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American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit
Section 3: MathematicsRevised Graphic Proofs 03.08.13
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic ElevationNot to Scale
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: MathematicsFinal Size: 35.4375” x 78”
As H
elen
Kel
ler’
s ed
ucat
ion
prog
ress
ed, s
he
mas
tere
d an
evo
lvin
g ar
ray
of c
ount
ing
and
calc
ulat
ing
tool
s de
velo
ped
for
blin
d le
arne
rs.
At th
e Pe
rkin
s In
stitu
tion,
she
dis
cove
red
“mod
ern”
mat
h ai
ds s
uch
as th
e Ta
ylor
sla
te
and
the
conc
ept o
f “m
enta
l ari
thm
etic
.”
To p
repa
re fo
r co
llege
, she
spe
nt m
ost o
f 18
97 s
tudy
ing
alge
bra,
geo
met
ry, a
nd s
cien
ce
at th
e Ca
mbr
idge
Sch
ool f
or Y
oung
Lad
ies.
Sh
e us
ed a
bra
illew
rite
r to
rec
ord
the
step
s
of h
er w
ork,
and
she
cre
ated
geo
met
ry
diag
ram
s us
ing
wir
es a
nd p
ins
on a
pad
ded
wor
k bo
ard.
Kelle
r’s
qual
ifyin
g ex
ams
for
adm
issi
on to
Ra
dclif
fe in
189
9 in
clud
ed g
eom
etry
and
al
gebr
a te
sts
in M
odifi
ed A
mer
ican
Bra
ille
— a
syst
em s
he h
ad le
arne
d on
ly th
e da
y be
fore
! Al
thou
gh h
er e
ffort
s w
ere
succ
essf
ul, s
he
ackn
owle
dged
that
ari
thm
etic
and
geo
met
ry
wer
e no
t her
favo
rite
sub
ject
s!
Mat
hem
atic
s
“Mis
s Su
lliva
n tr
ied
to t
each
me
to c
oun
t
by s
trin
ging
bea
ds in
gro
ups,
and
by
arra
ngin
g ki
nder
gart
en s
traw
s
I lea
rned
to
add
and
sub
trac
t.”
— H
elen
Kel
ler,
1903 Ph
oto
cour
tesy
Am
eric
an F
ound
atio
n fo
r th
e Bl
ind
MA
.01
– In
tro
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: MathematicsFinal Size: 35.4375” x 78”
Illus
trat
ion
of S
aund
erso
n’s
Boar
d &
Key
dev
ice
Saun
ders
on’s
Pal
pabl
e Ar
ithm
etic
dev
ice
was
abo
ut o
ne s
quar
e fo
ot,
divi
ded
into
sm
alle
r squ
ares
. Eac
h sq
uare
was
div
ided
into
four
eq
ual p
arts
by
perp
endi
cula
r lin
es. P
ins
inse
rted
in th
e sm
all s
quar
es
repr
esen
ted
num
bers
. Stu
dent
s w
orke
d th
eir m
ath
prob
lem
s on
the
boar
d in
the
sam
e w
ay n
umer
als
are
arra
nged
on
pape
r.
“Pal
pabl
e Ar
ithm
etic
”N
icho
las
Saun
ders
on, a
mat
hem
atic
ian
who
lost
his
sig
ht a
s a
boy,
cr
eate
d th
e fi r
st k
now
n ca
lcul
atin
g de
vice
for
peop
le w
ho w
ere
blin
d. H
e ca
lled
his
met
hod
“Pal
pabl
e Ar
ithm
etic
.” P
alpa
ble
mea
ns
“tan
gibl
e” o
r “c
apab
le o
f bei
ng to
uche
d or
felt.
”
Alth
ough
he
neve
r at
tend
ed c
olle
ge h
imse
lf, S
aund
erso
n ta
ught
at
Cam
brid
ge U
nive
rsity
in E
ngla
nd, e
arni
ng th
e di
stin
guis
hed
posi
tion
of L
ucas
ian
Prof
esso
r of
Mat
hem
atic
s.
Befo
re m
athe
mat
ical
aid
s w
ere
inve
nted
, peo
ple
who
wer
e bl
ind
used
sm
all,
com
mon
obj
ects
– s
hells
, peb
bles
, or
stic
ks –
for
coun
ting
and
calc
ulat
ing.
