d ocumfnt se 004 568 - ericdonovan johnson, can be obtained from j. weston walch, publisher, in...
TRANSCRIPT
=,
D OCUMFNT RF cl
ED 021 733 SE 004 568
By-McComb, PatriciaAPPLIED MATHEMATICS, SECONDARY SCHOOLS GRADES 10-12.Minnesota State Dept. of Education, St. Paul.Repor t No- CURR- BULL- 20APub Date 64Note- 115p.EDRS Price MF-$0.50 HC-14.68Descriptors-COMPUTERS, CURRICULUM, *CURRICULUM GUIDES INSTRUCTION, *MATHEMATICAL
APPLICATIONS, *MATHEMATICS, SECONDARY SCHOOL MATHEMATICSIdentifiers-Department of Education, Minnesota
This manual indicates a practical approach to the topic of general or consumermathematics as taught in senior high school. This course is intended for those pupilswho cannot succeed in the sequential high school mathematics course. The material forthis course has been selected to provide experiences which will tend to improve themathematical competence of future workers and citizens. The objectives of thesematerials are (1) to increase accuracy, understanding, and efficiency in computatonalskills, (2) to develop new computational skills and extend the understanding of numberand computational processes, (3) to provide skill in collecting, reading, organizing, andinterpreting data, (4) to develop an attitude of social-mindedness acquired through astudy of consumer problems, (5) to provide the mathematical skill and knowledgenecessary 'to cope with the problems of the consumer and citizen, (6) to provide thebasic mathematics needed by pupils in their future work and study in the trades andsemi-professional occupations, (7) to stimulate an interest in learning mathematics, and(8) to provide an opportunity to demonstrate such traits as creativity, imagination,curiosity, and vizualization. (RP)
4411
0101
110,
41.3
.
OF
FIC
E O
F E
DU
CA
TIO
N
TH
IS D
OC
UM
EN
I HA
S B
EE
NR
EP
RO
DU
CE
D E
XA
CT
LY A
S R
EC
EIV
ED
FR
OM
TH
E
PE
RS
ON
OR
OR
GA
NIZ
AT
ION
OR
IGIN
AT
ING
IT.
PO
INT
S O
F V
IEW
OR
OP
INIO
NS
ST
AT
ED
DO
NO
T N
EC
ES
SA
RIL
YR
EP
RE
SE
NT
OF
FIC
IAL
OF
FIC
E O
FE
DU
CA
TIO
N
PO
SIT
ION
OR
PO
LIC
Ya
a
4
AP
PLI
ED
rA
TH
EM
AT
ICS
114
ST
AT
E O
F M
INN
ES
OT
A
DE
PA
RT
ME
NT
OF
ED
UC
AT
ION
St.
Pau
l
1964
CU
RR
ICU
LUM
BU
LLE
TIN
NO
. 20A
Pro
pert
y of
Sch
ool
Dis
tric
t No.
Cou
nty
Min
neso
ta
APP
LIE
DM
AT
HE
MA
TIC
S
SE
CO
ND
AR
Y S
CH
OO
LS
GR
AD
ES
10-
12
CU
RR
ICU
LUM
BU
LLE
TIN
NO
. 20A
STA
TE
CO
PYST
AT
E O
F M
INN
ESO
TA
DE
PA
RT
ME
NT
OF
ED
UC
AT
ION
St. P
aul 1
1964
tka*
,41°
4
CO
NT
EN
T
INT
RO
DU
CT
ION
5
Uni
t IC
ompu
ting
and
Com
putin
g M
achi
nes
7
Uni
t II
Prob
abili
t,R
isk,
and
Ins
uran
ce17
Uni
t III
The
Lan
guag
e of
Sci
ence
31
Uni
t IV
Fact
s fr
om F
igur
es a
nd F
igur
ing
from
Fac
ts
Uni
t VFa
ct o
r Fa
ncy
6,5
Uni
t VI
Mak
ing
a M
illio
n or
Som
ethi
ng L
ess
67
Uni
t VII
Stre
tchi
ng Y
our
Dol
lar
73
Uni
t VII
I Fi
nanc
ing
Free
dom
81
Uni
t IX
Mon
ey a
nd Y
our
Man
or87
Uni
t XB
last
Off
to th
e Fu
ture
93
Bib
liogr
aphy
105
CO
MM
ITT
EE
ME
MB
ER
S
Mrs
. Bea
tric
e O
gren
, Cha
irm
anM
athe
mat
ies
Dep
artm
ent C
hair
man
,Fr
idle
y
Rob
ert J
acks
on, C
onsu
ltant
Inst
ruct
or, U
nive
rsity
of
Min
r es
ota
Don
ald
H. F
irl
Inst
ruct
or, R
oche
ster
Hig
h Sc
hool
Don
ovan
A. J
ohns
onPr
ofes
sor,
Mat
hem
atic
s, U
nive
rsity
Hig
h Sc
hool
, Min
neap
olis
Ferg
us V
. Nyg
aard
Prin
cipa
l, W
asec
a H
igh
Scho
ol
Har
old
A. S
chro
eder
Hea
d of
Mat
hem
atic
s D
epar
tmen
t,B
lue
Ear
th H
igh
Scho
ol
Gar
ld J
. Sm
ithA
dmin
istr
ativ
e A
ssis
tant
, Stil
lwat
erPu
blic
Sch
ools
RE
VIS
ION
CO
MM
ITT
EE
ME
MB
ER
S
Mrs
. Bea
tric
e O
gren
, Fri
dley
Rob
ert J
acks
on, U
nive
rsity
of
Min
neso
ta
Dav
id L
. Dye
, Sta
te D
epar
tmen
tof
Edu
catio
n
Rob
ert J
ohns
on, B
rook
lyn
Cen
ter
Hig
h Sc
hool
Don
ald
L. C
laus
on, D
irec
tor
ofC
urri
culu
m
Patr
icia
McC
omb,
Edi
tor
INT
RO
DU
CT
ION
i 1
APP
LIE
D M
AT
HE
MA
TIC
S11
th a
nd 1
2th
grad
es
IN T
RO
DU
G I
'I O
N
A n
ew a
nd p
ract
ical
app
roac
h to
the
topi
cof
gen
eral
or
cons
umer
mat
hem
atic
s as
taug
ht in
the
seni
or h
igh
scho
ol is
bei
ngat
tem
pted
in th
is c
ours
e of
stu
dy. T
o ov
erco
me
the
disi
nter
est p
reva
lent
am
ong
stud
ents
taki
ng s
uch
a co
urse
, cau
sed
by th
e be
lief
that
they
are
repe
atin
g th
e sa
me
subj
ect w
hich
they
did
not
unde
rsta
nd o
r ap
prec
iate
in th
e ei
ghth
or
nint
hgr
ades
, an
atte
mp;
: has
bee
n m
ade
to b
ase
the
cour
se o
n st
uden
t-ce
nter
ed a
ctiv
ities
cf.
ady
nam
ic a
nd d
iffe
rent
nat
ure
from
thos
e th
at a
rege
nera
lly f
ound
in s
uch
a co
urse
. No
text
book
has
bee
n fo
und,
to d
ate,
with
this
appr
oach
; the
refo
re, a
min
imum
num
ber
ofba
sic,
rea
dily
obt
aina
ble
refe
renc
es f
or e
ach
topi
c
have
bee
n lis
ted
upon
whi
ch th
e te
ache
r m
ay s
tart
build
ing
his
own
file
of
reso
urce
mat
eria
l.
Prov
isio
n fo
r in
divi
dual
dif
fere
nces
is d
iffi
cult
toin
dica
te in
an
outli
ne. T
he m
ater
ial i
s su
ch th
at a
thre
e-tr
act p
rogr
am in
this
cou
rse
coul
d ev
olve
fro
m th
e su
bjec
t mat
ter
liste
d.A
s no
w o
rgan
ized
, the
cou
rse
is d
irec
ted
tow
ard
the
mid
dle
trac
t.
Func
tion
of th
e C
ours
eT
his
cour
se is
inte
nded
for
thos
e pu
pils
who
can
not
succ
eed
in th
e se
quen
tial h
igh
scho
ol m
athe
mat
ics
cour
ses.
The
mat
eria
l has
bee
n
sele
cted
to p
rovi
de e
xper
ienc
es w
hich
will
tend
toim
prov
e th
e m
athe
mat
ical
com
pete
nce
of f
utur
ew
orke
rs a
nd c
itize
ns. B
y of
feri
ng th
is
cour
se n
ear
the
end
ofth
eir
high
sch
ool e
duca
tion,
it is
like
ly th
atst
uden
ts w
ill b
e re
cept
ive
to tb
e to
pics
of
the
cour
se.
Sele
ctio
n of
Pup
ilsA
lthou
gh th
ere
are
man
y ite
ms
in th
is o
utlin
e w
hich
wou
ld b
e va
luab
le to
all
pupi
ls, i
t is
reas
onab
le to
ass
ume
that
it w
ill b
e "g
eare
d"at
a lo
wer
leve
l tha
n th
e se
quen
tial c
ours
es. T
his
cour
sesh
ould
be
rese
rved
for
stu
dent
s w
ho ta
ke n
oot
her
mat
hem
atic
s co
urse
in g
rade
s10
-12.
Tho
se in
the
grou
p w
ho h
ave
the
abili
ty to
com
plet
e th
e se
quen
tial c
ours
es s
houl
d be
disc
oura
ged
from
enr
ollin
g in
this
cou
rse.
All
othe
r pu
pils
sho
uld
be a
dvis
ed to
enr
oll f
cr it
eith
er in
thei
r 11
th o
r 12
th y
ear,
pre
fera
bly
the
latte
r. T
hose
in th
e gr
oup
who
ran
kbe
low
des
irab
le s
tand
ards
on
a di
agno
stic
test
. suc
h as
one
of th
ose
liste
d be
low
, sho
uld
be e
ncou
rage
d to
enro
ll in
App
lied
Mat
hem
atic
s.
With
in th
is g
roup
of
pupi
ls th
e pr
oble
m o
fin
divi
dual
dif
fere
nces
is o
ne o
f m
ajor
impo
rtan
ce;
abili
ties
as w
ell a
s ne
eds
and
inte
rest
sw
ill v
ary
grea
tly. I
n or
der
to p
rovi
de f
or th
isw
idel
y di
verg
ent g
roup
, a v
ery
flex
ible
, rat
her
than
ari
gidl
y re
stri
cted
, cou
rse
is s
ugge
sted
.
One
item
of
maj
or im
port
ance
is th
e us
e of
a c
ompe
tent
diag
nost
ic te
st e
arly
in th
e ye
ar to
det
erm
ine
the
amou
nt a
nd ty
pe o
f re
med
ial
wor
k w
hich
will
be
.i.ec
essa
ry f
or e
ach
indi
vidu
alst
uden
t. So
me
sugg
este
d di
agno
stic
test
s fo
llow
:
Bas
ic S
kills
in A
rith
met
ic T
est,
by W
rink
le,
Sand
ers,
and
Ken
del.
Scie
nce
Res
earc
h A
ssoc
iate
s.
Surv
ey o
f A
rith
met
ic (
Mat
hem
atic
s) A
chie
vem
ent,
byT
iegs
and
Cla
rk. C
alif
orni
a T
est B
urea
u.
Und
erst
andi
ng th
e M
eani
ngs
in A
rith
met
ic, b
y D
avid
Rap
papo
rt. S
cien
ce R
esea
rch
Ass
ocia
tes.
Ari
thm
etic
Com
puta
tion
Tes
t, by
Mad
den
and
Peak
.Wor
ld B
ook
Com
pany
.
Org
aniz
atio
n of
the
Cou
rse
Sinc
e th
is c
ours
e is
to b
e a
flex
ible
out
line
of m
ater
ialu
sed
by th
e te
ache
r to
fit
the
wid
ely
dive
rgen
tnee
ds o
f pu
pils
, it i
s re
com
men
ded
that
a c
onst
ant a
nd th
orou
gh r
evie
w a
nd r
emed
ial p
rogr
ambe
mai
ntai
ned
thro
ugho
ut a
ll un
its to
bui
ldco
mpu
tatio
nal s
kill.
Thi
s pr
actic
e
and
drill
sho
uld
be in
form
al, f
requ
ent,
and
shor
t,us
ing
a va
riet
y of
act
iviti
es a
nd e
xper
ienc
es,
such
as
gam
es a
rK1
cont
ests
. A h
elpf
ulbo
okle
t, "G
ames
for
Lea
rnin
g M
athe
mat
ics"
by
Don
ovan
Joh
nson
, can
be
obta
ined
fro
m J
. Wes
ton
Wal
ch, P
ublis
her,
in P
ortla
nd,
Mai
ne. T
his
revi
ew w
ork
may
fre
quen
tly b
e re
late
d to
the
prac
tical
pro
blem
s of
the
cour
se. T
hei-
evie
w a
nd r
emed
ial w
ork
shou
ld b
eta
ilore
d to
the
need
s of
the
indi
vidu
al a
s w
ell a
s to
the
grou
p.
The
ince
ntiv
e co
nditi
ons
will
hav
e to
var
y as
the
mot
ivat
ion
of th
e pu
pils
is s
ense
d. A
ctiv
ities
sho
uld
be s
tude
nt-c
ente
red
and
the
teac
her
mus
t be
aler
t to
the
rate
of
lear
ning
and
the
diff
icul
ty o
f th
e m
ater
ials
. The
exa
mpl
es a
nd e
xerc
ises
sho
uld
be r
elat
ed to
the
past
,pr
esen
t, an
d fu
ture
life
exp
erie
nces
of
the
stud
ents
. For
thes
e st
uden
ts th
e le
arni
ng a
peri
ence
s sh
ould
em
phas
ize
labo
rat o
ryw
ork,
vis
ual-
izat
ion,
and
act
ive
part
icip
atio
n.T
he te
ache
r sh
ould
app
eal t
o th
e st
uden
t's im
agin
atio
n an
d to
his
bac
kgro
und
and
expe
rien
ce. L
et h
im c
ontr
ibut
e id
eas
duri
ng th
epr
esen
tatio
n of
mat
eria
ls. I
t bec
omes
impo
rtan
t the
n th
at a
s m
uch
activ
ity a
s po
ssib
le, w
hich
is n
ot s
olel
y pa
per-
and-
penc
il ac
tivity
,be
used
as
part
of
the
clas
sroo
m p
roce
dure
.T
he s
tude
nts
shou
ld p
rogr
ess
as f
ar a
s po
ssib
le w
ithin
eac
h un
it. I
t will
be
nece
ssar
y an
d fe
asib
le to
vee
r fr
om th
e ou
tline
,co
nsid
erin
gth
e st
uden
ts' e
nvir
onm
enta
l bac
kgro
und
or la
ck o
f pr
evio
us k
now
ledg
e. T
opic
s pr
evio
usly
cov
ered
sho
uld
be o
mitt
ed a
nd c
oord
inat
edw
ith th
e ot
her
depa
rtm
ents
of
the
scho
ol to
avo
id u
nnec
essa
ry r
epet
ition
. The
teac
her
shou
ld f
eel f
ree
to a
dd o
r su
bsitu
te o
ther
uni
ts o
rto
pics
. It w
ill b
e ne
cess
ai y
to b
uild
a f
ile o
f re
sour
ces
and
activ
ities
, dra
win
g up
on p
eopl
e in
the
com
mun
ity f
orin
form
atio
n an
d sa
mpl
eite
ms.
Som
e he
lpfu
l fre
e or
inex
pens
ive
pam
phle
ts a
s w
ell a
s so
me
basi
c re
fere
nces
whi
ch c
an b
e ob
tain
ed f
or e
ach
mem
ber
of th
e cl
ass
are
liste
d in
the
sugg
este
d ai
ds. T
here
sho
uld
be m
uch
pupi
l pai
ticip
atio
n an
d w
hen
it is
app
aren
t tha
t int
eres
t is
flag
ging
and
mot
ivat
ion
diff
icul
t, di
scre
tion
will
hav
e to
be
used
as
to th
e ad
visa
bilit
y of
mov
ing
on to
ano
ther
topi
c. P
ract
ice
in c
ompu
tatio
n sh
ould
be
bala
nced
with
the
lear
ning
of
fact
s an
d it
wou
ld b
e an
unu
sual
cla
ss w
hich
wou
ld b
e in
tere
sted
in a
ll th
e in
form
atio
n lis
ted
as s
ubje
ct m
atte
r or
indo
ing
com
puta
tiona
l wor
k in
volv
ing
all s
ugge
sted
topi
cs. M
any
topi
cs m
ay b
e co
nsid
ered
opt
iona
l, de
pend
ing
upon
stu
dent
abili
ty,
back
grou
nd, a
nd in
tere
st. F
or e
xam
ple,
the
unit
on F
act o
r Fa
ncy
may
be
omitt
ed e
ntir
ely
or in
sert
ed a
t any
tim
e.
Obj
ectiv
es1.
To
incr
ease
acc
urac
y, u
nder
stan
ding
, and
eff
icie
ncy
in c
ompu
tatio
nal s
kills
.2.
To
deve
lop
new
com
puta
tiona
l ski
lls a
nd e
xten
d th
e un
ders
tand
ing
of n
umbe
r an
d co
mpu
tatio
nal p
roce
sses
.3.
To
prov
ide
skill
in c
olle
ctin
g, r
eadi
ng, o
rgan
izin
g, a
nd in
terp
retin
g da
ta.
4. T
o de
velo
p an
atti
tude
of
soci
al-m
inde
dnes
s ac
quir
ed th
roug
h a
stud
y of
con
sum
er p
robl
ems.
5. T
o pr
ovid
e th
e m
athe
mat
ical
ski
ll an
d kn
owle
dge
nece
ssar
y to
cop
e w
ith th
e pr
oble
ms
of th
e co
nsum
er a
nd c
itize
n.6.
