first determine the number of squares in each level of the pyramid, 1 cubes in first level, 4 cubes in second level, 9 cubes in
third level, and so on. Then we divide each level into 3 parts, the top, the bottom and the middle. The top is the same as the
bottom part. They are just a bunch of squares in square format. When divided top or bottom into vertical or horizontal, the
equation for # of sticks is n(n+1). Times that by 2 and you get a top or bottom, and times it by 2 again to get the total for top
and bottom. The result is 4n(n+1). Then there is the middle which are straight columns, and they are n+1 by n+1 on the side, so
(n+1)^2. The equation for each level is 4n(n+1)+(n+1)^2, and use a recursive fuction and you can get the total sticks for the pyramid.

==Feedback II==

VMT Feedback on Session #2

You can load the old chat messages by clicking on the double arrow icon above the chat scroll bar.
You can look through the history of the whiteboard by using the scroll bar all the way on the left
(be sure to scroll all the way down to the present in order to draw anything new.)

You did a great job of defining a challenging problem and solving it by using a combination of methods
that were well suited to the problem. And you shared what you did with the other groups in the wiki.

You noticed that stating the problem and making it clear to everyone is a big part of working on a problem.
In going to 3-D, you selected a particular kind of pyramid. How would your problem change if you had two flat sides,
with each layer in a corner of the layer underneath, so that some cube faces and edges (sticks) were shared between layers?

Can you explain your formula for the number of sticks so that someone in a different group can see how you got it
by breaking each layer into its top surface, bottom and middle and then counting the horizontal
and vertical sticks separately?

Do you understand how team C got its formulae for the diamond pattern of squares?
What if they had a diamond pattern of diamonds (just rotate the squares 45 degrees)?

What shapes make mathematically interesting patterns in 2-D or in 3-D?

==Session III==

They take on the feedback mention of a "corner pyramid" but it gest too confusing so they make a bid to postpone it:

Aznx 5/16/06 7:24:08 PM EDT: Should we just go onto a different problem for now?
Quicksilver 5/16/06 7:24:20 PM EDT: This is todays problem isn't it?
Quicksilver 5/16/06 7:24:25 PM EDT: Discuss last week?
Quicksilver 5/16/06 7:24:50 PM EDT: And more problems I guess..
Aznx 5/16/06 7:24:55 PM EDT: Yeah
Aznx 5/16/06 7:25:05 PM EDT: WE can come back to this on Thursday

===Recovering last session's materials: we keep it and move onto a different sequence===

Aznx 5/16/06 7:26:08 PM EDT: This was our formula for this problem last time correct? [''Points to whiteboard's formula'']
bwang8 5/16/06 7:26:13 PM EDT: yeah
Quicksilver 5/16/06 7:26:13 PM EDT: Yes
Aznx 5/16/06 7:26:21 PM EDT: So how about we keep it
Quicksilver 5/16/06 7:26:29 PM EDT: Yes
Aznx 5/16/06 7:26:31 PM EDT: And move onto a different sequence?
p M M M

Who had mentioned squares???

Aznx 5/16/06 7:26:47 PM EDT: How about squares?
Aznx 5/16/06 7:26:52 PM EDT: As you had mentioned?
Quicksilver 5/16/06 7:26:55 PM EDT:
Aznx 5/16/06 7:26:56 PM EDT: bwang you have any ideas?
Quicksilver 5/16/06 7:27:01 PM EDT: These basically were squares
p M M
bwang8 5/16/06 7:27:24 PM EDT: squares of what?
p M M M
Quicksilver 5/16/06 7:27:45 PM EDT: The cubes were basically squares
Aznx 5/16/06 7:28:22 PM EDT: I'm sure we can come up with other sequences.
Quicksilver 5/16/06 7:28:41 PM EDT: How did team C get its formula for the diamonds?

