On the Shoulders of Giants New Approaches to Numeracy (1990) / Chapter Skim
Currently Skimming:
Dimension
Dimension
Pages 11-60
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From page 11... ...
The Edit meads association of sb~ _ Aces ~ ~ buses shoot 1~1 would become mom bang as quints Tampa nag 5~111S 1~ antis magic and mcasu~m~nl and 1~ in mom ~~ ha and geometry. in off lo peplum 1be im~nalions of gist hug sludenls, F~~1 present 1bem Tab a sequel of Then Sects far lbeir play in the Cbild~n's Ones.
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From page 12... ...
gem ~ stiff h~ blocks far Skins am film far Saline pagans on walers. ~ oPen, bong abed flays a~ IeL babied Ben chid pass in10 1be Odors Todd em abed ^ ~~ maw ^ ~ urn ~ ~~k e=~ ds=: but a student is laud 10 me albino Dimensional glean kindenga ^ n ju~dorb~ghscho |
From page 13... ...
˘~ t~= :~ mth all of:~.mic ~~) wew su~ posed to be ~1 mto ~ -~mgIe geometry -cur- ^M too ohen ~ Did ~~w mmpo~s ~ I; me~ as ~~ taw t~= ~ ~ ~~ -I ~~o h~ ~ bit of 1~ t:~.
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From page 16... ...
Of ~ 9~ ~ ~ ~ > ~ ~ ~ rI~= O tOur nest *
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From page 17... ...
17 ~ ~ ~ as tags of unit tbic~: ~ ~1 did in . ~~ ~_~a Ala by Its ~ a~ minip~c~ Dig Reck.
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From page 18... ...
~ - HO=E 9. ~ -I ~n of - ~s 'three .~ of ~t ~= but the ~e volu.~~ Decomposition modes :hustram deeper ideas than do mmpans-~ns of wIumes Aim.
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From page 19... ...
nGURE 10. =c ~ ~ of ~$ ~1 ~ ~ ~c ~ ~ a Spasm.
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From page 20... ...
2:e =~:~= ~ ~~y ~~:s rea~ for the Die techniques :~d ~r a~d ~~mat:~. We sh—d be ~~ ~ c=~ ~r -weir =~—pwp~tion wells ˘yramm r[ M:~v chdd~ ~e Acid by the ~ ~~ ~ ~~+ Th=e on~ summing wonder of t5e anment wona wew m~ ~ images ~ t60= 6~7 =6 ~ ~= I t~.
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From page 22... ...
th=e such boxes and st:~:~:~hew some water left overY. E;~enmen:ts mth dissent cvI~rs a~d related boxes mil quicMv sh~Y that th:s pattem wo~s ~r cviinders of a~v rad:~s or Theists The same mt:~, of cour=' re1~s the area of ~ circle to :ts I ~~.
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From page 23... ...
(~ucb filer students gill app=ci~ 1be limit comae hidden in 1bis demons~1ion.) s Un~una~y, abed scams lo be no sag nice ~~sp~den~ Cheer tube volume of ~ saber gad the volume .
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From page 24... ...
ED ~~ Has 51 Ever 10 ~~ a 1~~ ^~, ~ as long lace ~ aide, and Lice as him. , , Cur squaw tiles Farber ~ ~~ ~ lam Quaff ~~ as long Ad 1~= as ~de.
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From page 25... ...
A surmising ~~ is 1~= 1 are ~omel~c owns Abe -by egg gals ~ ~1 abbe Age. ague Me gists, gig bag a And ~ ~< .
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From page 26... ...
has ~ ~1 dIm=~n CAM - ~~n one and ALLA r the I~thm to ~e two. Ether ~~ p-~s Iead to 6~s Il~ ~e Mandelbrot set~ incIu~g some Of the :~t Stn~.~g eXa~S of ma~emadc~ an.5 O~ of the ~~ imp; s:kil:~:s w~e =n ~~e our ~~s :s the Ails id to- i; data ~:met:~+ The geometry of area =d volume can help students undemand concepts like rams~ accumulat:~3, and avem~ value.
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From page 27... ...
as too 27 ha_ ~~o BIKE 19. A ~r Mob elms art ~ ~ Id SEWS ~ ~~ Ages _ ~~ - ~~ He heifer of ~ sat ~~e : Abe amp ~= a~ Me me -~ A Isis m $40000 ~ ~ far ~1 year Id gem $46~000 Me mat 2 a.
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From page 28... ...
~~ :~o N~y bar ~ r:~.~Mi=: of rates ( - it :~s ~ nep fun=~) Iea~ to ~ a=~:~ation ~ph ~~—m straight 15~= (~' ~ ~~: ~~)
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From page 29... ...
FICUR~ 24 - ~O v) ~ of ~ cube ln general posliŁ~x :~ gmpt:C (~ Also i05~)
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From page 30... ...