To
prov
ide
the
basi
c m
athe
mat
ics
need
ed b
y pu
pils
in th
eir
futu
re w
ork
and
stud
y in
the
trad
es a
nd s
emi-
prof
essi
onal
occu
patio
ns.
7. T
o st
imul
ate
an in
tere
st in
lear
ning
mat
hem
atic
s.8.
To
prov
ide
an o
ppor
tuni
ty f
or th
e st
uden
ts to
dem
onst
rate
suc
h tr
aits
as
crea
tivity
, im
agin
atio
n, c
urio
sity
, and
vis
ualiz
atio
n.
6
COMPUTING ANDCOMPUTING MACHINES.
UN
IT I
CO
MPU
TIN
G A
ND
CO
MPU
TIN
G M
AC
HIN
ES
Intr
oduc
tion
Beg
inni
ng th
e co
urse
with
a u
nit o
n C
ompu
t:ngand Computing Machines is
an a
ttem
pt to
invo
lve
the
stud
ent i
mm
edia
tely
in s
omet
hing
new
and
inte
rest
ing
with
whi
ch to
sta
rt th
e sc
hool
yea
rin
stea
d of
t he
trad
ition
ally
dis
tast
eful
rev
iew
of
fund
amen
tals
. The
suc
cess
of
the
cour
se w
ill b
e de
pend
ent u
pon
the
reso
urce
fuln
ess
of th
e te
ache
r in
aro
usin
g an
d fo
ster
ing
this
inte
rest
and
by
givi
ng a
n ai
r of
fre
shne
ssan
d vi
talit
y to
the
less
on m
ater
ial a
nd c
lass
room
pro
cedu
res.
Ess
entia
lly th
e un
it, o
ther
than
giv
ing
the
stud
ent a
look
at c
ompu
ting
aids
as
a w
ell-
esta
blis
hed
part
of
our
pres
ent w
ayof
life
,se
rves
as
a ve
hicl
e to
rev
iew
, ren
ew, a
nd e
stab
lish
a ce
rtai
nba
sic
mat
hem
atic
s. T
he a
ctiv
ities
for
this
mat
eria
l sho
uld
be c
lass
room
act
iviti
esas
muc
h as
pos
sibl
e. T
hey
shou
ld n
ot b
e so
lely
hom
ewor
kac
tiviti
es, o
r th
e te
ache
r w
ill b
e lim
ited
to th
e ex
tent
upo
n w
hich
he
can
capi
ta-
lize
on th
e re
sults
of
the
activ
ities
. Pro
ject
s sh
ould
not
be
time-
cons
umin
g bu
t acc
ompl
ishe
d qu
ickl
y an
d ef
fici
ently
whi
le in
tere
stre
mai
nshi
gh. T
he e
ffec
t of
this
mat
eria
l mig
ht b
e am
plif
ied
by in
vitin
g on
e or
mor
e m
embe
rs o
f th
e co
mm
unity
who
hav
e an
acq
uain
tanc
e w
ithco
mpu
ting
mac
hine
s to
dem
onst
rate
or
spea
k to
the
clas
s ab
out t
his
equi
pmen
t.
Obj
ectiv
es1.
To
give
the
stud
ents
the
idea
that
the
cour
se h
as a
fre
sh a
ppro
ach
2. T
o gi
ve th
e st
uden
ts a
gre
ater
app
reci
atio
L f
or m
athe
mat
ics
as a
ser
vant
3. T
o un
ite th
e st
uden
ts in
to a
wor
king
ent
ity4.
To
revi
ew s
ome
arith
met
ic s
ubje
ct a
reas
as
a pr
epar
atio
n fo
r un
its to
fol
low
and
to s
tren
gthe
n th
e ar
ithm
etic
bac
kgro
unds
of
the
stud
ents
5. T
o te
ach
for
an u
nder
stan
ding
of
com
putin
g m
achi
nes
as th
ough
tful
inve
ntio
ns
Con
tent
A. M
otiv
atio
n fo
r th
e un
it
1. H
elp
the
stud
ent t
o se
e th
e va
lue
of c
ornt
-+
ing
mac
hine
s as
tim
esa
vers
and
as
bette
r in
sura
ito
war
d ac
cura
cy o
f re
sults
2. H
elp
the
stud
ent s
ee th
e co
mpu
t-in
g m
achi
ne a
s a
7-ea
l par
t of
our
pres
ent d
aily
life
3. H
elp
the
stud
ent t
o fe
el a
des
ire
to p
artic
ipat
e in
the
activ
ities
of
this
cou
rse
unit
Proc
edur
es a
nd A
ctiv
ities
Hav
e th
e st
uden
ts d
o so
me
time-
cons
umin
g,er
ror-
indu
cing
ari
thm
etic
com
puta
tions
, suc
h as
find
ing
the
sum
of
a lo
ng li
st o
f se
vera
l dig
it ad
-de
nds,
fin
ding
the
prod
uct o
r qu
otie
nt o
f tw
onu
mbe
rs h
avin
g si
x or
mor
e di
gits
in th
eir
nu-
mer
als,
rai
sing
a s
ingl
e di
git n
umbe
r to
a p
ower
grea
ter
than
ten,
for
exa
mpl
e. T
hen
com
pare
answ
ers
for
the
sam
e pr
oble
m a
nd d
iscu
ss ih
est
uC.a
its' r
eact
ions
to d
oing
thes
e pr
oble
ms
Mak
e a
clas
s bo
ok o
f pi
ctor
ial c
lippi
ngs
of a
ll or
som
e of
the
com
putin
g m
achi
nes
liste
d be
low
:
addi
ng m
achi
nes
cash
reg
iste
rsde
sk c
alcu
lato
rssl
ide
rule
sam
met
ers
voltm
eter
ssp
eedo
met
ers
num
eric
al ta
bles
odom
eter
sIB
M m
achi
nes
book
keep
ing
mac
hine
sel
ectr
onic
com
pute
rsno
mog
raph
s
7
Tea
chin
g A
ids
The
Mat
hem
atic
s T
each
er, M
arch
1958. pp.
162-170
The
IB
M p
ictu
re s
erie
s
Vor
wal
d, A
lan,
and
Cla
rk, F
rank
. Fro
mSa
nd T
able
to E
lect
roni
c B
rain
.. M
cGra
w,
1961
You
ng, F
rede
rick
H. D
igita
l Com
pute
rsan
d R
elat
ed M
athe
mat
ic. G
inn,
1961
In g
ener
al, m
ost r
efer
ence
title
s gi
Ven
els
e-w
here
for
this
uni
t whe
re th
e tit
le a
llude
sto
com
putin
g m
achi
nes
and
devi
ces
B. P
urpo
ses
of c
ompu
ting
mac
hine
s an
dde
vice
s
1. T
o ob
tain
a s
olut
ion
for
som
epr
oble
m w
hen
the
solu
tion
wou
ldot
herw
ise
be u
nobt
aina
ble
2. T
o ob
tain
a s
olut
ion
whi
ch w
ould
othe
rwis
e be
tim
e co
nsum
ing
toob
tain
3. T
o re
duce
and
elim
inat
e er
rors
4. T
o av
oid
the
bore
dom
of r
epea
ted
sim
ilar
oper
atio
ns n
eces
sary
toso
lve
cert
ain
prob
lem
s
Mou
nt s
ever
al la
rge
pcck
ets
onto
apl
ain
back
-gr
ound
of
som
e si
ze. L
abel
each
poc
ket w
ith o
neof
the
nam
es o
f th
e co
mpu
ting
mac
hine
s pr
e-vi
ousl
y lis
ted.
Hav
e th
est
uden
ts d
epos
it ph
oto-
grap
hs, c
lippi
ngs,
adv
ertis
ing,
vari
ous
dist
rib-
uted
mat
eria
ls, a
nd o
ther
rele
vant
item
s in
toth
e po
cket
s as
they
col
lect
them
. The
teac
her
may
see
d so
me
oral
l of
the
pock
ets
with
mat
e-ri
als
suita
ble
for
inst
ruct
ive
purp
oses
and
rela
-tiv
e to
her
pla
ns to
fol
low
Hav
e st
uden
ts m
ake
a bu
lletin
boar
d of
pic
ture
sof
com
pute
rs
Any
act
ivity
use
d sh
ould
be m
eani
ngfu
l in
term
sth
e un
it un
der
cons
ider
atio
nan
d sh
ould
con
-tr
ibut
e to
war
d th
e cl
ass
mem
bers
wor
king
to-
:.Ahe
r as
a g
roup
Eac
h ac
tivity
will
hav
e to
be o
rgan
ized
so
that
it do
es n
ot b
ecom
e to
o tim
eco
nsum
ing
in r
ela-
tion
to c
lass
tim
e. P
roje
cts
shou
ld b
e ac
com
p-lis
hed
quic
kly
and
effi
cien
tly
Invi
te o
ne o
r m
ore
pers
ons
in to
disc
uss
thei
rex
peri
ence
with
the
effe
ctiv
enes
sof
com
putin
gm
achi
nes
and/
or d
evic
es a
ndw
ith th
e ki
nd o
fjo
bs th
ey w
ill d
o
Bri
ng in
mac
hine
s th
at w
ill s
erve
as
exam
ples
of
each
pur
pose
list
ed. D
iscu
ssth
ese
mac
hine
s re
la-
tive
to e
ach
purp
ose
but d
ono
t lec
ture
Not
muc
h tim
e sh
ould
be
spen
t on
this
topi
c
5. T
o ob
tain
a c
onci
se o
rgan
ized
reco
rd o
f t.h
e so
lutio
n fo
r a
prob
-le
m s
o th
at a
ny s
tep
in th
e so
lutio
nm
ay b
e re
cove
red
at a
late
r tim
e
Som
e co
mpu
ting
devi
ces
of te
n ta
ken
for
gran
ted
1. N
ames
and
sym
bols
for
num
bers
2. T
he d
igita
l or
plac
e-va
lue
syst
emof
rec
ordi
ng a
r u
mbe
r w
ith a
num
eral
Con
side
r th
e te
diou
snes
s of
the
shep
herd
cou
nt-
ing
his
shee
p by
pla
cing
them
in a
one
-to-
one
corr
espo
nden
ce w
ith s
ticks
or
ston
es
Con
side
r th
e in
conv
enie
nce
of c
onve
ying
the
idea
of
a nu
mbe
r to
the
min
d of
ano
ther
by
al-
way
s re
sort
ing
to s
ome
sort
of
visu
al d
ispl
aysu
ch a
s ho
ldin
g up
fin
gers
Poin
t out
that
sym
bols
for
num
bers
mad
e it
pos-
sibl
e to
rec
ord
the
idea
of
a ce
rtai
n nu
mbe
r fo
rth
e us
e of
som
eone
not
pre
sent
Sym
bols
for
num
bers
hav
e le
ssen
ed th
e de
man
dsup
on o
ur m
emor
ies
Exp
lore
how
the
basi
c ar
ithm
etic
ope
ratio
nsw
ould
hav
e to
be
carr
ied
out w
hen
a di
ffer
ent
sym
bol i
s us
ed f
or e
ach
num
ber
(req
uire
s th
e us
eof
num
ber
line
oper
atio
ns).
Con
side
r w
hat o
ther
dem
ands
wou
ld b
e m
ade
upon
us
whe
n a
diff
eren
tsy
mbo
l is
used
for
eac
h nu
mbe
r in
stea
d of
usi
ngth
e di
gita
l sys
tem
Con
side
r in
wha
t way
giv
ing
mea
ning
to th
eor
der
of d
igits
in a
num
eral
is a
con
veni
ence
Wha
t is
the
valu
e of
hav
ing
a st
anda
rd b
ase
for
our
num
erat
ion
syst
em?
Not
e : T
his
is th
e lo
gica
l poi
nt a
t whi
ch to
exp
lore
num
erat
ion
syst
ems
with
bas
es o
ther
than
ten.
The
ext
ent t
o w
hich
one
goe
s in
to th
is to
pic
will
depe
nd u
pon
pupi
l int
eres
t, pu
pil b
ackg
roun
d,av
aila
ble
time,
for
exa
mpl
e. E
mph
asis
sho
uld
beon
the
mea
ning
of
nota
tion,
mea
ning
of
borr
ow-
ing
and
carr
ying
, and
on
the
arith
met
ic o
pera
-tio
ns. A
dditi
on a
nd m
ultip
licat
ion
tabl
es s
houl
d
9
If m
uch
wor
k is
don
e he
re in
num
erat
ion
syst
ems,
it m
ay p
rove
hel
pful
to m
ake
ase
t of
tray
ed s
trip
s of
tagb
oard
on
whi
chth
e va
riou
s un
it po
sitio
ns a
re la
belle
d bo
thto
the
righ
t and
the
left
of
a de
cim
al p
oint
.T
hen
vari
ous
num
eral
s m
ay b
e in
sert
edin
to th
e tr
ays
to f
orm
mul
ti-di
git n
umer
als.
With
a f
ew p
aper
ope
ratio
n sy
mbo
ls a
ndth
ese
tray
ed s
trip
s, o
ne c
an r
eadi
ly d
ispl
aybo
rrow
ing
and
carr
ying
, mul
tiplic
atio
n of
num
bers
, add
ition
of
num
bers
, and
so
on
Pape
r nu
mbe
r lin
es p
rove
use
ful i
n so
me
area
s he
re. T
hese
aid
s m
ay b
e st
retc
hed
over
the
fron
t cha
lkbo
ard
and
left
ther
edu
ring
the
entir
e da
ys u
sed
to c
over
the
mat
eria
l
The
Web
ster
Pub
lishi
ng C
o. (
1154
Red
oA
venu
e, S
t. L
ouis
, Mo.
) ha
s ha
d pr
inte
dnu
mbe
r lin
es a
vaila
ble
as p
art o
f th
eir
ad-
vert
isin
g ci
rcul
atio
n in
the
past
. An
in-
geni
ous
teac
her
can
mak
e th
ese
lines
do
for
both
the
posi
tive
and
nega
tive
port
ions
of a
num
ber
line
Film
s. H
ow M
an L
earn
ed to
Cou
nt, A
sso-
ciat
ed F
ilms.
A D
ay W
ithou
t Num
bers
, 10
min
Way
ne U
nive
rsity
. Ori
gin
of M
athe
-m
atic
s, 1
1 m
in U
nite
d W
orld
Film
s
Bas
e an
d Pl
ace.
30
min
Sta
te U
. of
Iow
a;E
arlie
st N
umbe
rs 3
0 m
in S
. U. I
.; al
so
3. M
emor
izat
ion
of a
dditi
on a
ndm
ultip
licat
ion
tabl
es
4. A
rith
met
ic a
lgor
ism
sa.
long
div
isio
nb.
long
mul
tiplic
atio
nc.
ext
ract
ing
a sq
uare
roo
td.
fin
ding
a g
reat
est c
omm
on d
i-vi
sor
5. E
xten
sion
of
one
set o
f nu
mbe
rsto
for
m a
larg
er s
et
a. e
xten
sion
to p
erm
it su
btra
c-tio
n of
all
poss
ible
pai
rs (
sign
ednu
mbe
rs a
nd z
ero)
b. e
xten
sion
to th
e se
t of
ratio
nal
num
bers
to p
erm
it al
l pos
sibl
edi
visi
ons
exce
pt b
y ze
ro
c. e
xten
sion
to th
e se
t of
num
ber
pair
s to
per
mit
the
pict
urin
g
be u
sed
in e
ach
syst
em, e
xcep
t the
dec
imal
and
bina
ry s
yste
ms.
A r
easo
nabl
e co
vera
ge m
ight
in-
clud
e th
e sy
stem
s w
ith b
ases
of
five
or
seve
n,tw
o an
d tw
elve
A g
ood
film
on
num
erat
ion
syst
ems
will
ser
vew
ell t
o em
phas
ize
the
idea
s yo
u ha
ve b
een
tryi
ngto
teac
h an
d w
ill a
lso
serv
e to
tie
idea
s to
geth
eran
d re
view
. If
the
stud
ents
hav
e ha
d co
nsid
er-
able
wor
k on
num
erat
ion
in th
e pa
st, t
he f
ilmw
ould
ser
ve e
xcel
lent
ly to
bri
ng to
min
d th
ose
basi
c id
eas
you
wis
h to
em
phas
ize
as p
art o
f th
eun
it at
this
poi
nt
The
maj
ority
of
your
stu
dent
s w
ill u
ndou
bted
lykn
ow th
eir
mul
tiplic
atio
n ta
bles
ver
y w
ell,
but
man
y m
ay n
eed
to in
crea
seth
eir
spee
d of
usa
geby
mea
ns o
f fl
ash
card
s, m
enta
l ari
thm
etic
, or
othe
r ga
mes
or
activ
ities
Rev
iew
tlts
e al
gori
sms
as s
eem
s us
eful
. If
con-
side
rabl
e w
ork
has
been
don
e w
ith. o
pera
tions
inth
e va
riou
s nu
mer
atio
n sy
stem
s, le
ss w
ork
will
need
to b
e do
ne h
ere
Dis
cuss
pri
me
num
bers
The
ext
ent t
o w
hich
wor
k is
don
e in
thes
e ar
eas
will
aga
in d
epen
d up
on th
e ne
eds
of th
e st
uden
ts.
A s
ubst
antia
l rev
iew
of
addi
ng, m
ultip
lyin
g,su
btra
ctin
g, a
nd d
ivid
ing
frac
tions
will
usu
ally
be b
enef
icia
l, ho
wev
er
If g
raph
ing
is s
tudi
ed le
t the
stu
dent
s fe
el th
atth
e re
al p
urpo
se o
f a
grap
h is
to p
icto
rial
ly r
ep-
rese
nt r
elat
ions
hips
and
that
eac
h ty
pe o
f gr
aph
has
its o
wn
usef
ulne
ss a
nd li
mita
tions
. Do
not
be a
frai
d to
ste
p ou
t fro
m tr
aditi
onal
type
s of
grap
hs to
get
thes
e no
tions
acr
oss.