===How do we access that [Team C's] formula===

Quicksilver 5/16/06 7:28:41 PM EDT: How did team C get its formula for the diamonds?
Quicksilver 5/16/06 7:28:46 PM EDT: How do we access that?
Aznx 5/16/06 7:29:00 PM EDT: Well, let's look at their problem.
bwang8 5/16/06 7:29:05 PM EDT: open browser
p M M M
bwang8 5/16/06 7:29:13 PM EDT: and click on the link
Quicksilver 5/16/06 7:29:14 PM EDT: oh nevrmind
p M
Quicksilver 5/16/06 7:29:15 PM EDT: i got it
bwang8 5/16/06 7:29:19 PM EDT: ok
p M M M M
Aznx 5/16/06 7:29:48 PM EDT: Wait, I think I messed up.

===How did they derive it?===

bwang8 5/16/06 7:33:30 PM EDT: how did they derive it
Aznx 5/16/06 7:33:50 PM EDT: There's the formula
bwang8 5/16/06 7:33:57 PM EDT: (n^2+(n-1)^2)*2+n*3-2
bwang8 5/16/06 7:34:08 PM EDT: n^2+(n-1)^2
Aznx 5/16/06 7:34:18 PM EDT: The 3n has to do with the growing outer layer of the pattern I think.

From Team's C Wiki:

We also found formulas for a diamond-like arangement of the squares:
sides:
(n^2+(n-1)^2)*2+n*3-2
squares:
n^2+(n-1)^2

Later on, Quicksilver who might know somebody in Team C (xander?) attempts to ask Team C directly, what they meant by N, through Nan but at some point he withdraws the question, aparently

{| class="wikitable" border="1" cellpadding="2"
|-
|
:bwang8 5/16/06 7:43:33 PM EDT: what is n in their equation
:p M M M M
:Aznx 5/16/06 7:43:56 PM EDT: Good question
:Aznx 5/16/06 7:44:01 PM EDT: I don't know. :P
:p M M
:Quicksilver 5/16/06 7:44:10 PM EDT: Sides, sticks or squares?
:p M
:Aznx 5/16/06 7:44:14 PM EDT: Gerry?
:Quicksilver 5/16/06 7:44:17 PM EDT: let ask xander
:p M M M M M M M M M M M M M M M M M M M M M M
:Gerry 5/16/06 7:45:18 PM EDT: I assume N is the stage in the pattern
:Aznx 5/16/06 7:45:29 PM EDT: What do you mean?
:Quicksilver 5/16/06 7:45:35 PM EDT: That is twelve extra squares in a five by five square
:Aznx 5/16/06 7:45:39 PM EDT: I'm confused how the pattern grows
:Aznx 5/16/06 7:45:50 PM EDT: Oh, I get it now
:Aznx 5/16/06 7:45:56 PM EDT: I was on the right track =P
:Gerry 5/16/06 7:45:59 PM EDT: Just like in the original problem on the Topic
:Gerry 5/16/06 7:47:12 PM EDT: Stage N=1 is one square
:Gerry 5/16/06 7:47:27 PM EDT: Stage N= 2 is a cross of 5 squares
:Quicksilver 5/16/06 7:47:37 PM EDT: Ok
:Gerry 5/16/06 7:48:00 PM EDT: Stage N=3 is the bigger figure with 13 squares
:Quicksilver 5/16/06 7:48:08 PM EDT: Stage N=3 is a cross of 9 squares or 13?
:Aznx 5/16/06 7:48:23 PM EDT: 13
:Aznx 5/16/06 7:48:32 PM EDT: i tihnk
:Quicksilver 5/16/06 7:48:42 PM EDT: Because those extra four are tricky
:Gerry 5/16/06 7:48:46 PM EDT: you could define the sequence either way
:Quicksilver 5/16/06 7:48:54 PM EDT: yeah
:Quicksilver 5/16/06 7:49:01 PM EDT: but it has to be constant
:Gerry 5/16/06 7:49:05 PM EDT: which did they do? Which is more interesting mathematically?
:Aznx 5/16/06 7:49:29 PM EDT: 13
:Aznx 5/16/06 7:49:34 PM EDT: is much more interesting =P
:Quicksilver 5/16/06 7:49:34 PM EDT: it doesn't actually have to be constant but it has to have some kind of patter
:bwang8 5/16/06 7:49:53 PM EDT: ok
:Aznx 5/16/06 7:50:23 PM EDT: Now what do we do?
:Quicksilver 5/16/06 7:50:33 PM EDT: this
:bwang8 5/16/06 7:50:38 PM EDT: (2n-1)^2= the # of squares in the big square
|
: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
|}