~ of ~ f our~} mens) Onal cube ln sc: en:~e 5=i on or ~~Y ilteratu re ~ such as Robe~ HeinIcin,~ stow ~ ~ ~and He Bu'6 ~ ~~—uSet0 or Madeleine L,509s ~ ~~e in nme:~4 or Even Abbot!
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From page 31... ...
~c ~m~cd Bum ted by castling I vents 1~ ~~= of ~ cut. ~ ~ .~ immune 3$ ~ cut Rabin ~ cut
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From page 32... ...
~2 EN be, To ~ , a C== o~.e -Gus m~ an ~~ I ~~ ~ SUi~C '~ - 3$ t50 Cow.
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From page 33... ...
~~r the =~e notions =n be =6d ~: SC~CS Wit ~~-\r~' flit tC=~C, whem the Vertical Onenm~iOn o~f Me ~~= emp~s ~ dimcti~iw. "~ bend—-en the tem~re ~s ~om 65 d~ to 40 de~ 2~- "~t ~ down -~v 25 .~- such ob~s c~ ta~ place tar ln adva.~-~ of fig s0~d :~.
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From page 34... ...
S 8~ i: kept t5~t 3~S - ~: and ow~ again in mat;~ema~;~ as well as ~~. As students b~e mme Hi I :n ~e kinds of numbed they use~ they =n Ire Actions or decimals into n~r lines =d nu - ~r mwIes~ ~~;g telep bo nc ~~e alO ng ~ ma ~ in ~ Aim: a=a Mali r~ 8' d: ~~t kind ~ ~ ~16~' using ~~S or real nu—e~ ~~nting a=~: distanced e numbers become more compli=~- tut the pro=~s rema:n the .
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From page 35... ...
Day 35 into two- p=~ o=~ the p:~e book or ~ picking ~ number. In wu =~W Or ~~s wish ^~t ~u want and ma~ ~ :~ ~~ W3t :$ in whichevet p8~ (~ 0r b~ow -~ur firm gue~)
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From page 36... ...
t.m mtt simple addition (~ Wh:~ ~~\r~:~; dCpCD~ OD th~ throw Of ~ bait ~ (LlC0) 8~d 13107 in mom =~pli=~ed ~es whem ~e mere can ~ In: ~ Vega;.
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From page 37... ...
D= If the Am: of ~ vllI~ is h=~g on ~ a, we :~n up the natum!
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From page 38... ...
:: a: I: i: ~ em~Y ~' wh~ Int - ~:s =~m t~ ~= :~t ~ ~ ~ when a:~t is ~d o~ ~ ie~ stde ~ ~ mmou~ s=~" It ~ - ~ the: ~::~ helm on me ~t ~* Thls Is: i: lo Ale::: ~ Me ratio dI~ w~ :~ ~ - !
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From page 39... ...
DMIE~ an - ~:n number of ~~. An I: ~~t is idea ~r Bins track.
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From page 40... ...
~ sound topic 1b~ Kneels= in a gem nice ~ is 1be Ion am. if ~ think of 1bis tb~eo~=m as a way of ~lcul~ing lbe lamb of be diurnal ~ a =~e bulb Men sides, then the extension 10 age dim~n~ons is im~mediale: Even a Slid bound by Granular sides, ~ f~ Bay the teem ~ one sue arid 1bon aDDlv il 10 ~ =~an~e ~~ ~ ~ ~ a.
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From page 41... ...
. Ibis makesi1 easy fbrtbe director 10 specify the 10cation ofanylighl, Lace she can indicaletbe position ofibc center oftbe disc using 1be same coordinates 1be1 The director of the play uses 10 give her ~ins1~ions Thai uses loo Ingress bus 1be Wing director needs a~nolber nu mber 10 represent lobe radius of tb~e all=.
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From page 42... ...
- - ~ - :~ ~~E 33* ~ ~~t w,~ ~ Ode ~ ~ ~ 45 -- ~- ~ ~1 t0~.m ~ ~t ~ ~e h~ Of ~ ~ - ~ =~s eq~s ~e b~ ~ ~˘ I~ AYE t~ ~beim t~ 10~ of t~ ~~r and ~ ~4 m:~ Be: - ins (~r O~ Skip ~, ~e :~In ~~s ~ ~ ~ ~~t ~ ~~.~n of ~~ in ~ pla~ is I
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From page 43... ...
~sw.x ~3~= or Iater eve~e hears thM time is the :~h d;rme~. That idea, however, Il.mitS the Am' of I ~ Alr=~v in the Iast centu~, ~~= realize that there a~re ma~ s:~ns in wh~ time can be viewed as ~ mash ~i=~$iO~' 6~t ~V ~O =~S 600-S it 60~6 anv special rOle as ~e ~~h dimension.
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From page 44... ...
~4 ^~= do. ~ = mitt ~ :~.~.~.~.