Try
som
e of
10
othe
rs o
f U
nder
stan
ding
Num
bers
ser
ies
Bak
st, A
aron
. Mat
hem
atic
al P
uzzl
es a
ndPa
stim
es. V
an N
ostr
and,
195
4, c
h. 5
Bak
st, A
aron
. Mat
hem
atic
s, I
ts M
agic
and
Mas
tery
. Van
Nos
tran
d, 1
952,
ch.
1 a
nd 2
Dan
tzig
, Tob
ias.
Num
ber,
the
Lan
guag
e of
Scie
nce.
Mac
mill
an, 1
954
Asi
mov
, Isa
ac. T
he R
ealm
of
Num
ber.
Hou
ghto
n, 1
959
Stei
n, E
dwin
I. S
uppl
emen
tary
Uni
ts in
Con
tem
pora
ry A
rith
met
ic a
nd E
lem
en-
tary
Alg
ebra
. Van
Nos
tran
d, 1
960
Som
e st
uden
ts w
ill h
ave
lear
ned
diff
eren
tap
proa
ches
to th
ese
oper
atio
ns. T
hete
ache
r sh
ould
inve
stig
ate
this
situ
atio
n
of c
erta
in r
elat
ions
hips
am
ong
num
bers
6. O
rien
tatio
n of
the
deci
mal
sys
tem
from
left
to r
ight
in it
s nu
mer
atio
n
D. S
hort
cut
s in
com
putin
g
1. M
ultip
licat
ions
a. b
y te
ns a
nd M
swer
s of
ten
b. b
y 5,
25,
50,
250
, 500
, and
so
onc.
by
num
bers
dif
feri
ng f
rom
apo
wer
of
ten
or o
ne o
f th
e fi
rst
nine
mul
tiple
s of
ten
by a
dif
-fe
renc
e of
one
or
two
the
follo
win
g:
a. N
on-p
erpe
ndic
ular
axe
sb.
Dif
fere
nt s
ized
uni
ts o
n th
e ax
esc.
Use
the
seco
nd c
oord
inat
e to
det
erm
ine
hori
zont
al d
ista
nce
and
the
firs
t coo
rdin
ate
to d
eter
min
e th
e ve
rtic
al d
ista
nce.
Com
pare
to s
tand
ard
proc
edur
eC
ompa
re b
lack
-and
-whi
te g
raph
s to
gra
phs
usin
gco
lor
to e
mph
asiz
e id
eas
cont
aine
d by
dat
a. C
on-
side
r ba
r gr
aphs
, pie
gra
phs,
bro
ken
line
grap
hs,
and
othe
r co
mm
on g
raph
sys
tem
s
Com
pare
thre
e di
men
sion
al g
raph
s to
two
di-
men
sion
al g
raph
s
Gra
ph s
impl
e in
equa
litie
s an
d co
mpa
re to
gra
phs
for
equa
litie
s
Inve
stig
ate
the
grap
hica
l sol
utio
n of
pro
blem
sK
eep
prob
lem
s si
mpl
e
The
res
ults
of
an o
ppos
ite o
rien
tatio
n co
uld
beex
plor
ed
Em
phas
is s
houl
d be
giv
en to
the
fact
that
eac
hte
chni
que
is f
or a
spe
cial
ized
situ
atio
n, a
nd th
atto
mak
e th
ese
met
hods
a p
art o
f on
e's
tool
s he
mus
t bec
ome
adep
t in
reco
gniz
ing
each
spe
cial
situ
atio
n. C
onsi
dera
ble
drill
will
be
nece
ssar
ybe
fore
this
will
be
brou
ght a
bout
. The
se d
rills
shou
ld b
e sh
ort,
freq
uent
, and
cum
ulat
ive
inte
chni
ques
use
d
An
exce
llent
clim
ax to
this
mat
eria
l wou
ld b
e a
spee
d co
ntes
t bet
wee
n op
posi
ng te
ams
or in
di-
vidu
als.
Suc
h a
cont
est s
houl
d em
ploy
som
e of
the
mor
e dr
amat
ic te
levi
sion
app
roac
hes
Exa
mpl
e: 58 X
72
= (
602)
(70
+ 2
)
11
Roa
d M
aps
of I
ndus
try.
Nat
'l. I
ndus
tria
lC
onfe
renc
e B
oard
, Inc
.46
0 Pa
rk A
venu
eN
ew Y
ork
22, N
. Y. F
ree
Bak
st, A
aron
. Mat
hem
atic
al P
uzzl
es a
ndPa
stim
es. V
an N
ostr
and,
195
4
Bak
st, A
aron
. Mat
hem
atic
s, I
ts M
agic
and
Mas
tery
. Van
Nos
tran
d, 1
952,
pp.
125
-13
3
Mey
ers,
Les
ter.
Hig
h Sp
eed
Mat
h. V
anN
ostr
and,
194
7
Stic
ker,
Hen
ry. H
ow to
Cal
cula
te Q
uick
ly.
Dov
er,
)5
John
son,
Don
ovan
A.,
and
Gle
nn, W
illia
mH
. Sho
rt C
uts
in C
ompu
ting.
Web
ster
,19
61
Film
: Qui
cker
Tha
n Y
ou T
hink
. Ass
ocia
ted
Film
s
2. S
quar
ing
num
bers
2. A
dditi
ona.
by
look
ing
for
pair
s w
hich
giv
eea
sy s
ums,
in th
edi
git c
olum
nsb.
add
ition
of
9, 9
9, 9
99 a
nd s
o on
e. a
dditi
on o
fth
e sa
me
num
ber
seve
ral t
imes
4. S
ubtr
actio
na.
of
9, 9
9, 9
99b.
8, 9
8, 9
98
5. D
ivis
ion
a. te
sts
for
divi
sibi
lity
by 2
, 3, 4
,5,
8, 9
, 10
b. te
sts
for
divi
sibi
lity
by 6
, 7, 1
1,13
6. C
heck
ing
of p
robl
ems
a. b
y ca
stin
g ou
tnin
esb.
by
use
of in
vers
e op
erat
ion
c. b
y ap
prox
imat
ion
(rou
ndin
gof
f)
E. S
ome
com
putin
g de
vice
s an
dsi
mpl
eco
mpu
ting
mac
hine
s
Dis
cuss
inve
rse
oper
atio
ns a
nd s
how
thei
r re
la-
tion
to c
heck
ing
App
roxi
mat
ion
tech
niqu
es c
an b
e us
ed f
ores
ti-m
atin
g an
swer
s pr
ior
to w
orki
ng a
prob
lem
and
as a
cru
de c
heck
for
acc
urac
y.B
oth
uses
sho
uld
be d
evel
oped
with
the
stud
ents
The
key
her
e lie
s in
hav
ing
the
stud
ents
actu
ally
mak
e an
d us
e so
me
of th
ese
devi
ces.
The
teac
her
may
wis
h to
pro
vide
som
eof
the
devi
ces
to s
ave
time,
but
she
sho
uld
not p
rovi
deth
em a
ll. A
good
dea
l of
lear
ning
will
take
pla
ceth
roug
h th
eir
cons
tnic
tion.
Any
idea
s a
stud
ent m
ay h
ave
tom
odif
y or
impr
ove
the
devi
ailL
and
mac
hine
sm
ay w
ell b
e bu
iltin
to h
is c
onse
rict
ion
If it
is d
esir
able
to h
ave
each
cla
ssm
embe
r ha
veon
e or
mor
e of
thes
e de
vice
s, m
ost o
f th
em c
anbe
mad
e in
expe
nsiv
ely
and
sim
ply
with
alit
tleth
ough
t
Perh
aps
a fe
w s
tude
nts
afe
inte
rest
edin
be-
com
ing
adep
t at o
pera
ting
an a
bacu
s or
soro
ban
and
wou
ld b
e w
illin
g to
put
on
adi
spla
y af
ter
alit
tle p
ract
ice.
Or
may
be th
ere
is s
ome
pers
onre
adily
ava
ilabl
e fr
om th
e co
mm
unity
who
is
12
Adl
er, I
rvin
g. T
hink
ing
Mac
hine
s. J
ohn
Day
, 196
1, c
h. v
i
Bak
st, A
aron
. Mat
hem
atic
al P
uzzl
es a
ndPa
stim
es. V
an N
ostr
and,
195
4, c
h. 1
3
N.C
.T.M
., 18
th Y
earb
ook,
Col
umbi
aT
each
ers
Col
lege
Pre
ss, 1
945,
pp.
164
-18
1, 1
17-1
29, 1
60-1
63
Bak
st, A
aron
. Mat
hem
atic
s, I
ts M
agic
and
Mas
tery
. Van
Nos
tran
d, 1
952,
pp.
113
-12
4, 2
64-2
98
John
son,
Don
ovan
A.,
and
Gle
nn, W
illia
mH
. Com
putin
g D
evic
es. W
ebst
er, 1
961
1. S
lidin
g ca
libra
ted
lines
2. T
he a
bacu
s an
d so
roba
n3.
Lin
kage
com
pute
rs (
requ
ire
the
use
of r
atio
n an
d pr
opor
tion)
4. F
inge
r co
mpu
ting
5. N
omog
raph
s6.
Nap
ier's
bon
es o
r ro
ds
F. E
xpon
entia
l com
putin
g
1. M
eani
ng o
f fa
ctor
and
exp
onen
t2.
Mul
tiply
ing
of p
ower
s of
the
sam
eba
se b
y ad
ditio
n of
exp
onen
ts a
ndth
e us
?..
of a
n ex
pone
ntia
l cha
rt3.
Div
isio
n of
pow
ers
of th
e sa
me
base
thro
ugh
the
subt
ract
ion
ofex
pone
nts
and
the
use
of th
e ch
art
4. R
aisi
ng a
pow
er to
a p
ower
by
mul
tiplic
atio
n of
exp
onen
ts5.
Sim
ple
thre
e-pl
ace
loga
rith
ms
usin
g th
e ba
se o
f te
n, in
clud
ing
use
of th
e ta
ble
6. S
lide
rule
s
G. T
wo
basi
c ty
pes
of c
ompu
ters
adep
t at o
ne o
f th
ese
mac
hine
s w
ho w
ould
be
will
ing
to e
el:n
:7:-
hi
Hec
togr
aph
or m
imeo
grap
h st
enci
ls m
ay o
e cu
tfo
r th
e ba
sic
piec
es n
eces
sary
to m
ake
slid
ing
calib
rate
d lin
es
Lat
tice
met
hod
of m
ultip
licat
ion
is a
goo
d in
-tr
oduc
tion
to N
apie
r's b
ones
The
ext
ent t
o w
hich
one
goe
s in
to th
is a
rea
will
depe
nd u
pon
the
mat
urity
of
the
stud
ents
in-
volv
ed
If th
e sl
ide
rule
s ar
e us
ed, o
ne s
houl
d as
soci
ate
them
with
slid
ing
calib
rate
d lin
es
Log
arith
mic
tabl
es m
ay b
e us
ed to
per
form
arith
met
ic o
pera
tions
eve
n th
ough
the
stud
ent
does
not
ent
irel
y un
ders
tand
why
the
syst
emw
orks
, but
hsh
ould
be
give
n to
und
erst
and
that
he
is m
akin
g us
e of
the
prop
ertie
s of
ex-
pone
nts
Fiel
d tr
ips
to p
lace
s us
ing
com
putin
g m
achi
nes
can
be u
z.ef
ul h
ere,
if th
e pe
rson
con
duct
ing
the
tour
is p
repa
red
to p
oint
out
suc
h th
ings
as
the
kind
of
job
each
mac
hine
will
do
and
will
em
pha-
size
the
fact
that
it is
the
man
ope
ratin
g th
em
achi
ne w
ho m
ust b
e th
e br
ain
and
not t
hem
achi
ne. T
he f
ollo
win
g tr
ips
are
exam
ples
of
wha
t mig
ht b
e do
ne:
1. V
isit
a la
rge
met
ropo
litan
ban
k
13
Stan
dard
alg
ebra
and
adv
ance
d al
gebr
ate
xts
John
son,
Don
ovan
A.,
and
Gle
nn, W
illia
mH
. Com
putin
g D
evic
es. W
ebst
er, 1
961
pp. 1
5-32
Adl
er, I
rvin
g. T
hink
ing
Mac
hine
s. J
ohn
Day
, 196
1 ch
xi
Eck
ert,
W. J
., an
d Jo
nes,
R. F
aste
r, F
aste
r.M
cGra
w, 1
955
Ber
kele
y an
d W
ainw
righ
t.C
ompu
ters
,T
heir
Ope
ratio
ns a
nd A
pplic
atio
ns, S
ec-
tion
1. R
einh
old
Pub.
, 195
6
1. D
igita
l com
pute
rsa.
def
initi
onb.
exa
mpl
esad
ding
mac
hine
s,de
sk c
alcu
lato
rs, c
ash
regi
ster
s,ab
acus
, mec
hani
cal b
ookk
eep-
ing
mac
hine
s, te
leph
one
dial
s,m
any
elec
tron
ic c
ompu
ters
2. A
nalo
g co
mpu
ters
a. d
efin
ition
b. e
xam
ples
loga
rith
mic
slid
eru
les,
aut
omob
ile s
peed
omet
ers,
elec
tric
al m
eter
s, o
dom
eter
s,th
erm
omet
ers,
clo
cks,
man
yel
ectr
onic
com
pute
rs
H. O
ther
idea
s in
volv
ed w
ith c
ompu
ting
mac
hine
s1.
Pun
ch c
ard
sele
ctio
n2.
Ele
ctri
city
and
mag
netis
m a
s us
edin
des
igni
ng a
mod
ern
com
pute
r3.
Fin
ding
phy
sica
l ana
logi
es o
f th
ebi
nary
num
ber
syst
em, s
uch
asel
ectr
ical
circ
uits
and
thei
rsw
itche
s4.
His
tori
cal o
rder
for
com
putin
gte
chni
ques
, com
putin
g de
vice
s,an
d co
mpu
ting
mac
hine
s
2. V
isit
an I
.B.M
. or
Bur
roug
hs m
anuf
actu
r-in
g pl
ant
3. V
isit
hist
oric
al d
ispl
ays
of o
lder
mod
els
ofco
mpu
ting
mac
hine
s. T
hese
old
er m
odel
sdo
not
hid
e th
e co
nstr
uctio
n of
the
mac
hine
sw
orki
ng p
arts
and
can
be
mos
t ins
truc
tive
A s
peed
con
test
usi
ng th
e de
sk c
alcu
lato
r an
d an
elec
tron
ic b
rain
wou
ld b
e in
tere
stin
g he
re
Hav
e pi
ctur
es o
f ea
ch ty
pe to
sho
w a
s th
ey a
rebe
irg
disc
usse
d
The
se a
re to
pics
som
e st
uden
ts m
ight
wis
h to
purs
ue o
n th
eir
own.
Or
they
may
be
used
toex
tend
the
unit
if th
e in
tere
st o
f th
e cl
ass
has
been
hig
h
14
Ric
hard
s, R
. K. A
rith
met
ic O
pera
tions
inD
igita
l Com
pute
rs. V
an N
ostr
and,
195
5
Mod
el e
lect
roni
c co
mpu
ters
are
ava
ilabl
e
Adl
er, I
rvin
g. T
hink
ing
Mac
hine
s. J
ohn
Day
Co.
, 196
1, c
h. x
, xii
Bak
st, A
aron
. Mat
hem
atic
al P
uzzl
es a
ndPa
stim
es. V
an N
ostr
and,
195
4, c
h. 4
Ber
kele
y an
d W
ainw
righ
t. C
ompu
ters
,The
irO
pera
tions
and
App
licat
ions
. Rei
nhol
d,19
56
B o
ehm
, Geo
rge
W. T
he N
ew W
orld
ofM
athe
mat
ics.
Dia
l Pre
ss, 1
959,
ch.
3
The
Mat
hem
atic
spp
. 169
-180
The
Mat
hem
atic
spp
. 471
-478
Tea
cher
. Mar
ch, 1
958.
Tea
cher
. Mar
ch, 1
959.
Rei
nfel
d, N
. Mat
hem
atic
al P
rogr
amm
ing.
Pren
tice-
Hal
l, 19
58
Cul
bert
son,
J. T
. Mat
hem
atic
s an
d L
ogic
for
Dig
ital D
evic
es. V
an N
ostr
and,
195
8.
John
son,
Don
ovan
A.,
and
Gle
nn, W
illia
mH
. Com
putin
g D
evic
es. W
ebst
er, 1
961.
pp. 3
7-51
New
man
, Jam
es. T
he W
orld
of
Mat
he-
mat
ics.
Sim
on a
nd S
chus
ter,
195
6. p
p.20
70-2
123
15
PROBABILITY, RISK, AND INSURANCE
UN
IT I
IPR
OB
AB
ILIT
Y, R
ISK
, AN
D I
NSU
RA
NC
E
Intr
oduc
tion
Thi
s un
it pr
opos
es th
e us
e of
the
mat
hem
atic
s of
pro
babi
lity
to m
ake
prac
tical
the
conc
epts
of
insu
ranc
e. E
ach
stud
ent h
asbe
en
or w
ill b
e fa
ced
with
the
prob
lem
of
the
impo
rtan
ce o
f va
riou
sty
pes
of in
sura
nce
bein
g so
ld f
or p
rote
ctio
nag
ains
t 100
1 th
ings
, inc
ludi
ng
the
risk
s of
life
itse
lf. T
he s
tude
nts
shou
ld n
ot o
nly
be m
ade
awar
e of
the
risk
s in
volv
ed in
man
y ev
ents
of
life,
but
also
be
mad
e to
real
ize
the
prob
abili
ty o
f ch
ance
as
it af
fect
sth
e su
cces
s or
fai
lure
of
each
eve
nt. M
any
inte
rest
ing
mat
hem
atic
al e
xper
imen
ts s
houl
dbe
cond
ucte
d by
the
stud
ents
so
that
they
may
see
firs
t han
d th
e re
sults
of
chan
ce. A
ll of
the
expe
rim
ents
with
pro
babi
lity
mus
tbe
clos
ely
guid
ed b
y th
e te
ache
- so
as
to tr
ansm
itm
eani
ng to
the
stud
y of
var
ious
type
s of
insu
ranc
e. I
t mig
ht b
e ad
visa
ble
to d
ivid
e
the
clas
s in
to w
ork
grou
ps f
orex
peri
men
ting
with
var
ious
eve
nts
conc
erni
ngpr
obab
ility
. The
se g
roup
s w
ould
then
be
resp
onsi
ble
for
repo
rtin
g th
eir
resu
lts to
the
who
le c
lass
.Man
y co
ncep
ts c
once
rnin
g th
e co
llect
ion
of d
ata
can
be ta
ught
in th
is u
nit t
hat w
illbe
hel
pful
in te
achi
ng U
nit I
V. P
rovi
de th
e cl
ass
with
as
man
y te
achi
ng a
ids,
act
iviti
es,
and
reso
urce
peo
ple
as p
ossi
ble
to m
ake
the
mat
eria
l
mea
ning
ful.