Hum
an fi
nger
s, h
owev
er, h
ave
alw
ays
been
our
mos
t con
veni
ent c
ount
ing
tool
s, fo
r bl
ind
and
sigh
ted
lear
ners
alik
e! C
ount
ing
on te
n fi n
gers
led
to th
e de
velo
pmen
t of
the
deci
mal
sys
tem
(bas
ed o
n th
e nu
mbe
r 10
) and
the
coun
ting
tool
cal
led
the
abac
us.
Co
un
tin
g
Nic
hola
s Sa
unde
rson
(168
2-17
39)
MA_
A01
(Tex
as S
late
)
Texa
s Sl
ate
By
turn
ing
the
tact
ile ty
pes
in th
e sl
ate’
s oc
tago
nal
hole
s, u
p to
32
diffe
rent
mat
h sy
mbo
ls c
ould
be
repr
e-se
nted
.
MA
.02
– In
ter
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: MathematicsFinal Size: 35.4375” x 78”
Late
r m
athe
mat
ics
devi
ces
calle
d sl
ates
us
ed p
egs
rath
er th
an p
ins.
Inve
ntor
s fo
cuse
d on
rep
rese
ntin
g nu
mbe
rs in
the
sim
ples
t way
w
ith th
e fe
wes
t peg
s.Ari
thm
etic
Sla
tes
The
Tayl
or
Slat
e
The
Reve
rend
Will
iam
Tay
lor
intr
oduc
ed h
is a
rith
met
ic b
oard
in
the
early
185
0s. W
ith th
is s
late
, stu
dent
s co
uld
mak
e 32
diff
eren
t sy
mbo
ls, i
nclu
ding
sig
ns, l
ette
rs, a
nd n
umbe
rs. T
o so
lve
a pr
oble
m,
user
s po
sitio
ned
type
in o
ctag
onal
cel
ls. T
he s
ymbo
l’s m
eani
ng w
as
dete
rmin
ed b
y th
e or
ient
atio
n of
the
type
. The
Tay
lor
Slat
e w
as th
e fi r
st a
rith
met
ic d
evic
e to
gai
n w
ides
prea
d ap
prov
al.
The
Texa
s Sl
ate
The
Texa
s Sl
ate
had
squa
re c
ells
and
use
d ty
pes
with
Ar
abic
num
eral
s si
mila
r to
thos
e us
ed in
pri
ntin
g. T
he
Texa
s sl
ate
fam
iliar
ized
stu
dent
s w
ho w
ere
blin
d w
ith
the
shap
es o
f Ara
bic
num
eral
s.
The
Pen
tag
on
al S
late
In 1
829,
Sco
tsm
an W
illia
m L
ang
of G
lasg
ow d
evel
oped
a s
late
w
ith a
pen
tago
nal p
eg a
nd h
ole
that
allo
wed
for
form
atio
n of
all
ten
fi gur
es. T
he A
mer
ican
Pri
ntin
g H
ouse
pro
duce
d an
upd
ated
ve
rsio
n of
the
Lang
des
ign
in 19
32.
Ari
thm
etic
Sla
te
Arith
met
ic S
late
, “Bo
oks
for t
he B
lind,
” Ca
ssel
l’s Il
lust
rate
d Fa
mily
Pap
er, 1
860
The
Tayl
or S
late
Key W
illia
m L
ang’
s pe
ntag
onal
sla
te
Gir
l wit
h Te
xas
Slat
e, N
ew Y
ork
Inst
itut
ion
for
the
Educ
atio
n of
the
Blin
d, 1
939
MA
.03
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: MathematicsFinal Size: 35.4375” x 78”
Cu
bar
ith
m
The
cuba
rith
m, a
bra
ille
mat
h sl
ate
from
Par
is, F
ranc
e us
ed a
pla
stic
gri
d to
hol
d a
set o
f sm
all c
ubes
. Bra
ille
sym
bols
mar
ked
the
six
face
s of
eac
h cu
be, a
nd n
umer
als
wer
e fo
rmed
by
rota
ting
the
cube
s.