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From page 45... ...
ltC St8t() Sp8~6 iti; I: t~i=S0~S5;~a}~ 8~6 ~ paint =~:~g 80 S0~i~: t~ ~ 'Is physi(~7` i=~- iil(0 ~ 53~} tt055i~g ~~p Char I WAS ~ ~~:~r~ ~~ii describe cu - e in th~ state space Similarly ~ point mos ::~g Ash ~ ci=3sC~ 11~-C
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From page 46... ...
F1OURE 35. F^bc~s kindest included basic cabals shot could ~ bung Ham Lyle at dim ~iliOD.S~ 1ben view ^ Siren per$~ctive$ to see vadous c~~ctio~1 ~apes.
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From page 47... ...
~~ the ~~ m:~n of Cros~tional silces~ One w~ to vi~e ~~s wi~:~t a:~v ap~g ~ kn:i~ to ~ mal: model: {s to l.~:e w~t wou~ hap 90n :f ~'e Ah =bme~ed the 51~cy :n waw. H~ ~fYiii the ~~ of the water Iced cha~9 l.~e ex.~.~e 1,~t, is A..
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From page 48... ...
48 Mar ~.~O,~= ~ N~Y .
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From page 49... ...
~< ~5 39 ^~Ł~8 BY ~ try \50 ~~— Beg any / ~Ye=~n of the a=~s ~~ I'm ap~.~t i=~h ~ ~ the: tI~= wI~:~t ch~ngl~g ~~r a=~: :~h ~-e inv~tiw~on -of Wick O pOl750~ O~-~tS l=6S tO ~ l~g uuzZIe. If wee ~.~e ~ manned.= did ~ ~ '~e o~!
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From page 50... ...
appmm~e t~ power of sIt=ng technics=— ~r ex:~' tn relatIng: Me volume of ~ su - ~e of mvolut:~n to the chan~g amas of bits circular cm~ ~~s or in hi ~e contour lineS on the seduce Of 3~ ~h ~ ~space. ~ng bets-= ~~n s are 1.~d to t~ notions o~f cntI~ poi:~:t th~' they =n ~~V nd and apprecIate s11~ phenomena t~t relate dI~nt dimensions~ What ha~s If Ate brie ~ ~~.~t 0: ~ band in different tions7 It ~s =~v to =~: o~t ~e actu~ expenments and see that the =e positions—~e the slice Melds ~ pair of Circles.
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From page 51... ...
W: =ample, the number of ewes fo~d Alit six points is ~ ~ ~ ~2 ~ Ax. ~ ~ ~ ~^ ~ 5- some students maY know t6e ~~:~a n;n ~ iffy tor t~e sum o: the 6~t ~ integers, pertaps ~n co~on with the famous story o; :~e Amp Gauss wh~o us ~ thi COOL up all the n~ mom ~ to :~O AnOther t~e Of patte:m :~s ~~ed by the table—that the number of ~~es at an~v stage is t~0 ~~ 0: the :~s number of ed~s a~ ~~ Awry In, - ,,
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From page 52... ...
To 611 in 1be missing value he In Mason mom powers, maw of Ibid am just like those 1b~ relay ewes lo points. Since abed bare as maw leant ~ abed am d~i~incl 1dples of bedims, 1be 1otal Ember of trances is jug tbc combinations of a curtain neuter of omens taken 1b~ al a time.
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From page 53... ...
. ^ 1 ne tomb ~> for examples glans ln SuC~SSlOn 10{ 1 1 1 ~ 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 ~ 1 4 1bc num~= of -~1s ~ilb ~ ~i~ ~ ~m 1he ~ur ~inl~ d~s.
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From page 54... ...
54 an/ .# ~< ~ ~ : , ~ ., __ ~ ~ ~/ FIGURE Ha. Sb~i~ beeps idiom ~ by gum ~ ~~ 1~1 flu_ in ~c ups.
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From page 55... ...
to .. C; :::::::::::::: ~ : HG ~E, 4~,5 ~ ~p ~ f~t Irk, ~= \n ;~ ~ detc~.~.~ ~v the -~.~-t d.~.~03~t of ~e In; t DISK ˘~= ~5 3~6 ~t ~ ~ - ~n ~ :~= am Lemon:` as In th~ IC ~r 68~.
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From page 56... ...
56 I:: ~ art The cmcm! - I: ~~ ~ - ~~ is ~M it is sO Memo that every po:~ :~ Every - er ~~: fir ~ ~~w 6~ns ~ one vmex' ~ Ken Twin harems By: ~~ ~$.
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From page 58... ...
BE gasps Be mod Gail obse~1= 1b~ junior lids ~~ is tag we may divide Be sides of an -I into tab emus pans Abe Regions divide tam- cud into 3' s~1 cubes (ague 46~. He ~s~1 is ~ smog cub Mung ~= c~ gem of the of cub> one gem ebb edge one gem _ ~~-nsion~ ~~> and ~ on.
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From page 59... ...
^~ ^~S I ~ ~^ ~~ ma, #: ~~, 1911r 16 ~> Pence and Panic, Ads. ~~ ^~^ Peg Allo, CA: Dan gout ~i~` 1978.
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