Obj
ectiv
es1.
To
reco
gniz
e th
e ri
sks
in d
aily
life
and
lear
n ho
w to
mee
t the
se r
isks
2. T
o le
arn
how
to d
eter
min
e th
epr
obab
ility
of
an e
vent
3. T
o le
arn
the
role
of
prob
abili
ty in
indu
stry
, sci
ence
, gov
ernm
ent,
and
part
icul
arly
in in
sura
nce
4. T
o le
arn
the
purp
ose,
type
s, a
ndbe
nefi
ts o
f va
ried
insu
rauc
e :;o
ntra
cts
5. T
o bu
ild c
ompe
tenc
e an
d in
tere
st in
inde
pend
ent r
eadh
ig o
f m
athe
mat
ical
mat
eria
l and
bus
ines
s co
ntra
cts
Con
tent
Proc
edur
es a
nd A
ctiv
ities
A. I
ntro
duct
ion
1. T
he )
urpo
se a
nd o
bjec
tives
of
Dis
cuss
the
obje
ctiv
es o
f th
e un
it. U
se th
efi
lm
this
uni
tH
ow' s
Cha
nces
as
an in
trod
uctio
n
2. P
revi
ew o
f th
e to
pics
and
ctiv
-iti
es o
f th
e un
ita.
The
ris
ks in
the
wor
ldin
whi
chw
e liv
eb.
The
det
erm
inat
ion
of r
isk
byex
peri
men
tc.
The
det
erm
inat
ion
of A
sk b
yan
alys
is
3. A
ssig
nmen
ts o
f un
it le
arni
ng a
c-tiv
ities
Rea
d se
ctio
n of
How
to T
ake
A C
hanc
e.Il
lust
rate
the
topi
c w
ith h
isto
rica
l inc
iden
ts
Tea
chin
g A
ids
Film
: How
's C
hanc
es
Boo
k: H
uff,
Dar
rell.
How
to T
ake
a O
zanc
e
Pam
phle
t: Jo
hnso
n, D
onov
an. P
roba
bilit
y,R
isk
and
Cha
nce
Hav
e st
uden
ts li
st th
e ri
sks
of th
e w
orld
inw
hich
New
s an
d da
ta f
rom
new
spap
ers
and
pe-
we
live:
acc
iden
t,ill
ness
, wea
ther
, cat
astr
ophe
,ri
odic
als
spor
ts, p
urch
ases
Find
,exa
mpl
es o
f th
e ri
sks
in th
e ne
ws
of th
e da
y
Con
side
r w
hat o
ccup
atio
ns o
r in
dust
ries
invo
lve
chan
ce a
nd w
hat r
isks
or
chan
ces
are
invo
lved
:po
litic
ian,
far
mer
, law
yer,
teac
her,
bio
logi
st,
wea
ther
for
cast
er, p
ilot,
doct
or, f
or e
xam
ple
Out
line
the
sour
ce o
f in
form
atio
n fo
r th
eun
it,th
e as
sign
men
ts a
nd r
espo
nsib
ilitie
sof
the
stud
ents
17
Bul
letin
boa
rd d
ispl
ay o
n pr
obab
ility
or
insu
ranc
e
Cha
rts:
Fam
ily N
eeds
for
Lif
e In
sura
nce
a. P
erfo
rmex
peri
men
ts a
nd k
eep
a re
cord
of
the
resu
lts in
a n
ote-
book
b. C
olle
ct a
rtic
les,
new
s re
port
s,or
adve
rtis
emen
ts r
elat
ed to
chan
ce a
nd in
sura
nce
c. C
olle
ctin
sura
nce
polic
ies
for
anal
ysis
and
stu
dyd.
Mak
e a
disp
lay,
mod
el,e
xhib
it,or
gam
eill
ustr
atin
g ch
ance
,pe
rmut
atio
ns, m
athe
mat
ical
expe
rim
enta
tion
or in
sura
nce
B. P
roba
bilit
yThe
Mea
sure
of C
hanc
e1.
Wha
t is
chan
ce?
Wha
t is
the
defi
nitio
n of
prob
abil-
ity?
Wha
t are
equ
ally
like
ly e
vent
s?
2. H
ow is
the
prob
abili
tyof
an
even
t by
expe
rim
enta
tion
dete
r-m
ined
?a.
Equ
ally
likel
y ev
ents
A s
tude
nt n
oteb
ook
for
supp
lem
enta
ry m
ate-
rial
s ca
n be
ver
y us
eful
A c
olle
ctio
n of
pic
ture
s,in
sura
nce
polic
ies
orne
ws
mat
eria
l,m
ount
ed a
nd la
bele
d ca
n be
an
appr
opri
ate
part
of
this
note
book
Hav
e a
vari
ety
of p
amph
lets
and
book
s on
pro
b-ab
ility
and
insu
ranc
e in
the
clas
sroo
m
Mak
e a
bulle
tin b
oard
dis
play
with
obj
ects
whi
chca
n be
man
ipul
ated
to f
orm
per
mut
atio
ns
Hav
e an
insu
ranc
ere
pres
enta
tive,
a h
ighw
aypa
trol
man
, or
scie
ntis
t des
crib
eth
e ro
le o
f pr
ob-
abili
ty o
r in
sura
nce
in th
eir
wor
k
Beg
in w
ith a
dis
cuss
ion
of c
hanc
eD
isco
ver
a w
ay to
mea
sure
chan
ceG
ener
aliz
e to
the
form
ula
s
fs
or n
umbe
r of
way
sfo
r a
succ
essf
ul e
vent
to o
ccur
divi
ded
by th
e to
tal n
umbe
rof
way
s th
e ev
ent
may
occ
ur.
Est
imat
e th
e pr
obab
ility
of
ava
riet
yof
eve
nts
such
as
toss
edco
ins,
dic
e, o
r ca
rds
If th
ese
obje
cts
are
not
appr
opri
ate
for
use
inyo
ur c
omm
unity
use
spin
dial
s,do
min
oes,
chec
kers
, or
hom
e m
ade
card
s or
eve
nfl
ash
card
s
Mak
e re
gula
r po
lyhe
dron
ssu
ch a
s oc
toge
n or
dode
cahe
dron
or
pyra
mid
s fo
r us
ein
stea
d of
dice
Hav
e th
e st
uden
ts d
eter
min
ew
hat t
he p
rob-
abili
ty s
houl
d be
for
eac
h ev
ent.
Thi
s sh
ould
be
emph
asiz
ed
Exp
erim
ent w
ith th
e to
ssin
g of
coin
s, c
ards
,bl
ocks
, dic
e, c
heck
ers.
Spi
n di
alF-
,to
ps, b
alls
Dra
w m
arbl
es, c
ards
, dom
inoe
s,le
tters
, -fo
r ex
-ex
ampl
e, f
rom
bag
s
Wri
te th
e pr
obab
ility
of
the
even
ts18
Spea
ker:
pat
rolm
an, i
nsur
ance
rep
rese
n-ta
tive,
or
rese
arch
sci
entis
t
.
Obj
ects
for
ana
lysi
s of
eve
nts:
coi
ns,
dice
,ca
rds,
pol
yhed
ra, s
pin
dial
, thu
mb
tack
s
John
son,
Don
ovan
. Pra
bcbi
lity,
Ris
k,an
dC
hanc
e
Mos
telle
r, R
ourk
e, a
nd T
hom
as.
Prob
-ab
ility
and
Sta
tistic
s
Seco
nd y
ear
alge
bra
text
book
with
sect
ion
on p
roba
bilit
y
b. E
vent
s no
t equ
ally
like
ly
3. D
eter
min
e th
e po
ssib
le w
ays
dif-
fere
nt e
vent
s ca
n oc
cur
4. T
he c
ompa
riso
n of
pre
dict
edev
ents
and
sam
ple
resu
lts
C. P
roba
bilit
y an
d E
very
day
Eve
nts
1. T
he in
terp
reta
tion
of p
roba
oilit
y
The
law
of
aver
ages
Com
pare
the
obta
ined
pro
babi
lity
with
the
esti-
mat
e of
pro
babi
litie
s
Thi
s sh
ould
be
emph
asiz
ed to
sho
w m
athe
mat
i-ca
lly h
ow o
ne c
an c
lose
ly d
eter
min
e th
e ac
tual
resu
lt
Exp
erim
ent w
ith e
vent
s no
t equ
ally
like
ly s
uch
as to
ssin
g a
thum
b ta
ck, t
ossi
ng a
dou
ble
dice
(tw
c',i
ce g
lued
toge
ther
), o
dd s
hape
d bl
ocks
,sp
inni
ng a
dia
l, an
d ir
regu
lar
poly
hedr
ons-
-gu
ess
firs
t wha
t the
pro
babi
lity
mig
ht b
e
Mak
e an
ana
lysi
s of
pos
sibl
e ev
ents
to d
eter
min
eth
e po
ssib
le e
vent
sa.
one
coi
n, tw
o co
ins,
thre
e co
ins
b. o
ne d
ice,
one
coi
n an
d on
e di
ce, t
wo
dice
c. m
arbl
es in
a b
agd.
fam
ilies
of
one
child
, tw
o ch
ile-
en, t
hree
child
ren
a. M
ake
pred
ictio
ns o
f ev
ents
bas
ed o
n th
epo
ssib
le e
vent
s lis
ted.
Com
pare
the
pred
ic-
tions
with
act
ual r
esul
ts in
toss
ing
coin
s,di
ce, d
raw
ing
mar
bles
or
card
s fr
om a
bag
b. C
ompa
re p
redi
ctio
ns o
f w
eath
er o
r sp
orts
even
ts w
ith th
e ac
tual
eve
nts
c. C
ompa
re th
e pr
edic
tions
of
even
ts n
oteq
ually
like
ly s
uch
as ta
ck to
ssin
gd.
Com
pare
the
pred
ictio
ns o
f co
mpo
und
even
ts s
uch
as to
ssin
g a
tota
l of
7 w
ith tw
odi
ce w
ith o
btai
ned
resu
lts.
Com
poun
dev
ents
can
invo
lve
any
two
of th
e ob
ject
sof
the
expe
rim
ents
des
crib
ed a
bove
Com
pare
the
resu
lts o
f ex
peri
men
ts o
f in
divi
dual
sin
coi
n to
ssin
g w
ith th
e co
mbi
ned
resu
lts o
f al
lto
sses
to il
lust
rate
the
law
of
aver
ages
Find
pro
babi
litie
s of
eve
nts,
suc
h as
wea
ther
,th
umb
tack
toss
es, p
ropo
rtio
n of
e's
in a
par
a-gr
aph,
car
ds in
a d
eck
and
com
pare
the
pred
ic-
tions
with
res
ults
obt
aine
d by
sm
all a
nd la
rge
num
bers
19
Scho
ol M
athe
mat
ics
Stud
y G
roup
Mat
he-
mat
ics
for
Juni
or H
igh
Scho
olV
olum
e II
,Pa
rt I
I
Che
ck th
e ne
wsp
aper
s fo
r in
form
atio
n on
pred
ictio
ns a
nd la
ter
for
the
resu
lts. U
sew
eekl
y w
eath
er p
redi
ctio
ns a
nd p
redi
ctio
nsof
foo
tbal
l gam
es
2. T
he m
orta
lity
tabl
e an
dth
e pr
ob-
abili
ty o
f de
ath
Use
the
mor
talit
y ta
ble
topr
edic
t dea
th a
t agi
ven
age
Com
pile
dea
th r
ecor
ds to
com
pare
the
tabl
epr
edic
tions
with
act
ual e
vent
s.Pr
edic
t the
num
-be
r of
cla
ss m
embe
rs w
ho a
relik
ely
to d
ie in
5,
10, o
r 15
yea
rs
3. A
ccid
ent r
ates
and
pred
ictic
n of
Com
pare
the
stat
e ac
cide
nt r
ates
with
the
pre-
acci
dent
sdi
ctio
ns f
or th
e w
eeke
nd, m
onth
or
year
Pred
ict t
he n
umbe
r of
acc
iden
tsex
pect
ed f
orth
e cl
ass
mem
bers
dur
ing
teen
age
driv
ing
Mak
e a
surv
ey o
f th
e co
mm
unity
to c
ompa
reac
tual
acc
iden
t rat
es w
ith th
epr
edic
ted
rate
4. P
roba
bilit
y an
d od
dsC
ompa
re th
e pr
obab
ility
with
odds
in c
oin
toss
-in
g, ta
ck to
ssin
g, d
ice
toss
ing,
dial
spi
nner
s,ac
cide
nt r
ates
, car
d dr
awin
g,bi
rths
, dai
ly e
vent
s
D. P
erm
utat
ions
or th
e w
ays
anev
ent
may
occ
ur1.
The
per
mut
atio
ns o
fal
l mem
bers
of a
set
Perm
utat
ions
by
expe
rim
ent
Perm
utat
ions
ccm
pute
d by
form
ula
The
mea
ning
of
fact
oria
l
2. T
he p
erm
utat
ions
of
som
e m
em-
bers
of
a se
t
Perm
utat
ion
by e
xper
imen
t
Com
bina
tions
com
pute
d by
afo
rmul
a
Use
stu
dent
s, le
tters
, boo
ks, d
igits
,spo
rts
equi
p-m
ent,
gift
s, g
ame
assi
gnm
ents
,dr
ess
acce
ssor
ies,
hous
ehol
d fu
rnis
hing
s, c
ards
, to
form
per
mut
a-tio
ns u
sing
all
mem
bers
of
a se
t
Exa
mpl
e: U
se a
foo
tbal
l tea
m1.
Det
erm
ine
all t
he p
ossi
ble
way
s a
star
ting
11 c
ould
be
arra
nged
2. D
eter
min
e al
l the
pos
sibl
e w
ays
33 p
laye
rsco
uld
be u
sed
by k
eepi
ng th
emin
thei
rri
ght p
ositi
ons
Col
lect
dat
a to
fur
nish
a p
atte
rn f
ordi
scov
erin
gth
e fo
rmul
a
Com
pute
fac
tori
als
and
use
it to
mak
e a
tabl
eof
fac
tori
als
Use
the
sam
e m
ater
ials
as
innu
mbe
r on
e to
dete
rmin
e th
e nu
mbe
r of
per
mut
atio
nsfo
r so
me
mem
bers
of
a se
t
20
Mor
talit
y ta
ble
Gat
her
this
dat
a fo
r at
leas
t one
mon
thfr
om a
maj
or n
ewsp
aper
Insu
ranc
e da
ta o
r W
orld
Alm
anac
Dat
a on
acc
iden
ts a
nd d
eath
Con
tact
you
r lo
cal p
olic
e de
part
men
t
Vita
l sta
tistic
s of
loca
l new
spap
er
3. T
he c
ombi
natio
ns o
fob
ject
s
Com
bina
tions
by
expe
rim
ent
Com
bina
tions
com
pute
d by
a,f
orm
ula
E. T
he P
roba
bilit
y of
Tw
o or
Mor
eE
vent
s1.
The
pro
babi
lity
of tw
oev
ents
occu
rrin
g to
geth
er o
r si
mul
tane
-ou
sly
(The
inte
rsec
tion
of -
two
even
ts A
and
B)
2.ile
pro
babi
lity
of o
ne e
vent
or
anot
her
even
t (T
he u
nion
of
two
even
ts A
or
B)
F. R
ando
m S
ampl
ing
and
Prob
abili
ty1.
Way
s of
3el
ectin
gsa
mpl
es
Use
the
sam
e m
ater
ial a
sin
num
ber
one
to il
lus-
trat
e th
e di
ffer
ence
betw
een
perm
utat
ions
and
com
bina
tions
Mat
eria
l and
topi
cs o
npe
rmut
atio
ns a
nd c
om-
bina
tions
can
be
exte
nded
to m
ore
adva
nced
topi
cs f
or h
igh
abili
tyst
uden
ts. T
his
topi
c is
suita
ble
for
a va
riet
y of
orig
inal
illu
stra
tions
and
com
mun
ity a
pplic
atio
ns
Form
the
latti
ce f
or th
e su
mof
two
dice
Find
the
prob
abili
ty o
fto
ssin
g a
give
n su
m
Com
pare
this
pro
babi
lity
with
expe
rim
enta
l re-
sults
.