===Le'ts explain it together right here===
This is something Aznx did in Session II also when team members were refusing the task of writing on the Wiki.
(Aznx to the rescue)

Aznx 5/16/06 8:03:11 PM EDT: Le'ts explain it together right here

but bwang has to leave and Qs and Az struggle with creating an explanation, so they '''project''' it to next session:

====then lets pick it up next time when bwang can explain it====

Aznx 5/16/06 8:05:21 PM EDT: I'm not sure if we got the first formula though.
Quicksilver 5/16/06 8:05:33 PM EDT: i don't think we did
Aznx 5/16/06 8:06:37 PM EDT: (2n-1)^2=the # of squares in the big square
Aznx 5/16/06 8:06:46 PM EDT: Correct?
Quicksilver 5/16/06 8:06:57 PM EDT: Yes
Quicksilver 5/16/06 8:07:27 PM EDT: But that is not in theri equation
Aznx 5/16/06 8:07:29 PM EDT: So isn't that our first formula right there?
Aznx 5/16/06 8:07:36 PM EDT: Hold on.
Quicksilver 5/16/06 8:07:39 PM EDT: no....we want the sides
Aznx 5/16/06 8:08:14 PM EDT: Read above.
Aznx 5/16/06 8:08:24 PM EDT: bwang kind of explains it.
Aznx 5/16/06 8:08:31 PM EDT: I somehow get it, but can't explain it.
Quicksilver 5/16/06 8:08:59 PM EDT: then lets pick it up next time when bwang can explain it
Aznx 5/16/06 8:09:12 PM EDT: So leave for today?
Quicksilver 5/16/06 8:09:18 PM EDT: i suppose

==Bwang's message in between sessions III and IV==

bwang8 joins the room 5/16/06 8:42:43 PM EDT
p M M M M M M M M M M M M M M M M M
bwang8 5/17/06 12:18:23 AM EDT: Sorry i have to leave early
bwang8 5/17/06 12:18:56 AM EDT: i updated our wiki page to explain the equation from the last session
p M M M
bwang8 5/17/06 12:34:54 AM EDT: i also added some stuff, change anything you want
bwang8 5/17/06 12:39:55 AM EDT: i hope you can read this

==Feedback III==

We started doing the problems you were doing and got very interested in them.
We did not always arrive at the same answers and we weren't always able to decide if we were looking at it
in the same way that you were, but we liked working "with you" in this sense.

From our perspective, the goal has not been for you to answer “our” problems,
but for you to figure out math questions that interest you and to pursue them.
That is why we set the challenge for the iPod as having your group excel at collaboration,
not at finding some answers.

You seem to have pursued some interesting questions and are all contributing and also
making use of each other’s ideas. It is sometimes hard for us to tell what you are writing and thinking.
It seems that there are times when you say you are following each other,
but it is not clear that you are really in agreement or completely understand each other.
You might actually discover some more math if you state things in more detail –
to be completely sure you are in agreement.

For session four, you could revisit a problem you were working on before,
in order to state more clearly for other groups in the wiki: (a) a definition of your problem,
(b) a solution and (c) how you solved the problem. Or you could try a new variation of these pattern problems,
like a 3-D version of group C’s diamond pattern.

It is up to you to pursue whatever most interests you and what enables you to improve and enjoy your ability to work together.
As you know, one hour goes by pretty quickly, so it’s easy to run out of time for a complicated problem.
Be creative and enjoy the session.