Ho
ff A
id
In 19
46, P
aul H
off,
a te
ache
r at
the
Min
neso
ta B
raill
e an
d Si
ght S
avin
g Sc
hool
, pat
ente
d an
upw
ard-
wri
ting
brai
lle
slat
e fo
r w
ritin
g m
ath
prob
lem
s. T
he H
off A
id p
ocke
t de
vice
wro
te b
raill
e fr
om le
ft to
rig
ht, l
ike
a br
aille
wri
ter,
rath
er th
an fr
om r
ight
to le
ft lik
e ot
her
mat
h sl
ates
.
Pate
nt il
lust
rati
on fo
r a
Brai
lle w
riti
ng d
evic
e de
sign
ed b
y Pa
ul W
. Hof
f
The
Cyp
her
ing
Bo
ard
As b
raill
e be
cam
e m
ore
wid
ely
used
, dev
ices
usi
ng b
raill
e co
de r
epla
ced
othe
r de
vice
s w
ith th
eir
own
uniq
ue s
ymbo
ls. I
n hi
s 18
67 b
ook,
Bra
ille
Syst
em fo
r the
Blin
d,
Hen
ry R
obyn
of t
he M
isso
uri S
choo
l for
the
Blin
d pi
ctur
ed a
n “A
ppar
atus
for
Wri
ting,
Cy
pher
ing,
and
Mus
ic.”
“Cy
pher
ing”
(cip
heri
ng) m
eans
cal
cula
ting
or fi
guri
ng.
Usi
ng a
Hof
f Aid
A T
aylo
r Sl
ate
By tu
rnin
g th
e ta
ctile
type
s in
the
slat
e’s
octa
gona
l hol
es,
up to
32
diffe
rent
mat
h sy
mbo
ls c
ould
be
repr
esen
ted.
A
cub
arit
hm b
raill
e sl
ate
A Ta
ylor
Sla
teA
cuba
rithm
bra
ille
slat
e
MA_
A04
(Cub
arith
m)
MA_
A02
(Tay
lor
Slat
e)
MA
.04
– Fl
ip/In
ter
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: MathematicsFinal Size: 35.4375” x 78”
Men
tal A
rith
met
ic
Blin
d st
uden
ts w
ere
also
taug
ht to
so
lve
prob
lem
s m
enta
lly, o
r “i
n th
eir
head
s,”
befo
re th
ere
was
a s
ubst
itute
for
penc
il an
d pa
per
to r
ecor
d th
em. A
rith
met
ic b
ooks
w
ere
mad
e up
of w
ord
prob
lem
s.
Abra
ham
Nem
eth,
a p
rofe
ssor
at t
he U
nive
rsity
of
Det
roit
who
was
blin
d, c
reat
ed a
bra
ille
code
for u
se
in a
dvan
ced
mat
hem
atic
s. In
195
2, h
is in
vent
ion
was
acc
epte
d as
the
nat
iona
l sta
ndar
d br
aille
m
athe
mat
ics
code
. Alth
ough
Nem
eth
Code
use
s th
e sa
me
brai
lle s
ymbo
ls a
s lit
erar
y br
aille
, the
y no
w
expr
ess
mat
hem
atic
al a
nd s
cien
tifi c
mea
ning
s.
The
Nem
eth
Code
rep
rese
nts
Arab
ic n
umer
als
by
drop
ping
bra
ille
sym
bols
into
the
low
er tw
o-th
irds
of th
e br
aille
cel
l. Be
caus
e of
the
posi
tion
of th
e nu
mbe
rs, t
he b
raill
e nu
mbe
r si
gn is
not
alw
ays
need
ed to
dis
tingu
ish
num
bers
from
lette
rs. B
raill
e sy
mbo
ls r
epre
sent
mat
hem
atic
al o
pera
tions
suc
h as
plu
s, m
inus
, mul
tiply
, and
div
ide.
“…[W
]e g
ive
our
pupi
ls c
onsi
dera
ble
drill
in m
enta
l or
al w
ork.
The
cla
ss is
cal
led
to a
tten
tion
and
the
wor
k is
co
nduc
ted
in m
ilita
ry fa
shio
n. T
he p
upils
, if s
itti
ng, t
ake
erec
t pos
itio
ns a
nd w
hen
stan
ding
, hav
e bo
th fe
et o
n th
e fl o
or a
nd h
ands
by
thei
r si
des.