Wri
te th
e po
ssib
le e
vent
sfo
r to
ssin
g a
coin
and
a di
e
Find
the
prob
abili
ty o
f a
give
n co
mbi
natio
nC
ompa
re th
e co
mpu
ted
prob
abili
ty w
ith e
xper
i-m
enta
l res
ults
Find
the
prob
abili
ty o
f ev
ents
for
seve
ral c
oins
or o
ther
obje
cts
such
as
two
dial
spin
ners
or
two
card
pac
ks
Dic
e, p
olyh
edra
, irr
egul
arsh
aped
obj
ects
Use
the
even
ts d
escr
ibed
in n
umbe
r on
e ab
ove
Dou
ble
dice
for
med
by
glui
ng tw
o di
ce
to il
lust
rate
the
unio
nof
two
even
tsto
geth
er
Glu
e tw
o di
ce to
geth
er a
nd to
ss
Col
lect
dat
a fo
r th
e to
tals
turn
ed u
p
Com
pare
thes
e re
sults
with
unat
tach
ed d
ice
toill
ustr
ate
depe
nden
t and
inde
pend
ent e
vent
s
Col
lect
info
rmat
ion
from
scie
nce
and
curr
ent
new
s of
c3m
poun
dev
ents
Exp
erim
ents
in s
elec
ting
sam
ples
21
Form
sev
eral
set
s of
dou
ble
dice
by
glui
ngdi
ffer
ent f
aces
toge
ther
Tab
le o
f ra
ndom
num
bers
Why
sam
plin
g is
nee
ded,
e.g;
food
, cro
ps, r
ain,
T.V
. aud
ienc
e,so
il
2. T
he r
ando
m s
elec
tion
of s
ampl
es
3. R
ando
m n
umbe
rs
4. R
ando
m s
ampl
es a
nd p
redi
ctio
n
G. A
pplic
atio
n of
Pro
babi
lity
1. M
athe
mat
ical
exp
ecta
tion
the
prod
uct o
f a
prob
abili
ty a
nd a
mea
sure
of
mon
etar
y va
lue
The
for
mul
a fo
r m
athe
mat
ical
expe
ctat
ion
The
use
of
mat
hem
atic
al e
xpec
ta-
tions
in in
dust
ry, l
otte
ries
, far
m-
ing,
insu
ranc
e
The
Min
neso
ta P
ollM
inne
apol
is T
ribu
neD
iscu
ss r
easo
ns w
hy s
mal
l sam
ples
are
take
n.D
iscu
ss w
ays
this
sho
uld
be d
one
to g
et b
est
resu
lts
The
com
pari
son
of s
ampl
es r
esul
ts a
nd th
e po
pu-
latio
n ch
arac
teri
stic
s by
coi
n to
ssin
g, c
ard
draw
-in
g, d
aily
sta
tistic
s
Sam
ple
rand
om n
umbe
rs a
nd c
ompa
re th
e re
-su
lts.
Poss
ible
activ
ity: H
ave
a la
rge
popu
latio
nof
num
eral
s in
a b
ox. S
elec
t sam
ples
fro
m th
ebo
x. S
tudy
the
smal
l sam
ples
. Com
pare
the
re-
sults
of
man
y sm
all s
ampl
es w
ith th
e re
sults
of
the
who
le p
opul
atio
n
Surv
ey th
e st
uden
ts b
y sa
mpl
es o
f bi
rthd
ays,
wei
ghts
, hei
ghts
, T.V
. vie
N in
g, o
utsi
de a
ctiv
ities
,ho
bbie
s, a
llow
ance
s, f
or e
xam
ple.
Cou
nt le
tters
or
vow
els
in s
ampl
e pa
ragr
aphs
and
com
pare
res
ults
Find
new
s ite
ms
or a
dver
tisem
ents
quo
ting
sam
ple
resu
lts. S
tudy
com
mun
ity p
robl
ems
orop
inio
ns s
uch
as tr
affi
c by
sam
plin
g
Find
out
how
pub
lic o
pini
on p
olls
sel
ect s
ampl
esSp
eake
r: P
ublic
opi
nion
pol
l wor
ker
Det
erm
ine
the
mat
hem
atic
al e
xpec
tatio
n fo
r a
Cro
p re
port
s of
U.S
. Agr
icul
ture
Dep
art-
vari
ety
of e
vent
s in
spo
rts,
lotte
ries
, ins
uran
ce,
men
tcr
ops,
col
lege
edu
catio
n
If a
per
son
is to
rec
eive
"M
" do
llars
in c
ase
ace
rtai
n ev
ent o
ccur
s, a
nd if
the
prob
abili
ty th
atth
e ev
ent w
ill o
ccur
is "
p" th
en th
e va
lue
of h
isex
pect
atio
n is
"M
p" d
olla
rs
2. U
ses
of p
roba
bilit
y an
d sa
mpl
ing
Surv
ey th
e cl
ass
to d
eter
min
e ex
pect
ed p
urch
ases
in in
dust
ryof
gas
olin
e, c
andy
, sho
es, o
ther
art
icle
s
22
INK
Inve
ntor
ies,
con
stru
ctio
n, a
dver
-tis
ing,
qua
lity
cont
rol
3. U
ses
of p
roba
bilit
y in
sci
ence
Err
ors
of m
easu
rem
ent
Dis
trib
utio
n of
rai
ndro
ps, l
eave
s,pe
bble
s
Seed
pol
linat
ion
and
crop
s
Popu
latio
n pr
edic
tions
Supe
rstit
ion
and
rand
om e
vent
s
4. U
ses
of p
roba
bilit
y an
dsa
mpl
esin
pol
itics
Publ
ic o
pini
on p
olls
Prej
udic
e an
d de
cisi
on
5. P
roba
bilit
ies
and
aver
ages
Exp
ecte
d de
viat
ions
in w
eigh
ts,
heig
hts,
test
sco
res
Dev
iatio
ns in
rea
ctio
n tim
e,st
reng
th, j
umps
, hea
rt b
eat
H. L
ife
Insu
ranc
e1.
The
pur
pose
of
life
insu
ranc
em
eetin
g th
e ri
sks
of d
eath
Sam
ple
a pr
oduc
t suc
h as
cha
lk, b
all
poin
t pen
s,pa
per
to p
redi
ct th
equ
ality
of
the
prod
uct
sam
ple
the
info
rmat
ion
and
read
ing
habi
ts o
fth
e cl
ass
to d
eter
min
e th
eef
fect
iven
ess
of a
d-ve
rtis
ing,
pub
licity
, com
mun
ityin
form
atio
n
Mak
e a
seri
es o
f m
easu
rem
ents
of
a se
tof
dig
itsan
d ta
bula
te th
e re
sults
to s
how
the
rang
e of
erro
rs
Mak
e di
stri
butio
n of
mea
sure
s of
suc
hite
ms
asle
aves
, ker
nels
of
corn
on
a co
b, li
nes
in s
hellf
ish,
shee
ts o
f pa
per
in a
pur
chas
ed r
eam
,for
exa
mpl
e
Mak
e an
opi
nion
pol
l of
clas
s m
embe
rs,
teac
hers
,or
par
ents
on
som
esc
hool
pol
icy
or s
uper
stiti
on
Com
pile
mea
sure
s of
cla
ss m
embe
rs to
com
pare
the
dist
ribu
tions
with
pop
ulat
ion
norm
s
If a
ppro
pria
te e
xten
d th
e id
eas
ofm
athe
mat
ical
expe
ctat
ion
to g
ambl
ing
devi
ces
tote
ach
that
gam
blin
g do
es n
ot p
ay. T
his
mus
t be
done
with
extr
eme
caut
ion
and
prop
erex
ampl
es. C
om-
mun
ity o
pini
on w
ill n
eed
to b
e ch
ecke
dto
det
er-
min
e w
heth
er th
ese
activ
ities
are
feas
ible
The
topi
cs a
nd p
robl
ems
of th
is u
nit a
redi
scus
sed
with
exe
rcis
es in
the
pam
phle
t:T
he M
athe
-m
atic
s of
Lif
e In
sura
nce.
Thi
s pa
mph
let
coul
dw
ell b
e th
e te
xt f
or th
is u
nit
Dis
cuss
the
role
of
life
insu
ranc
e in
term
s of
mee
ting
the
need
s of
fam
ily f
or s
uppo
rt,
educ
a-
23
Obj
ects
for
sam
plin
g an
d m
easu
rem
ent,
incl
udin
g co
rnco
bs, s
ea s
hells
, peb
bles
,fo
r ex
ampl
e
Polic
e de
mon
stra
tion
of g
ambl
ing
devi
ces
as u
sed
indi
spat
chin
g pa
trol
car
s
Film
stri
p: H
ow L
tfe
Insu
ranc
e B
egan
How
Lif
e In
sura
nce
Ope
rate
s
Pam
phle
t: T
he M
athe
mat
ics
of L
ife
In-
sura
nce
Lif
e In
sura
nce
Fact
Boo
k
Cha
rt: F
amily
Nee
ds f
or L
ife
Insu
ranc
e
2. T
he ty
pes
of li
fe in
sura
nce-
-ter
m,
ordi
nary
life
, end
owm
ent,
paid
-up
insu
ranc
e
3. T
he m
orta
lity
tabl
e an
d th
e co
stof
insu
ranc
e
4. T
he b
enef
it pa
ymen
ts o
f lif
e in
-su
ranc
eca
sh s
urre
nder
val
uepa
id-u
p in
sura
nce
paym
ent a
t dea
thre
tirem
ent p
aym
ents
read
ing
the
smal
l pri
nt o
f th
epo
licy
5. L
ife
insu
ranc
e as
an
inve
stm
ent
prog
ram
min
g lif
e in
sura
nce
getti
ng th
e m
ost o
ut o
f yo
urin
sura
nce
dolla
r
tion,
or
busi
ness
est
ablis
hmen
t. Il
lust
rate
with
spec
ific
fam
ily s
ituat
ion
Det
erm
ine
the
cost
s in
volv
ed w
hich
sho
uld
de-
teri
nine
the
amou
nt o
f in
sura
nce
need
ed
Illu
stra
te th
e ty
pes
of li
fe in
sura
nce
with
pol
icie
sL
ife
insu
ranc
e po
licie
sfu
rnis
hed
by s
tude
nts
Dis
cuss
the
funa
ion
of te
rm, o
rdin
ary
life,
en-
dow
men
t, or
oth
er p
olic
ies
Rel
ate
the
cost
of
insu
ranc
e to
mat
hem
atic
alT
he m
orta
lity
tabl
eex
pect
atio
n
Rel
ate
insu
ranc
e ra
tes
to c
hang
es in
life
spa
n,ac
cide
ntal
dea
ths
and
incr
ease
in v
alue
of
insu
r-an
ce c
ompa
ny in
vest
men
ts
Use
spe
cifi
c in
sura
nce
polic
ies
to c
ompu
te b
enef
itFi
lmst
rip:
How
Lif
e In
sura
nce
Polic
ies
Wor
kpa
ymen
ts
Com
pare
the
bene
fits
with
pre
miu
m p
aym
ents
Com
pare
the
amou
nt r
ecei
ved
unde
r ea
ch b
ene-
fit p
lan
Find
the
exce
ptio
ns f
or w
hich
no
bene
fit p
ay-
men
ts a
re m
ade
Dis
cuss
the
spec
ific
doc
umen
ts n
eede
d to
qua
lify
for
bene
fits
Com
pare
insu
ranc
e be
nefi
ts w
ith in
com
e fr
omFi
lmst
rip:
Pla
nnin
g Fa
mily
Lif
e In
sur-
inve
stm
ents
Rel
ate
the
cost
of
insu
ranc
e to
the
purp
ose
for
whi
ch in
sura
nce
is p
urch
ased
Con
side
r ap
prop
riat
e bu
dget
allo
catio
ns to
life
insu
ranc
e
The
topi
c of
per
sona
l bud
gets
and
rec
ord
keep
ing
is_d
iscu
ssed
in a
late
r un
it in
mor
e de
tail
ance
I.R
eal P
rope
rty
Insu
ranc
e1.
The
pur
pose
of
prop
erty
insu
r-an
ce
2. S
tatis
tics
and
insu
ranc
e ra
tes
3. T
ypes
and
rat
es o
f po
licie
sre
adin
g th
e sm
all p
rint
settl
ing
insu
ranc
e cl
aim
s (t
hero
le o
f ad
just
men
t)
Bri
ng a
life
insu
ranc
e re
pres
enta
tive
to th
e cl
ass
to d
iscu
ss p
olic
ies
and
answ
er q
uest
ions
Vis
it th
e of
fice
of
a la
rge
insu
ranc
e co
mpa
ny
Org
aniz
e a
mut
ual l
ife
insu
ranc
e co
mpa
ny f
orth
e cl
ass
Use
the
mor
talit
y ta
ble
and
oper
atio
n co
sts
as a
basi
s fo
r de
term
inin
g ra
tes
Wri
te a
sam
ple
polic
y
Est
ablis
h a
mea
ns o
f re
cord
kee
ping
illus
trat
e th
e ne
ed f
or p
rope
rty
insu
ranc
e in
Spea
ker:
Ins
uran
ce c
ompa
ny r
epre
sent
a-tiv
eer
ms
of c
omm
unity
eve
nts
How
is in
sura
nce
on a
mov
ie s
tar
sim
ilar
topr
oper
ty, i
nsur
ance
?
Wha
t are
exa
mpl
es o
f th
e m
easu
re o
f pr
oper
tyin
sura
nce?
Whe
re a
nd u
nder
wha
t cir
cum
stan
ces
is p
rop-
erty
insu
ranc
e re
quir
ed?
Wha
t is
the
prob
abili
ty o
f pr
oper
ty d
amag
e be
-ca
use
of f
ire,
ear
thqu
akes
, flo
od, t
ides
, sto
rms,
light
ning
, fal
lout
, dus
t, in
sect
s, d
ecay
, hea
t,fr
eezi
ng, p
lant
life
?
Wha
t kin
ds o
f in
sura
nce
polic
ies
can
be o
b-ta
ined
? Il
lust
rate
with
exa
mpl
es
Film
: Ins
uran
ce A
gain
st F
ire
Los
ses
For
wha
t kin
d of
des
truc
tion
is it
impo
ssib
le to
insu
re?
Use
pol
icie
s to
illu
stra
te e
xclu
sion
s
Rel
ate
cost
to m
athe
mat
ical
exp
ecta
tion
of lo
ssPr
oper
ty in
sura
nce
polic
ies
Det
erm
ine
wha
t doc
umen
ts, e
stim
ates
, or
stat
e-m
ents
are
req
uire
d fo
r co
llect
ion
Dis
cuss
the
role
and
cos
t of
adju
ster
s, a
ttorn
eys,
agen
ts, c
ourt
set
tlem
ent 25
4. P
rogr
amm
ing
prop
erty
insu
ranc
e
J. A
utom
obile
Ins
uran
ce1.
The
pur
pose
of
auto
mob
ile in
sur-
ance
2. T
he ty
pes
of p
olic
ies
liabi
lity
colli
sion
]co
mpr
ehen
sive
med
ical
_pay
men
t
How
do
com
pani
es s
hare
ris
ks?
How
do
com
pani
es m
ake
it im
poss
ible
topr
ofit
from
pro
pert
y da
mag
e?
How
can
insu
ranc
e ta
xes
be r
educ
ed b
y pr
oper
tylo
sses
?
Rel
ate
insu
ranc
e co
sts
to r
isks
and
bud
gets
Org
aniz
e a
casu
alty
insu
ranc
e co
mpa
ny f
or c
lass
mem
bers
Insu
re a
gain
st r
isks
, suc
h as
loss
or
brea
kage
of
book
s, p
enci
ls, g
lass
es, s
port
s eq
uipm
ent,
dini
ngha
ll di
shes
Col
lect
dat
a to
det
erm
ine
the
risk
s
Cal
cula
te th
e m
athe
mat
ical
exp
ecta
tion
and
dete
rmin
e in
sura
nce
rate
s
Det
erm
ine
basi
s fo
r pr
oof
of lo
ss
Col
lect
a v
arie
ty o
f pa
mph
lets
, cha
rts,
dat
a, a
ndpo
licie
s to
illu
stra
te p
robl
ems
in a
utom
obile
insu
ranc
e
Use
film
s, in
sura
nce
repr
esen
tativ
es, o
rtr
affi
cof
fice
rs to
pre
sent
the
prob
lem
to th
e cl
ass
Illu
stra
te th
e pu
rpos
e of
aut
omob
ile in
sura
nce
in te
rms
of lo
cal a
ccid
ents
Dis
cuss
why
insu
ranc
e is
man
dato
ry in
som
est
ates
Deb
ate
the
issu
e of
req
uire
d ve
rsus
opt
iona
l in-
sura
nce,
sta
te v
ersu
s pr
ivat
ein
sura
nce
Film
: Cas
ualty
Ins
uran
ce
Spea
ker:
Tra
ffic
off
icer
Use
pol
icie
s to
illu
stra
te th
e va
riou
s ty
pes
ofA
utom
obile
insu
ranc
e po
licie
sin
sura
nce
26
3. S
tatis
tics
and
insu
ranc
e ra
tes
4. P
rogr
amm
ing
auto
mob
ile in
sur-
ance
K. H
ealth
and
Acc
iden
t Ins
uran
ce1.
The
impo
rtan
ce o
f he
alth
and
acci
dent
insu
ranc
es
2. T
he ty
pes
of m
edic
al in
sura
nce
hosp
ital a
nd s
urgi
cal
med
ical
inco
me
cata
stro
phe
Illu
stra
te b
enef
it pa
ymen
tsan
a th
e ba
sis
for
settl
emen
t of
insu
ranc
e cl
aim
s
Col
lect
dat
a re
gard
ing
loca
l acc
iden
t rat
es
Rel
ate
acci
dent
rat
es to
insu
ranc
e co
sts
atdi
f-fe
rent
age
leve
ls a
nd f
or d
iffe
rent
type
sof
in-
sura
nce
Use
loca
l new
s an
d co
urt d
ecis
ions
toill
ustr
ate
paym
ents
Rel
ate
loca
l acc
iden
t aw
ards
to th
eam
ount
of
insu
ranc
e ne
eded
Rel
ate
cost
of
insu
ranc
e to
aut
omob
ileco
sts
ofop
erat
ion
and
inst
allm
ent p
aym
ents
Com
pare
the
rate
s of
dif
fere
nt in
sura
nce
com
-pa
nies
and
det
erm
ine
the
reas
on f
or th
ese
dif-
fere
nces
. How
are
rat
es r
elat
ed to
the
loca
tion
of r
esid
ence
?