==Session IV==

===Recovering their "plan"===

The three members had postponed the "flat sided pyramid" and two of them (Az and Qs) postponed explaining their work (or bwang's formula) on the diamond pattern. They recover the pyramid problem

Quicksilver 5/18/06 7:03:44 PM EDT: for this session, it says to revisit an old problem
p M M
Aznx 5/18/06 7:04:10 PM EDT: Hm.
Aznx 5/18/06 7:04:16 PM EDT: Let's make a new problem
p M
bwang8 5/18/06 7:04:18 PM EDT: the pryamid one that we didn't finish last time
Aznx 5/18/06 7:04:21 PM EDT: Not necessarily 3-D
Aznx 5/18/06 7:04:26 PM EDT: Yeah
Quicksilver 5/18/06 7:04:31 PM EDT: ok
Aznx 5/18/06 7:04:32 PM EDT: let's do the pyramid one
Quicksilver 5/18/06 7:04:35 PM EDT:

After this they start reading the feedback and later on the decision of what to do, resurfaces:

bwang8 5/18/06 7:14:38 PM EDT: let's just focus on diamond problem today
Aznx 5/18/06 7:14:44 PM EDT: So let's really focus on the pyramid.
bwang8 5/18/06 7:14:48 PM EDT: we almost got the solution last seesion
Quicksilver 5/18/06 7:14:49 PM EDT: lol
Aznx 5/18/06 7:14:49 PM EDT: Diamond or pyramid?
bwang8 5/18/06 7:14:55 PM EDT: diamond
Aznx 5/18/06 7:14:59 PM EDT: aditya?
bwang8 5/18/06 7:15:02 PM EDT: because we worked on it longer
Quicksilver 5/18/06 7:15:03 PM EDT: diamond
Aznx 5/18/06 7:15:08 PM EDT: agreed then
bwang8 5/18/06 7:15:11 PM EDT: ok
Quicksilver 5/18/06 7:15:13 PM EDT: we have a more thorough understanding of it
bwang8 5/18/06 7:15:18 PM EDT: yeah

...

Aznx 5/18/06 7:16:03 PM EDT: I'd say, we work on the pyramid problem, solve it thoroughly, and then state the solution as they suggested in the feedback. Then, if we have enough time, which probably will ,we'll sytart on the pyramid problem.
Quicksilver 5/18/06 7:16:21 PM EDT: u said two pyramid problems?
Quicksilver 5/18/06 7:16:27 PM EDT: read ur thing again
Aznx 5/18/06 7:16:27 PM EDT: OOps
Aznx 5/18/06 7:16:34 PM EDT: I meant in the first part
Aznx 5/18/06 7:16:37 PM EDT: the diamond problem
Aznx 5/18/06 7:16:41 PM EDT: not the pyramid
bwang8 5/18/06 7:16:41 PM EDT: lol

===As usual Routine===

Aznx 5/18/06 7:12:00 PM EDT: Let's first discuss about the feedback.
bwang8 5/18/06 7:12:06 PM EDT: ok
Aznx 5/18/06 7:12:06 PM EDT: As usual, what do you guys think?

===So where were we?===

Quicksilver 5/18/06 7:15:57 PM EDT: so where were we?
bwang8 5/18/06 7:16:00 PM EDT: so right now we know that we must calculate the number of squares on each level by making a big square and minus the 4 extra corners

Notice how they orient to process/steps:

Aznx 5/18/06 7:16:57 PM EDT: we pretty much solved it didnt we?
bwang8 5/18/06 7:17:09 PM EDT: yeah
Aznx 5/18/06 7:17:11 PM EDT: Well 50% of it I should say.
Quicksilver 5/18/06 7:17:15 PM EDT: lets just recap the process
Quicksilver 5/18/06 7:17:27 PM EDT: from the point of view who had never seen this problem
bwang8 5/18/06 7:17:32 PM EDT: we know how to calculate the big square in a level
Quicksilver 5/18/06 7:17:44 PM EDT: ok hold on
bwang8 5/18/06 7:17:50 PM EDT: as in this
bwang8 5/18/06 7:17:56 PM EDT: whole thing
Quicksilver 5/18/06 7:17:57 PM EDT: our objective is to find the amount of squares and sticks in each level righrt?
bwang8 5/18/06 7:18:03 PM EDT: yeo