It is
not
icea
ble
that
whe
n a
girl
beg
ins
to fi
nger
her
cur
ls a
nd fi
x th
e bo
w o
n he
r ha
ir
that
her
min
d is
ther
e al
so. W
hat w
e ai
m fo
r is
per
fect
po
ise
wit
h th
e m
ind
who
lly c
ente
red
on th
e w
ork.”
—
Mis
s M
ary
B. S
choo
nmak
er, T
each
er
New
Yor
k In
stitu
te fo
r the
Blin
d, 19
20
Page
Rep
rodu
ctio
n fr
om T
he M
enta
l Ari
thm
etic
of W
arre
n Co
lbur
n
Page
Rep
rodu
ctio
n fro
m T
he M
enta
l Ari
thm
etic
of W
arre
n Co
lbur
n,
Amer
ican
Prin
ting
Hou
se fo
r the
Blin
d, 18
74
MA_
A05
(Men
tal A
rith
met
ic)
Nem
eth
key
and
prob
lem
usi
ng T
ACK
-TIL
ES®
Use
thes
e TA
CK-T
ILES
® to
dis
cove
r a m
ath
prob
lem
writ
ten
in th
e N
emet
h Br
aille
Cod
e. T
ACK-
TILE
S® a
re m
ovab
le ti
les
desi
gned
to h
elp
stud
ents
lear
n br
aille
.
TACK
TIL
ES® c
ourt
esy
of T
ACK
TILE
S® B
raill
e Sy
stem
s
MA_
A06
(Tac
k-til
es)
Abr
aham
Nem
eth,
Ph.
D. (
born
191
8)
The
Nem
eth
Bra
ille
Co
de
Nem
eth
key
and
prob
lem
usi
ng T
ack-
tiles
Use
thes
e TA
CK
-TIL
ES®
to d
isco
ver a
mat
h pr
oble
m
writ
ten
in th
e N
emet
h B
raill
e C
ode.
TA
CK
-TIL
ES®
are
mov
able
tile
s de
sign
ed to
hel
p st
uden
ts le
arn
brai
lle.
TAC
K T
ILES
® c
ourt
esy
of T
AC
K T
ILES
® B
raill
e Sy
s-te
ms
Page
Rep
rodu
ctio
n fr
om T
he M
enta
l Arit
hmet
ic o
f War
ren
Col
burn
Page
Rep
rodu
ctio
n fr
om T
he M
enta
l Arit
hmet
ic o
f War
ren
Col
burn
, A
mer
ican
Prin
ting
Hou
se fo
r the
Blin
d, 1
874
RW.0
5
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: MathematicsFinal Size: 35.4375” x 78”
Pla
ne
Geo
met
ry
In 1
846,
John
Als
ton
desc
ribe
d m
akin
g a
tact
ile g
eom
etry
di
agra
m “
by fi
a th
ick
thre
ad u
pon
the
pape
r by
m
eans
of g
lue,
with
spl
it pe
as a
t the
ang
les
to r
epre
sent
th
e le
tters
.” T
acks
and
str
ing
on a
boa
rd w
ere
also
use
d to
ill
ustr
ate
geom
etri
c fi g
ures
. Des
pite
occ
asio
nal p
in-p
rick
ed
fi nge
rs, p
ins
and
stri
ngbo
ards
are
stil
l use
d to
day
to
teac
h pl
ane
geom
etry
.
Soli
d G
eom
etry
Solid
geo
met
ry m
odel
s fo
r te
achi
ng h
ave
been
pro
duce
d si
nce
the
mid
-1800
s. G
eom
etri
c fi g
ures
em
boss
ed o
n pa
per
wer
e al
so u
sefu
l to
geom
etry
stu
dent
s.
The
Ab
acas
Des
pite
em
phas
is o
n “m
enta
l ari
thm
etic
” an
d m
athe
mat
ics
slat
es,
blin
d st
uden
ts h
ad n
o m
etho
d co
mpa
rabl
e to
sig
hted
stu
dent
s’
penc
ils a
nd p
aper
unt
il th
e in
trod
uctio
n of
the
Cran
mer
Aba
cus
in
the
1960
s. A
lthou
gh n
ot m
ade
prim
arily
for
use
by b
lind
stud
ents
, th
e ab
acus
pro
vide
d th
em w
ith a
han
ds-o
n w
ay to
cal
cula
te w
ith
spee
d an
d ac
cura
cy.