Hav
e st
uden
ts b
ring
insu
ranc
e ad
vert
isem
ents
,po
licie
s, o
r co
ntra
cts
to th
e cl
ass
Dis
cuss
the
cost
s of
ill-
heal
th o
r ac
cide
nts
Rel
ate
the
occu
rren
ce o
f ill
ness
and
acc
iden
t to
loca
l hos
pita
l rat
es a
nd m
edic
al c
osts
Use
pol
icie
s or
con
trac
ts to
illu
stra
te th
e ty
pes
of in
sura
nce
Det
erm
ine
bene
fits
pay
able
Und
er w
hat c
ondi
tions
will
pay
men
t be
mad
e?
Whe
n ar
e be
nefi
ts n
ot p
aid?
Wha
t are
max
imum
ben
efits
?
Wha
t pro
ofs
or d
ocum
ents
mus
t be
furn
ishe
dw
ith c
laim
s?
27
Loc
al a
ccid
ent n
ews
and
cour
t cas
es
Aut
omob
ile in
sura
nce
com
pany
pam
phle
ts
Col
lect
info
rmat
ion
abou
t loc
al h
ospi
tal
char
ges
and
med
ical
cos
ts
Pam
phle
t: So
urce
Boo
k of
Hea
lth I
nsur
-an
ce D
ata
Film
: For
Som
e M
ust W
atch
Hea
lth a
nd a
ccid
ent i
nsur
ance
pol
icie
s
3. S
tatis
tics
and
insu
ranc
e ra
tes
Wha
t is
the
risk
of
illne
ss o
r ac
cide
nt?
4. S
elec
ting
the
appr
opri
ate
insu
r-an
ce
L. G
over
nmen
t Ins
uran
ce1.
The
pur
pose
of
soci
al s
ecur
ity
2. T
he s
ocia
l sec
urity
tax
Wha
t is
the
mat
hem
atic
al e
xpec
tatio
n of
insu
r-an
ce p
olic
ies?
Why
do
insu
ranc
e ra
tes
vary
?
Who
are
una
ble
to o
btai
n he
alth
or
acci
dent
insu
ranc
e?
Why
is a
ir f
light
insu
ranc
e ex
pens
ive?
Col
lect
dat
a to
det
erm
ine
insu
ranc
e pr
emiu
ms
paid
by
empl
oyer
s or
bus
ines
s es
tabl
ishm
ents
Wha
t kin
d of
exa
min
atio
ns o
r in
spir
atio
ns a
rere
quir
ed b
y in
sura
nce
com
pani
es?
Rel
ate
insu
ranc
e co
sts
to s
tand
ards
of
livin
g,bu
dget
s, a
nd r
isk
It is
usu
ally
too
cost
ly to
insu
re f
or a
ll ri
sks.
How
larg
e a
savi
ngs
acco
unt w
ith c
atas
trop
hein
sura
nce
coul
d be
use
d as
a s
ubst
itute
for
hea
lthor
acc
iden
t ins
uran
ce?
The
loca
l off
ice
of th
e So
cial
Sec
urity
Boa
rdw
illsu
pply
you
with
a v
arie
ty o
f pa
mph
lets
, cha
rts,
film
s, a
nd s
peak
ers
abou
t the
topi
c
The
teac
her
will
nee
d to
use
goo
d ju
dgm
ent
inth
e se
lect
ion
of m
ater
ials
, mos
t of
whi
ch a
re n
otm
athe
mat
ical
Why
do
we
need
gov
ernm
ent i
nsur
ance
?
Com
pute
the
soci
al s
ecur
ity ta
x on
dif
fere
ntw
age
paym
ents
Com
pute
the
min
imum
and
max
imum
tax
pay-
men
ts m
ade
by e
mpl
oyee
s
Rel
ate
thes
e pa
ymen
ts to
the
amou
nt o
f re
tire-
men
t or
heal
th in
sura
nce
whi
ch c
ould
1)
e pu
r-ch
ased
by
thes
e am
ount
s fr
om a
com
mer
cial
com
pany
28
Col
lect
dat
a ab
out t
rave
l ins
uran
ce a
t an
airp
ort
Con
tact
sev
eral
insu
ranc
e co
mpa
nies
for
answ
ers
to th
e qu
estio
ns in
this
sec
tion
Pam
phle
ts f
rom
the
Soci
al S
ecur
ity B
oard
Film
s fr
om th
e So
cial
Sec
urity
Boa
rd
3. T
he b
enef
it pa
ymen
tsfa
mily
ben
efits
in c
ase
of d
eath
cash
pay
men
ts a
t dea
thre
tirem
ent b
enef
itsun
empl
oym
ent b
enef
itslia
bilit
y be
nefi
tsm
edic
al p
aym
ents
4. E
ligib
ility
req
uire
men
ts
Hav
e ea
ch s
tude
nt d
eter
min
e th
e po
ssib
lebe
ne-
fit p
aym
ents
for
his
fam
ily
Det
erm
ine
the
requ
irem
ents
fel
' elig
ibili
tyfo
rth
ese
bene
fits
Und
er w
hat c
ondi
tions
are
pay
men
ts m
ade?
Who
is n
ot e
ligib
le f
or b
enef
its?
Ref
eren
ces
Boo
ks Huf
f, D
arre
ll. H
ow to
Tak
e a
Cha
nce.
Nor
ton,
1959
Mos
telle
r, F
re _
2,ri
ck, a
nd o
ther
s. P
roba
bilit
y an
dSt
atis
tics.
Add
ison
-Wes
ley,
196
1Pr
ice,
Ray
, and
Mus
selm
an, V
erno
n. G
ener
alB
usin
ess.
McG
raw
, 195
4R
osen
berg
, R. R
ober
t, an
d L
ewis
, Har
ry. B
usin
ess
Mat
hem
atic
s. M
cGra
w, 1
958
Scho
ol M
athe
mat
ics
Stud
y G
roup
(SM
SG).
Mat
hem
atic
s fo
r Ju
nior
Hig
h Sc
hool
. Vol
ume
II,
Part
II.
Yal
e U
niv.
Pre
ssW
ilhel
ms,
Fre
d T
., an
d H
eim
erl,
Ram
on P
.C
onsu
mer
Eco
nom
ics.
McG
raw
, 195
9
Film
stri
psH
ow L
ife
Insu
ranc
e B
egan
How
Lif
e In
sura
nce
Ope
rate
sH
ow L
ife
Insu
ranc
e Po
licie
s W
ork
Plan
ning
Fam
ily L
ife
Insu
ranc
e
Inst
itute
of
Lif
e In
sura
nce
488
Mad
ison
Ave
nue
New
Yor
k, N
ew Y
ork
Cha
rts
Fam
ily N
eeds
for
Lif
e In
sura
nce
Four
Bas
ic _
Lif
e In
sura
nce
Polic
ies
Free
: Ins
titut
e of
Lif
e In
sura
nce
Film
s Cas
ualty
Ins
uran
ce. E
BF
IA"f
e In
sura
nce-
-Wha
t it M
eans
and
How
it W
orks
.Ass
ocia
ted
Film
sIn
sura
nce
Aga
inst
Fir
e L
osse
s. E
BF
Shar
ing
Eco
nom
ic R
isk.
Cor
onet
Mea
sure
of
a M
ean.
Ass
ocia
ted
Film
s29
Soci
al S
ecur
ity. T
each
ing
Film
s C
usto
dian
sFr
om E
very
Mou
ntai
n Si
de. M
oder
n T
alki
ng P
ictu
re S
ervi
ceH
ow's
Cha
nces
. Ass
ocia
ted
Film
sFo
r So
me
Mus
t Wat
ch. A
ssoc
iate
d Fi
lms
Pam
phle
tsC
liffo
rd, K
ieff
er, a
nd S
obel
. The
Mat
hem
atic
s of
Lif
e In
sura
nce.
Edu
catio
nal D
ivis
ion,
Ins
titut
e of
Lif
e In
sura
nce,
488
Mad
ison
Ave
nue
New
Yor
k 22
, New
Yor
k. T
his
is s
uita
ble
as a
text
for
this
uni
t
John
son,
Don
ovan
. Pro
babi
lity,
Ris
ks, a
nd C
hanc
e.W
ebst
er P
ublis
hing
Com
pany
, 196
3. T
his
is a
sui
tabl
e te
xt f
or th
e pr
obab
ility
por
tion
of th
e un
it
Wha
t Ldf
e In
sura
nce
Mea
nsIn
stitu
te o
f L
ife
Insu
ranc
e, 4
88 M
adis
on A
venu
eN
ew Y
ork
22, N
ew Y
ork
Blu
epri
nt f
or T
omor
row
Edu
catio
nal D
ivis
ion,
Ins
titut
e of
Lif
e In
sura
nce
488
Mad
iSon
Ave
nue,
New
Yor
k 22
, New
Yor
k
Mon
ey M
anag
emen
t: Y
our
Hea
lth a
nd R
ecre
atio
n D
olla
rIn
stitu
te o
f H
ouse
hold
Fin
ance
Cor
pora
tion
Prud
entia
l Pla
za, C
hica
go 1
, Illi
nois
Lif
e In
sura
nce
Fact
Boo
kIn
stitu
te o
f L
ife
Insu
ranc
e, 4
88 M
adis
on A
venu
eN
ew Y
ork
22, N
ew Y
ork
Sour
ce B
ook
of H
ealth
Inst
itute
of
Lif
e In
sura
nce,
488
Mad
ison
Ave
nue
New
Yor
k 22
, New
Yor
k
TNE LANGUAGE OF SCIENCE
UN
IT I
II
TH
E L
AN
GU
AG
E O
F SC
IEN
CE
Intr
oduc
tion
Thi
s un
it is
des
igne
d to
aid
the
stud
ent i
n im
prov
ing
his
skill
s an
d te
chni
ques
of
com
puta
tion.
The
met
hod
used
to a
ttain
this
goal
is th
e in
form
al u
se o
f th
e co
ncep
tsof
mod
ern
alge
bra.
It i
s de
velo
ped
from
the
very
fam
iliar
eve
ryda
y si
tuat
ions
tow
ards
the
mor
e m
athe
mat
ical
appl
icat
ions
. Em
phas
is s
houl
dbe
pla
ced
on th
e di
scov
ery
and
unde
rsta
ndin
g of
eac
h ne
w id
ea a
s it
is ta
ught
.
Man
y of
the
topi
cs m
ay b
e ne
w to
you
,th
e te
ache
r. I
f so
, a li
ttle
hom
ewor
k on
you
r pa
rt w
ill p
repa
re y
ou f
or th
eta
sk. T
his
is o
ne
unit
that
mus
t be
wel
l pla
nned
inad
vanc
e in
ord
er to
kee
p th
eco
ntin
uity
thro
ugho
ut. M
uch
dupl
icat
ion
ofm
ater
ials
will
be
nece
ssar
y.
Mak
e su
re y
our
prob
lem
s co
ver
the
topi
cs a
s yo
u ha
ve p
lann
ed.M
any
of th
e to
pics
are
bas
ic to
an
elem
enta
ry c
ours
ein
alg
ebra
. Mak
e
free
use
of
the
recr
eatio
nal
prob
lem
s re
late
d to
the
unit.
The
y w
ill a
dd m
uch
to th
e in
tere
st o
f th
eun
it. I
nclu
ded
in th
e un
it is
wor
k
with
sig
ned
or d
irec
ted
num
bers
.T
his
shou
ld b
e de
velo
ped
from
the
intr
oduc
tion
of s
igne
d nu
mbe
rs to
the
mas
tery
of
the
four
pro
c-
esse
s of
ari
thm
etic
.T
he w
ork
of th
e en
tire
unit
culm
inat
es in
the
uses
of
mat
hem
atic
sin
sci
entif
ic f
orm
ulas
. Bef
ore
this
uni
t is
begu
n,
muc
h pr
epar
atio
n w
ill b
e re
quir
edin
wri
ting
less
on m
ater
ials
and
prob
lem
s, g
athe
ring
pro
ject
idea
s, a
ndse
arch
ing
for
fres
h id
eas
for
each
less
on.
Obj
ectiv
es1.
To
help
the
stud
ent o
btai
n an
dm
aint
ain
skill
in c
ompu
ting
with
und
erst
andi
ng, a
ccur
acy,
and
eff
icie
ncy
2. T
o fu
rthe
r de
velo
p th
e st
uden
t'sab
ilitie
s an
d te
chni
aues
in g
ener
alpr
oble
m s
olvi
ng
3. T
o fu
rthe
r de
velo
p th
est
uden
t's in
tuiti
on (
shre
wd
gues
sing
)an
d es
timat
ion
4. T
o co
ntin
ue o
r pr
esen
t the
met
hod
of d
isco
very
5. T
o he
lp th
e st
uden
t, w
ith th
e us
eof
sym
bols
and
mat
hem
atic
alco
ncep
ts, m
ake
gene
raliz
atio
ns a
fter
disc
over
ing
a po
ssib
le tr
uth
by v
erif
ying
an
intu
ition
by
expe
rim
enta
tion
(ind
uctiv
e m
etho
d).T
his
shou
ld h
elp
the
stud
ent t
o be
com
e m
ore
"sym
bol"
min
ded.
Con
tent
A. H
ow w
e m
ake
use
of s
ymbo
ls1.
Sym
bols
use
d in
eve
ryda
ysi
tua-
tions
Proc
edur
es a
nd A
ctiv
ities
One
met
hod
of g
ettin
g th
is u
nit
star
ted
wou
ldbe
to p
rint
the
nam
e of
som
est
uden
t, sa
y"J
ohn,
" on
the
boar
d an
d as
k th
ecl
ass
if th
atis
Joh
n. M
ost o
f th
e cl
ass
will
agr
eeth
at w
hat
you
put o
n th
ebo
ard
is J
ohn.
The
n as
k so
me
stud
ent t
o co
me
to th
e fr
ont o
f th
e ro
om.
Whe
nth
e st
uden
t arr
ives
at t
he f
ront
,as
k hi
m to
shak
e ha
nds
with
Joh
n. I
t sho
uld
not t
ake
the
clas
s ve
ry lo
ng to
dis
cove
r w
hat y
ou m
ean,
that
is, w
hat y
ou p
ut o
n th
e bo
ard
was
not
John
, but
a gr
oup
of f
our
sym
bols
rep
rese
ntin
g th
e id
ea o
fJo
hn. H
ave
the
clas
s gi
ve o
ther
exam
ples
inev
eryd
ay s
ituat
ions
whe
re w
e us
esy
mbo
ls to
conv
ey c
erta
inid
eas.
Som
e ex
ampl
es w
hich
mig
ht b
e br
ough
t out
are
: bar
ber
pole
s, s
hort
-ha
nd n
otat
ion,
Mor
se c
ode,
Bra
ille
syst
em f
orth
e bl
ind,
mus
ical
not
es, s
hape
san
d de
sign
s of
vari
ous
sym
bols
use
d in
one
'sre
ligio
n, r
efer
eean
d um
pire
des
igna
ting
vari
ous
situ
aVon
s
31
Tea
chin
g A
ids
The
Gro
wth
of
Mat
hem
atic
al I
deas
, Gra
des
K-1
2, T
wen
ty-F
ourt
h Y
earb
ook,
N.
C.T
.M.,
Was
hing
ton,
D.C
., 19
59. p
p.32
7-36
9
2. S
ymbo
ls u
sed
in m
athe
mat
ics
B. O
pen
and
Clo
sed
Sent
ence
s1.
Eng
lish
sent
ence
sa.
true
b. f
alse
c. d
on't
know
If th
e cl
ass
does
not
bri
ng u
psi
tuat
ions
invo
lv-
ing
sym
bols
in m
athe
mat
ics,
the
teac
her
can
easi
ly in
sert
som
e id
eas
whi
ch s
houl
d le
ad to
som
e si
mpl
esy
mbo
ls u
sed:
Rom
an n
umer
als,
7r, 1
,5,
17,
3'2
, A f
or a
rea,
for
exam
ple.
At t
his
time
put a
larg
e tw
o (2
) an
d a
smal
ler
thre
e (3
)on
the
boar
dan
d as
k th
e cl
ass
whi
ch is
larg
er,
two
or th
ree.
Typ
ical
repl
ies
wou
ld h
ave
som
esa
ying
two
and
othe
rs th
ree
and
fina
lly s
ome
stud
ent w
ill a
sk, "
Wha
t do
you
mea
n?"
Thi
ssh
ould
then
be
disc
usse
dbr
ingi
ng o
ut th
at 2
isju
st a
sym
bol r
epre
sent
ing
the
idea
of
two.
The
re-
fore
, if
we
wan
t to
talk
abo
utth
e sy
mbo
l, w
esh
ould
hav
e a
spec
ial n
ame
for
it. T
he s
peci
alna
me
whi
ch w
egi
ve to
the
sym
bol r
epre
sent
ing
a nu
mbe
r is
"num
eral
"
Dev
elop
a le
sson
aro
und
the
diff
eren
ces
in c
on-
cept
of
"num
eral
" an
d"n
umbe
r"
In o
rder
to g
et th
e st
uden
ts to
unde
rsta
nd w
hat
we
mea
n by
ope
nan
d cl
osed
sen
tenc
es, t
hete
ache
r co
uld
pres
ent t
o th
est
uden
ts o
n th
ebl
ackb
oard
or
on ty
ped
shee
ts, a
gro
upof
Eng
-lis
h se
nten
ces
whi
ch w
ould
cont
ain
both
ope
nan
d cl
osed
sen
tenc
es. T
his
grou
psh
ould
con
tain
thre
e ty
pes
of s
ente
nces
- tr
ue,
fals
e, c
an't
tell.
Exa
mpl
e: 1
) St
. Pau
l is
the
capi
tal
of M
inne
sota
true
. 2)
A R
ussi
an d
isco
vere
dA
mer
icaf
alse
.3)
He
play
ed f
or th
e M
inne
sota
Tw
insc
an't
tell.