===We had already figured that out/We can use the equation from Session 1===

Pointing to a textbox created in Session 1!!!

bwang8 5/18/06 7:20:48 PM EDT: what is the pattern
Aznx 5/18/06 7:20:56 PM EDT: Triagnular numbers.
Quicksilver 5/18/06 7:20:58 PM EDT: triangular numbers!
bwang8 5/18/06 7:21:00 PM EDT: yep
bwang8 5/18/06 7:21:00 PM EDT: yep
Aznx 5/18/06 7:21:03 PM EDT: We had already figured that out.
bwang8 5/18/06 7:21:10 PM EDT: we can use the equation from session 1
Quicksilver 5/18/06 7:21:11 PM EDT: yes
Aznx 5/18/06 7:21:20 PM EDT: Yup.
bwang8 5/18/06 7:21:36 PM EDT: n(n+1)/2
p M M M
bwang8 5/18/06 7:21:56 PM EDT: 4*n(n+1)/2= the four corners
Quicksilver 5/18/06 7:21:57 PM EDT: this right? ['''Points to a textbox from Session 1 with formula''']
bwang8 5/18/06 7:22:03 PM EDT: yes
Aznx 5/18/06 7:22:06 PM EDT: Yeah
bwang8 5/18/06 7:22:28 PM EDT: (2n-1)^2-2n(n-1)
bwang8 5/18/06 7:22:48 PM EDT: this is the equation for each level

'''Is it here that bwang8 does his subtle transformation of "ns" and changes the equations so that the n in thefour corners from (7:21:56 PM EDT: 4*n(n+1)/2) which is 2n(n+1) matches that in the central square: 2n(n-1)? And integrates it into the complete formula (7:22:28 PM EDT: (2n-1)^2-2n(n-1))

The final textbox and the Wiki report end up with different formulas:
big square: (2n-1)^2
4 corners: n(n+1)/2*4

(2n-1)^2-n(n+1)/2*4

Wiki: (2n+1)^2-n(n+1)/2*4

===But that is not what it ENDS UP TO BE===

Aznx 5/18/06 7:25:43 PM EDT: But that's not what it ends up to be.
Aznx 5/18/06 7:25:56 PM EDT: If you double check with our already-given formula
Quicksilver 5/18/06 7:26:00 PM EDT: why?
Aznx 5/18/06 7:26:07 PM EDT: It's this
Quicksilver 5/18/06 7:26:12 PM EDT: oh yeah
Quicksilver 5/18/06 7:26:14 PM EDT: it doesn't work
p M
Aznx 5/18/06 7:26:16 PM EDT: The first one [''Points to Team C's formula for sticks which is on the whiteboard'']

They clarify which formula is for squares and Aznx accepts that he was confused

===What is the actual solution===

Aznx 5/18/06 7:28:22 PM EDT: So is that all?
Quicksilver 5/18/06 7:28:37 PM EDT: what is the actual solution then? those equations?
Aznx 5/18/06 7:28:43 PM EDT: Yeah.
p M
Quicksilver 5/18/06 7:28:59 PM EDT: but when we put in the wiki how we did it....what will we write
Aznx 5/18/06 7:29:20 PM EDT: Um.

This is what makes it to the Wiki:

In session 4, we continued our progress on the diamond problem. We found that if we filled up the diamound
with more squares and get an easier square with 2n+1 as the dimension. So the number of squares
in the big square is (2n+1)^2. We then minus the squares that we added on which was at the 4 corners,
which grow in the same pattern as the triangle number in the first session. We used the formula for # of squares
from the first session and times it by 4 to calculate the 4 corners that we add on to make the big square.
The final formula for the # of squares in the diamond is (2n+1)^2-n(n+1)/2*4. We tested it several times to check if it works.