An a
ncie
nt to
ol, t
he a
bacu
s fi r
st a
ppea
red
arou
nd
800
year
s ag
o in
Chi
na. D
r. Te
rran
ce V
. Cra
nmer
, a
blin
d m
an, a
dapt
ed a
Japa
nese
aba
cus
so th
at
the
bead
s w
ould
sta
y in
pla
ce a
nd n
ot m
ove
acci
dent
ally
. The
Cra
nmer
Aba
cus
was
fi rs
t pr
oduc
ed b
y th
e Am
eric
an P
rint
ing
Hou
se fo
r th
e Bl
ind
in 19
63.
Geo
met
ric
Mod
el
This
woo
den
sphe
re is
cut
into
sec
tions
atta
ched
by
leat
her
hing
es. E
ach
sect
ion
can
be o
pene
d. T
his
mod
el, p
roba
bly
mad
e at
Cor
nell
Uni
vers
ity, w
as re
fere
nced
in th
e 18
72 a
nnua
l re
port
for t
he K
entu
cky
Scho
ol fo
r the
Blin
d.
Wor
king
wit
h an
aba
cus
Dr.
Ter
ranc
e Cr
anm
er
Geo
met
ry
MA_
A07
Tabl
etop
(Pin
& R
ubbe
r Ba
nd G
eom
etry
)M
A_A0
8 Ta
blet
op(M
itche
ll W
ire
Form
s)
MA_
A09
Tabl
etop
(Cra
nmer
Aba
cus)
MA_
A11
Tabl
etop
(Tal
king
Cal
cula
tor)
MA
.06
– Ta
ble/
Inte
r
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Encased Object LabelFinal Size: 8” x 3”
Texas SlateThe Texas Slate had square-shaped cells and used types with Arabic numerals.
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Tabletop Object LabelFinal Size: 6” x 3”
Cranmer AbacusUse the abacus to add and subtract.
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Tabletop Object LabelFinal Size: 6” x 3”
Mitchell Wire FormsMitchell Wire Forms with Matched Planes and Volumes, American Printing House for the Blind, 1951-1990
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Tabletop Object LabelFinal Size: 6” x 3”
Rubber Bands and PinsTry your hand at constructing a geometric fi gure. Take a rubber band from the bin and place it over the stationary pins on the board. What shape have you made?
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Tabletop Instructional/Information PanelFinal Size: 7” x 14”
Beads are stationary to illustrate the correct positions. The columns of beads from right to left represent ones, tens, hundreds, and so on. The single bead above the bar represents 5 units, and the 4 beads below the bar represent single units. Beads are in use only when set against the center bar.
Now, follow the steps to do a calculation on the abacus. Your fi rst problem: 274+91 = 365.
Abacus Calculations: Step by Step
Step 1: Set 2 beads in the hundreds column for 200.
Step 2: Set 2 beads in the tens column plus the 5 bead from the tens column to make 270.
Step 3: Set 4 beads from the ones column to make 274.
Step 4: To add 91 to 274, fi rst add in the 90. Since there are not enough unused beads left in the tens column, you must add 1 bead from the hundreds column.
Step 5: Then, subtract the extra 10 to make a total addition of only 90 (by removing one bead from the tens column).
Step 6: Finally, add in the 1 by adding the 5 beads in the ones column and subtracting the 4 single units in that column. This leaves the correct answer of 365.
American Printing House for the Blind
“Child in a Strange Country” Traveling Exhibit Concept
Schultz
11-261 / Revisions 13-131
03-08-13
—
Mathematics Revised Graphic Proof Package
11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Tabletop Information Panel and Object LabelFinal Size: 8” x 7”
Although the Cranmer abacus is still used, technology also plays a major role in helping blind or visually impaired students learn mathematics today. For example, talking calculators provide answers to calculations at the push of a button.
Talking Calculators
Orion TI-36X Talking Scientifi c CalculatorThe Orion TI-36X Talking Scientifi c Calculator is based on the popular TI-36 from Texas Instruments®.