Ins
truc
t the
stu
dent
s to
read
eac
h se
nten
ceca
refu
lly a
nd th
en in
dica
te w
hich
sent
ence
s ar
etr
ue a
nd w
hich
are
fal
se.
Aft
er g
ivin
g th
emen
ough
tim
e to
com
plet
e th
is g
roup
of s
ente
nces
,go
thro
ugh
and
disc
uss
each
sen
tenc
e. I
f so
me-
one
in th
e gr
oup
orth
e gr
oup
itsel
f ha
s no
t dis
-co
vere
d th
at s
ome
sent
ence
s ca
nnot
be il
assi
fied
eith
er tr
ue o
r fa
lse,
the
teac
her
can
dire
ct th
edi
scus
sion
by
care
ful q
uest
ioni
ng, s
oth
at th
ecl
ass
will
rec
ogni
ze th
is f
act.
At t
his
time
have
each
stu
dent
wri
te a
t lea
st o
ne s
ente
nce
of e
ach
of th
e th
ree
type
s
A s
ente
nce,
whi
ch is
fou
nd to
be e
ither
true
or
fals
e w
ill b
e co
nsid
ered
a c
lose
dse
nten
ce. I
f it
isim
poss
ible
to te
ll w
heth
er a
sen
tenc
eis
true
or
32
Muc
h of
the
mat
eria
lin
volv
ing
open
and
clos
ed s
ente
nces
cou
ld b
efo
und
in a
l-m
ost a
ny m
oder
nel
emen
tary
alg
ebra
book
2. M
athe
mat
ical
or
Num
ber
Sent
ence
a. tr
ueb.
fal
sec.
don
't kn
ow
3. P
atte
rns
in N
umbe
r Se
nten
ces
a. M
akin
g op
en s
ente
nces
true
inte
rms
of a
dditi
on, s
ubst
rac-
tion,
mul
tiplic
atio
n an
d di
vi-
sion
b. S
olvi
ng r
elat
ing
oper
atio
ns1)
add
ition
and
sub
trac
tion
2) m
ultip
licat
ion
and
divi
sion
c. T
he n
atur
e of
mul
tiplic
atio
n1)
in te
rms
of a
dditi
on2)
in te
rms
of d
ivis
ion
fals
e, it
will
be
cons
ider
ed a
n op
en s
ente
nce.
In
solv
ing
open
sen
tenc
es in
mat
hem
atic
s, c
oexp
lete
lyav
oid
"tel
l em
" te
achi
ng
The
pre
viou
s m
ater
ial s
houl
d pr
esen
t an
oppo
r-tu
nity
for
the
teac
her
to p
oint
out
that
mat
he-
mat
ical
sen
tenc
es c
an b
e cl
assi
fied
as
eith
er o
pen
or c
lose
d ju
st a
s w
ell a
s w
asdo
ne p
revi
ousl
y w
ithE
nglis
h se
nten
ces.
It s
houl
d be
em
phas
ized
that
in m
athe
mat
ics,
an
expr
essi
on o
f th
e fo
rm3
+ 4
= 7
whi
ch is
to r
ead
"thr
ee p
lus
four
equa
ls (
or is
equ
al to
) se
ven"
is c
alle
d a
sent
ence
.T
he s
ymbo
l " =
" is
reg
arde
d as
the
verb
of
the
sent
ence
. Hav
e th
e st
uden
ts c
onst
ruct
sev
eral
clos
ed (
true
or
fals
e) s
ente
nces
of
mat
hem
atic
s.A
t thi
s po
int a
set
of
exer
cise
s in
volv
ing
only
sim
ple
addi
tion,
sub
trac
tion,
mul
tiplic
atio
n, a
nddi
visi
on c
ould
be
give
n to
the
stud
ent t
o ha
vehi
m d
eter
min
e w
heth
er o
r no
t the
y ar
e tr
ue o
rfa
lse
Ask
the
stud
ents
if th
ey c
ould
thin
k of
any
met
hod
of w
ritin
g a
mat
hem
atic
al s
ente
nce
soth
at it
wou
ld b
e cl
assi
fied
as
"ope
n."
A s
en-
tenc
e of
the
form
6 +
? =
8 is
cal
led
an o
pen
sen-
tenc
e. W
e co
uld
also
wri
te th
at s
ame
mat
hem
a-tic
al s
ente
nce
6 +
fl =
8. W
e ag
ree
that
"0"
may
be
repl
aced
by
any
num
eral
(num
ber
sym
-bo
l). T
hese
sen
tenc
es c
ould
be
read
"si
x pl
usso
me
num
ber
is e
qual
to e
ight
."W
e ha
ve n
o w
ayof
kno
win
g w
heth
er o
r no
t thi
s is
true
unt
il w
ere
plac
e th
e "?
" or
"0"
by
som
e nu
mer
al. I
f w
ere
plac
ed th
e "0
" in
6 +
LI
= 8
with
var
ious
num
eral
s, m
any
sent
ence
s ca
n be
for
med
. Ask
the
clas
s "H
ow m
any
num
eral
s w
ill m
ake
6 +
LI
= 8
true
?" f
ollo
wed
clo
sely
by
the
ques
-tio
n, "
How
man
y nu
mer
als
will
mak
e6
+=
8 f
alse
?"
It s
houl
d be
com
e ap
pare
nt th
at th
e "
"is
sim
ply
hold
ing
the
plac
e fo
r a
num
ber.
The
re a
re m
any
othe
r sy
mbo
ls w
hich
may
acc
ompl
ish
the
sam
epu
rpos
e. F
or e
xam
ple,
the
sent
ence
s 6
+ E
J =
8,
6 +
A =
8,G
+D
=8,
and6
+O
= 8
cou
ldbe
use
d in
terc
hang
eabl
y. I
t is
impo
rtan
t for
the
33
4. O
rder
of
oper
atio
nsa.
Add
ition
and
subt
ract
ion
alon
gw
ith m
ultip
licat
ion
and
divi
-si
on
stud
ents
to u
nder
stan
d th
at th
ese
sym
bols
are
just
hol
ding
a p
lace
and
are
to b
e re
plac
edby
num
eral
s to
for
m c
lose
d se
nten
ces.
We
call
figu
res
like
0, A
, 0, a
nd 0
num
ber
fram
es
It c
anno
t be
emph
asiz
ed to
o of
ten
that
ther
ew
ill b
e tim
es w
hen
we
will
nee
d to
use
mor
eth
anon
e sy
mbo
l to
hold
the
plac
e fo
r a
num
ber
in a
nop
en s
ente
nce.
It i
sag
reed
in m
athe
mat
ics
that
whe
n th
e sy
mbo
l hol
ding
the
plac
e fo
r a
num
ber
is r
epla
ced
by a
num
eral
, the
sym
bol
shou
ld b
ere
plac
ed in
oth
er p
lace
s w
here
it o
ccur
s in
the
sam
e se
nten
ce w
ithth
e sa
me
num
eral
. For
in-
stan
ce,
El +
=--
A +
=, w
e m
ay r
epla
ce th
e"0
" by
"2"
and
the
"A"
by "
7" a
nd h
ave
2 +
7 =
7 +
2, w
hich
is a
true
sen
tenc
e
Hav
ing
stud
ents
mak
e tr
ue s
ente
nces
out
ofop
en s
ente
nces
and
then
stu
dyin
g th
e pa
ttern
s,sh
ould
hel
p th
e st
uden
t to
see
the
rela
tion
ofad
ditio
n an
d su
btra
ctio
n to
eac
h ot
her.
In
like
man
ner,
the
rela
tion
that
mul
tiplic
atio
n an
d di
-vi
sion
are
als
o in
vers
e op
erat
ions
of
one
anot
her.
Exa
mpl
es:
f43
A =
16
f46
= 2
116
= 4
32
X E
l --=
46
By
mak
ing
use
of p
atte
rns
such
as:
2 x
6=
6+
63
X 6
= 6
+ 6
+ 6
4 X
6 =
6 +
6-1
-6
+ 6
and
rela
ted
exer
cise
s, w
e ca
n po
ssib
ly g
etth
est
uden
ts to
see
the
conn
ectio
n be
twee
n m
ulti-
plic
atio
n an
d ad
ditio
n
In o
rder
to s
how
a n
eed
for
"ord
er o
f op
era-
tions
," th
e te
ache
r co
uld
pres
ent s
ever
al m
athe
-m
atic
al e
xpre
ssio
ns u
sing
mor
e th
an o
ne o
pera
-tio
n, s
uch
as 4
+ 5
X 6
and
hav
e th
e st
uden
tsev
alua
te. T
here
mig
ht b
e so
me
disa
gree
men
t on
whe
ther
this
sho
uld
equa
l 34
or 5
4. I
n or
der
toav
oid
conf
licts
, som
e ag
reem
ent m
ust b
e m
ade.
It h
as b
een
agre
ed u
pon
in m
athe
mat
ics
that
we
will
per
form
all
of th
e m
ultip
licat
ion
and
divi
-si
on f
irst
and
aft
er th
ese
have
bee
n ca
rrie
d ou
t,w
e pe
rfor
msu
btra
ctio
n an
d ad
ditio
n. I
n ca
sew
e ha
ve a
ser
ies
ofad
ditio
ns a
nd s
ubtr
actio
ns,
34
b. M
ultip
licat
ion
and
divi
sion
alon
g w
ith s
quar
ing
and
squa
rero
otin
g
1) p
ositi
ve, n
egat
ive,
and
zer
oex
pone
nts
2) in
tegr
al a
nd f
ract
iona
l ex-
pone
nts
we
perf
orm
thes
e op
erat
ions
in o
rder
fro
m le
ftto
rig
ht. T
he s
ame
is tr
ue in
cas
e a
seri
es o
fm
ultip
licat
ions
and
div
isio
ns a
re in
volv
ed in
the
sam
e pr
oble
m. (
Squa
ring
and
squa
re r
ootin
gsh
ould
pre
cede
mul
tiply
ing
and
divi
ding
als
o.)
If th
e te
ache
r so
des
ires
, thi
s m
ight
be
an o
ppor
-tu
ne ti
me
to le
ad th
e cl
ass
into
the
idea
of
ex-
pone
nts,
whi
ch c
an b
e th
ough
t of
as m
akin
g us
eof
sim
plif
ying
not
atio
ns5
+ 5
+ 5
+ 5
= 4
(5)
10 -
1- 1
0 -1
- 10
= 3
(10)
In th
e pr
eced
ing
mat
hem
atic
al s
ente
nces
, the
four
and
thre
e w
ere
used
to w
rite
eac
h se
nten
cein
a m
ore
conc
ise
form
(5)
(5)
(5)
(5)
= (
5)4
(10
(10)
(10
) =
(10
)3
Her
e th
e fo
ur a
nd th
ree
are
once
mor
e us
ed to
sim
plif
y th
e m
athe
mat
ical
sta
tem
ent.
In s
en-
tenc
es (
1) a
nd (
2), t
he f
our
and
thre
e in
dica
ted
the
num
ber
of f
ives
or
tens
incl
uded
in th
e re
-sp
ectiv
e pr
oduc
ts10
4=
1000
010
1=
?10
3=
1000
10°
?
102
=10
010
-1=
?10
-2=
?21
=?
74=
2401
2°?
73=
343
2-1
=?
72=
492-
2=
?71
=?
14=
?70
?13
?
7-1
=?
12=
?7-
2?
11=
?
24=
161°
=?
23=
81-
1=
?
22=
41-
2=
?
Indu
ctiv
ely,
this
sho
uld
lead
to th
e id
ea th
at1
1
x1=
x; x
0=1
whe
n x
0; a
nd x
-1-1
7; x
-2=
x7
etc.
Sim
ple
way
to w
rite
6,0
00,0
00,0
00,0
00,0
00 is
6 X
101
5
35
c. U
se o
f pa
rent
hese
s
Usi
ng 7
3 as
an
exam
ple:
The
sm
all n
umer
al th
ree
(3),
whi
ch is
abo
ve th
elin
e, is
cal
led
an e
xpon
ent
and
a nu
mer
al c
on-
tain
ing
an e
xpon
ent,
such
as
73,
is a
n "e
xpon
en-
tial n
umer
al."
The
num
ber
bein
g m
ultip
lied,
the
7 of
the
73, i
s th
e "b
ase"
of th
e ex
pone
ntia
lnu
mer
al
A m
uch
sim
pler
way
of
usin
g re
peat
ed m
ulti-
plic
atio
n is
by
givi
ng th
e de
sire
dnu
mbe
r to
ace
rtai
n po
wer
. Thu
s, 7
3 is
"sev
en to
the
thir
dpo
wer
" or
mol
e si
mpl
y, "
seve
n to
the
thir
d"
Thi
s pr
esen
ts th
e qu
estio
n of
how
cer
tain
mul
ti-pl
icat
ion
and
divi
sion
pro
blem
s ca
nbe
sim
plif
ied
16 =
(2)
(2)
(2)
(2)
=24
8 =
(2)
(2)
(2)
=23
16/8
= 2
or
2'32
,/8=
4 o
r22
32 =
258
=23
(16)
(8)
= (
21)
(2)3
= 2
7
Hav
ing
the
stud
ents
not
ice
that
the
sum
(if
mul
tiply
ing)
or
diff
eren
ce (
if d
ivid
ing)
of th
eex
pone
nts
of th
e tw
oex
pone
ntia
l num
eral
s is
equa
l to
the
expo
nent
of
the
answ
er.
Thi
s sh
ould
be e
xten
ded
so a
s to
incl
ude
answ
ers
whi
ch w
illha
ve a
bas
e to
a n
egat
ive
or z
ero
expo
nent
Thi
s is
an
addi
tiona
lco
mpu
tatio
nal s
kill
that
will
aid
the
stud
ents
. Oth
ers
wer
est
udie
d in
Uni
t I
Pare
nthe
ses,
as
used
to d
eter
min
eor
der
and
tohe
lp s
epar
ate,
cou
ld b
e in
tere
stin
gly
pres
ente
dby
put
ting
seve
ral E
nglis
hse
nten
ces
on th
ebo
ard.
The
n by
inse
rtin
g pu
nctu
atio
nm
arks
,sh
ow h
ow th
e m
eani
ng c
anbe
com
plet
ely
chan
ged.
Sev
eral
exa
mpl
es w
hich
may
be u
sed:
1) N
o pr
ice
too
high
! 2)
Slow
men
wor
king
.3)
Joh
n sa
id th
e te
ache
r is
ver
yin
telli
gent
Whe
neve
r a
mat
hem
atic
al s
ente
nce
or e
xpre
s-si
on c
onta
ins
one
of th
e gr
oupi
ngsy
mbo
ls (
pa-
rent
hese
s, b
rack
ets,
bra
ces)
this
beco
mes
a "
sig-
36
5. D
isco
veri
ng th
e so
lutio
n of
ope
nse
nten
ces
a. o
ne o
pera
tion
b. tw
o op
erat
ions
6. M
issi
ng D
igit
Prob
lem
s
a. F
aded
doc
umen
t
b. S
erie
s
c. C
rypt
ogra
ms
nal"
say
ing
"do
me
firs
t." I
f in
the
expr
essi
on,
4 +
6 X
2, w
e w
ante
d to
add
fir
st in
stea
d of
the
indi
cate
d m
ultip
licat
ion,
we
coul
d si
mpl
y w
rite
this
exp
ress
ion
(4 +
6)
Use
the
disc
over
y m
etho
d in
get
ting
the
stu-
dent
s to
see
a g
ener
al m
etho
d of
sol
ving
ope
nse
nten
ces.
Ope
n se
nten
ces
of o
ne o
pera
tion,
mak
ing
use
of a
dditi
on, s
ubtr
actio
n, m
ultip
lica-
tion,
and
div
isio
n sh
ould
be
som
ewha
t mas
tere
dbe
fore
beg
inni
ng o
pen
sent
ence
s w
ith tw
o op
era-
tions
Whe
n th
e in
tere
st o
f th
e cl
ass
seem
s to
be
wan
-in
g, s
ome
type
of
mat
eria
l whi
ch w
ould
act
as
ach
ange
of
pace
cou
ld b
e he
lpfu
l. M
akin
g us
e of
fade
d do
cum
ents
, num
ber
seri
es, c
rypt
ogra
ms,
mag
ic s
quar
es a
nd m
issi
ng o
pera
tion
prob
lem
sca
n be
intr
igui
ng, s
timul
atin
g,an
d ch
alle
ngin
g
Exa
mpl
es:
Fade
d D
ocum
entr
epla
ce th
e st
ars
with
dig
itsto
mak
e th
e st
atem
ents
true
49*
8**
+ *
*4*3
2
678
384
37X
2* 2*6
7* 6N
umbe
r se
ries
com
plet
e th
e nu
mbe
r se
ries
:1,
3, 7
, 15,
31,
**,
***
86, 7
8, 6
7, 5
3, 3
6,**
, **
Cry
ptog
ram
ssol
ve b
y su
bstit
utin
g a
num
eral
in p
lace
of
each
lette
r. E
ach
lette
r re
pres
ents
one
num
eral
37
Bak
st, A
aron
. Mat
hem
atic
s, I
ts M
agic
and
Mas
tery
. Van
Nos
tran
d, 1
952
Bak
st, A
aron
. Mat
hem
atic
al P
uzzl
es a
ndPa
stim
es. V
an N
ostr
and,
195
4
Bal
l, W
. W. R
ouse
, and
Cox
eter
, H. S
. M.
Mat
hem
atic
al R
ecre
atio
ns a
nd E
ssay
s.M
acm
illan
Co.
, 194
7
DeG
razi
a, J
osep
h. M
ath
is F
un. E
mer
son
Boo
ks I
nc.,
1954
Frie
nd, J
. New
ton.
Num
bers
: Fun
and
Fact
s. C
harl
es S
crib
ner's
Son
s, 1
954
Gar
dner
, Mar
tin. M
athe
mat
ics,
Mag
ic a
ndM
yste
ry. D
over
Pub
licat
ions
, 195
6
Gill
es, W
. F. M
agic
and
Odd
ities
of
Num
-be
rs. V
anta
ge P
ress
, 195
3
Jone
s, S
. I. M
athe
mat
ical
Clu
bs a
nd R
ec-
reat
ions
. S. I
. Jon
es C
o., 1
940
Kra
itchi
k, M
auri
ce. M
athe
mat
ical
Rec
rea-
tions
. Dov
er P
ublic
atio
ns, 1
942
d. M
agic
Squ
are
e. M
issi
ng o
pera
tion
7. E
quiv
alen
t sen
tenc
es a
nd th
est
ruct
ure
of a
rith
met
ica.
Dis
cove
ry o
f th
ead
ditio
n, s
ub-
trac
tion,
mul
tiplic
atio
n, a
nddi
visi
on a
xiom
s
b. D
isco
ver
the
"pri
ncip
le o
for
der"
(co
mm
utat
ivity
) fo
rad
ditio
n an
d m
ultip
licat
ion
SEN
D (
one
SAV
E (
four
pos
sibl
eM
OR
Eso
lutio
n)M
OR
Eso
lutio
ns)
MO
NE
YM
ON
EY
Mag
ic N
umbe
r Sq
uare
s H
ave
stud
ents
wor
kw
ith th
e na
tura
l num
bers
in s
olvi
ngm
agic
squa
res
of th
ird
and
four
th o
rder
. Aft
er a
cer
tain
amou
nt o
f tim
e, s
ome
hint
s or
key
num
bers
may
be g
iven
. The
sum
s of
eac
h ro
w, c
olum
n,an
dth
e tw
o di
agon
als,
mus
t be
the
sam
ein
a tr
uem
agic
squ
are
Mis
sing
ope
ratio
nwha
t num
ber
sign
als
orop
erat
ion
will
mak
e th
e fo
llow
ing
true
?18
39
= 1
5(9
5)7
= 2
8
Prov
ide
the
stud
ents
with
eno
ugh
exce
rcis
es,
hase
them
dis
cove
r a
met
hod,
oth
erth
an tr
ial
and
erro
r, to
fin
d th
e so
lutio
n se
tfo
r an
ope
nse
nten
ce. T
hey
shou
ld b
e ab
le to
disc
over
the
addi
tion,
sub
trac
tion,
mul
tiplic
atio
n, a
nddi
vi-
sion
axi
oms
In h
avin
g th
e st
uden
ts d
isco
ver
the
"pri
ncip
leof
ord
er"
(com
mut
ativ
ity)
for
addi
tion
and
mul
tiplic
atio
n, r
evie
w th
e id
ea th
at 1
3?
= 8
beco
mes
a c
lose
d se
nten
ce w
hen
the
? is
repl
aced
by s
ome
num
eral
. We
may
exp
ress
the
sam
eid
eaby
usi
ng a
fra
me
inst
ead
of th
e qu
estio
nm
ark.
The
fra
me
may
be
of d
iffe
rent
shap
es s
uch
asA
, 0, 0
, 0. S
omet
imes
, we
need
to u
sem
ore
than
one
fra
me
in a
sen
tenc
e to
sta
nd f
ordi
ffer
ent n
umbe
rs. D
iffe
rent
fra
mes
with
in a
sent
ence
mus
t be
repl
aced
by
diff
eren
tnum
bers
,bu
t ide
ntic
al f
ram
es w
ithin
a s
ente
nce
mus
t be
repl
aced
by
the
sam
e nu
mbe
rFo
r ex
ampl
e: E
t + 9
= 1
9E
l A +
= 4
+ A
Hav
e st
uden
tsth
e id
ea th
ata
+ b
= b
+ a
.tio
n8
=
do a
set
of
prob
lem
s le
adin
g to
addi
tion
is c
omm
utat
ive,
i.e.
,H
ave
stud
ents
try
it fo
r su
btra
c-
8
38
Mer
rill,
Hel
en A
bbot
. Mat
hem
atic
al E
xcur
-si
ons.
Dov
er P
ublic
atio
ns, 1
957
Mey
er, J
erom
e. F
un w
ithM
athe
mat
ics
Dov
er P
ublic
atio
ns, 1
952
Ran
som
, Will
iam
P. O
ne H
undr
ed M
athe
-m
atic
al C
urio
sitie
s. J
. Wes
ton
Wal
ch,
1954
c. D
isco
ver
the
"pri
ncip
les
ofgr
oupi
ng"
(ass
ocia
tivity
) fo
rad
ditio
n an
d m
ultip
licat
ion
d. D
isco
ver
the
"dis
trib
utiv
epr
oper
ty"
(dis
trib
utio
n of
mul
tiplic
atio
n an
d di
visi
onov
er a
dditi
on a
ndsu
btra
ctio
n)
1) M
ultip
lyin
g a
two
plac
enu
mer
al b
y on
e pl
ace
num
-er
al2)
Mul
tiply
ing
a tw
o pl
ace
num
eral
by
two
plac
e nu
m-
eral
C. C
hang
ing
Eng
lish
to M
athe
mat
ical
Phra
ses
Aga
in b
y us
ing
fram
e al
gebr
a an
d in
duct
ive
reas
onin
g, tr
y to
get
the
stud
ents
to s
eeth
atm
ultip
licat
ion
is c
omm
utat
ive.
Hav
e st
uden
tstr
y di
visi
on f
or c
omm
utat
ivity
102
= 2
:10
In m
akin
g us
e of
exa
mpl
es a
nd s
ets
ofpr
oble
ms,
see
if th
e st
uden
ts c
ange
nera
lize
or d
isco
ver
that
++
0)
=+
+ 0
and
als
o th
atA
(E
0)
= (
A1E
)
Thi
s is
cal
led
the
prin
cipl
e of
gro
upin
g fo
rad
di-
tion
and
mul
tiplic
atio
n (a
ssoc
iativ
ity).
Do
you
thin
k su
btra
ctio
n an
d di
visi
on a
re a
ssoc
iativ
e?
Hav
e st
uden
ts d
o a
set o
f pr
oble
ms
usin
gfr
ame
alge
bra
whi
ch le
ads
them
to th
e id
ea th
at m
ulti-
plic
atio
n an
d di
visi
on c
an b
e di
stri
bute
d ov
erad
ditio
n an
d su
btra
ctio
n. D
evel
op th
e id
ea th
atad
ditio
n an
d su
btra
ctio
n ca
nnot
be
dist
ribu
tive
over
mul
tiplic
atio
n or
divi
sion
. In
othe
r w
ords
,E
l(A
+=
El
A +
El
.(>
is a
lway
s tr
uebu
t ED
+ (
A0)
=+
(1E
1 +
(>
) is
not
alw
ays
true
At t
his
time
it sh
ould
be
prof
itabl
e to
sho
ww
hat
we
real
ly m
ean
wIl
en w
e m
ultip
ly s
ixte
en b
yth
ree
3 X
(16
) =
3 X
(10
+ 6
)=
3 X
10
+ 3
X 6
=3
X 1
0+
18
=3
X 1
0+
(10
+ 8
)=
3 X
10
+ 1
X (
10)
+ 1
X 8
=(3
+ 1
)10
+ 1
X 8
4 x
10 +
1 x
8=
40
+ 8
= 4
8
In li
ke m
anne
r, if
the
teac
her
deem
s it
advi
sabl
e,w
e ca
n de
mon
stra
tew
hat i
s m
eant
whe
n w
em
ultip
ly a
two
digi
t num
ber
times
a tw
o di
git
num
ber
In b
oth
sect
ions
C a
nd D
, the
maj
or s
tres
ssh
ould
be
plac
ed u
pon
the
rela
tions
hip
betw
een
sent
ence
s in
the
Eng
lish
lang
uage
and
inth
e
lang
uage
of
mat
hem
atic
s. A
n un
ders
tand
ing
ofth
is r
elat
ions
hip
will
be
of g
reat
hel
p to
the
stu-
dent
in r
eadi
ng s
cien
tific
mat
eria
lan
d in
the
solv
ing
of v
erba
l pro
blem
s
The
exp
ress
ion,
"fi
ve m
ore
than
agi
ven
num
-be
r" c
ould
be
wri
tten
6 -I
- 5,
if th
e gi
ven
num
ber
wer
e si
x. I
t cou
ldbe
wri
tten
as 1
1 +
5 if
the
give
n nu
mbe
r w
ere
elev
en. I
t cou
ldbe
wri
tten
23 +
5 if
the
give
n nu
mbe
r w
ere
twen
ty-t
hree
.It
cou
ld a
lso
be w
ritte
n+
5 w
here
the
expr
es-
sion
+ 5
rep
rese
nts
five
mor
e th
anth
e gi
ven
num
ber
and
the
CD
sta
nds
for
the
give
nnu
mbe
r
Giv
e th
e cl
ass
man
y ex
ampl
es o
fE
nglis
h ph
rase
san
d th
e re
late
d m
athe
mat
ical
phr
ases
.T
he e
x-am
ples
cou
ld b
e si
mila
r to
the
follo
win
g:if
Mot
her
bake
d 30
coo
kies
and
pla
ced
six
cook
ies
in e
ach
bag,
then
she
mus
t hav
eus
ed 5
bag
s.30
6 =
5
Exa
mpl
es o
f bo
th c
lose
d an
d op
enE
nglis
h-m
athe
mat
ical
sen
tenc
es s
houl
d he
lp th
est
uden
tde
term
ine
som
e of
the
rela
tions
hips
whi
ch e
xist
amon
g nu
mbe
rs.
The
stu
dent
sho
uld
then
be
read
y to
sup
ply
the
mat
hem
atic
alph
rase
or
sent
ence
whe
n gi
ven
an E
nglis
hph
rase
or
sen-
tenc
e
A s
tude
nt s
houl
d, d
evel
op s
ome
grow
th in
his
prob
lem
-sol
ving
abi
lity
if h
e ca
n an
swer
the
follo
win
g qu
estio
ns p
erta
inin
g to
his
prob
lem
1. W
hat d
o yo
u kn
ow to
be
true
inth
e pr
ob-
lem
?2.
Wha
t is
aske
d fo
r in
the
prob
lem
?3.
Cou
ld y
ou w
rite
the
prob
lem
inth
e fo
rmof
a n
umbe
r se
nten
ce?
4. C
an y
ou s
olve
the
num
ber
sent
ence
?5.
Is
the
answ
er to
the
prob
lem
reas
onab
le?
D. C
hang
ing
Mat
hem
atic
al P
hras
es to
A g
ood
deal
of
thin
king
take
s pl
ace
whe
n a
Eng
lish
Phra
ses
stud
ent i
s gi
ven
a nu
mbe
r ph
rase
or
sent
ence
and
then
has
to w
rite
an
Eng
lish
phra
se o
r se
n-te
nce
corr
espo
ndin
g to
it 40
E. I
nequ
atio
ns o
r in
equa
litie
s(
and
solu
tion
sets
1. E
very
day
situ
atio
ns2.
Mat
hem
atic
al s
ituat
ions
F. D
iiect
ed N
umbe
rs a
nd th
e N
umbe
rL
ine
1. N
eed
for
them
and
whe
re th
eyar
e us
ed
Pres
ent t
o th
e cl
ass
the
sym
bol "
="
and
ask
the
clas
s w
hat i
t mea
ns a
nd g
ive
som
e ex
ampl
es.
The
n as
k th
em w
hat t
hey
thin
k th
e sy
mbo
lm
eans
. It s
houl
d no
t be
very
long
bef
ore
som
e st
uden
t or
the
clas
s de
velo
ps th
e id
ea o
f"n
ot e
qual
." B
oys
not e
qual
in a
bilit
y, o
ne c
ityla
rger
than
ano
ther
, for
exa
mpl
e, s
houl
d be
dis
-cu
ssed
to s
how
the
stud
ents
that
man
y th
ings
are
"not
equ
al"
Intr
oduc
e "<
" to
rep
rese
nt "
is le
ss th
an"
and
">"
to r
epre
sent
"is
gre
ater
than
." H
ave
the
stud
ents
mak
e bo
th tr
ue a
nd f
alse
sen
tenc
esus
ing
thes
e in
equa
lity
sym
bols
Dis
cuss
som
e of
the
gam
es th
ey a
ll ha
ve p
laye
dth
at r
equi
re k
now
ledg
e of
sig
ned
num
bers
Show
ing
a ne
ed f
or s
igne
d nu
mbe
rs1,
3,5,
?2,
4,6,
?7,
12,
17, ?
17,
13,
9, ?
25,
15,
5, ?
75,
50,
25, ?
123,
63,
32,
?In
ord
er to
con
tinue
the
patte
rns
liste
d, s
ome
type
of
num
ber
othe
r th
an p
ositi
ve is
nee
ded.
It
is v
ery
esse
ntia
l to
show
a "
need
" fo
r th
e ne
ga-
tive
num
bers
. A n
umbe
r of
situ
atio
ns w
here
the
nega
tives
are
enc
ount
ered
will
hel
p th
e st
uden
tsto
kee
p up
perm
ost i
n th
eir
min
ds th
e "w
hy"
wili
ch w
ill s
how
res
ults
of
the
"how
" la
ter.
For
exam
ple:
1) o
verd
raw
n ba
nk a
ccou
nts
2) te
mpe
ratu
res
abov
e an
d be
low
zer
o3)
com
pari
ng a
bove
and
bel
ow s
ea le
vel
4) s
core
s of
var
ious
gam
es5)
gai
ns a
nd lo
sses
in f
ootb
all y
arda
ge6)
dat
es o
f B
.C. a
nri A
.D.
7) p
rofi
ts a
nd lo
sses
8) n
orth
and
sou
th la
titud
es9)
eas
t and
wes
t lon
gitu
de10
) ab
ove
and
belo
w a
cer
tain
wat
er le
vel
pred
eter
min
ed w
heth
er o
r no
t a f
lood
ing
cond
ition
exi
sts
41
Meh
l, W
illia
m G
. Nov
el D
rill
Mat
er&
jor
Sign
ed N
umbe
rs. T
he M
athe
mat
ics
Tea
cher
,Yol
ume
52, N
o. 4
, Apr
il, 1
959.
pp. 2
47-2
49
2. T
he n
umbe
r lin
e
3. P
rope
rtie
s of
dir
ecte
d nu
mbe
rs
Intr
oduc
e by
pic
turi
ng a
str
aigh
t lin
ew
ithar
row
s on
bot
h en
ds.
Thi
s in
dica
tes
that
the
line
cont
inue
s en
dles
sly
in b
oth
dire
ctio
ns.
Cho
ose
any
poin
t on
the
line
and
cons
ider
this
the
star
ting
poin
t or
orig
in. L
abel
this
poi
nt 0
.M
ark
off
equa
l uni
ts to
the
righ
t and
toth
e le
ftof
this
0
5 4
3 2
1 0
1 2
3 4
5--
>
We
can
talk
abo
ut th
e tw
o at
the
righ
t of
zero
or th
e tw
o at
the
left
of
zero
and
thus
des
igna
tew
hich
of
the
two
iden
tical
num
bers
we
wan
t.W
e co
uld
also
des
igna
te th
e le
ftnu
mbe
rs b
y a
sym
bol s
uch
as 5
, 4, 3
, 2, 1
,and
the
righ
t num
bers
by
+1,
+2,
+3,
+4,
+5.
The
sear
e ca
lled
"di.-
ecte
dnu
mbe
rs."
It i
s no
w p
ossi
ble
to a
nsw
er n
ot o
nly
"how
man
y,"
but "
whi
chdi
rect
ion.
" W
e ca
n no
w e
stab
lish
a"o
ne-t
o-on
eco
rres
pond
ence
" to
the
poin
ts o
n th
e lin
ean
dth
e di
rect
ed n
umbe
rs
nega
-po
si-
tive
CE
HG
A F
BK
D ti
vedi
rec-
4 3
2 1
0 +
1+2
+3
+4
dire
c-tio
ntio
n
The
sym
bols
+ a
ndno
w h
ave
a du
al m
ean-
ing.
The
+ c
an m
ean
eith
er a
dditi
on o
rdir
ectio
nof
a n
umbe
r, w
hile
coul
d m
ean
subt
ract
ion
or d
irec
tion
of a
num
ber
The
"pr
oper
ty"
of o
ppos
ites
orin
vers
es c
an b
ede
mon
stra
ted
by li
stin
g se
vera
l pai
rs o
f nu
mbe
rssu
ch a
s: + 7
,7
8, +
82,
+ 2
AIn
com
pari
ng le
ft w
ith r
ight
, not
ice
each
one
has
an "
oppo
site
. ' N
otic
e th
at b
oth
dire
cted
num
bers
are
the
sam
e di
stan
ce f
rom
the
orig
inbu
t opp
osite
dir
ectio
n. T
hus:
(+ 7
) +
( 7
) =
0T
his
lead
s to
the
intu
itive
idea
ofa
bsol
ute
valu
e.B
oth
+ 7
and
7 ar
e th
e sa
me
dist
ance
fro
mth
e or
igin
, tha
t is,
sev
en u
nits
. If
we
don'
t car
e
42
whi
ch d
irec
tion
from
the
orig
in b
ut a
re in
ter-
este
d on
ly in
the
num
ber
of u
nits
aw
ay w
e us
eth
e sy
mbo
l "I
" to
des
igna
te th
is.
Exa
mpl
es:
I
77
+ 5
512
= 1
2
In o
rder
to u
se th
e in
tuiti
ve a
ppro
ach
in o
per-
atin
g w
ith s
igne
d nu
mbe
rs, g
ive
the
stud
ents
ase
ries
of
prob
lem
s su
ch a
s:a)
Add
ition
(+3)
+ (
+2)
=?
(-I-
3)+
(2)
= ?
(+3)
+ (
+=
?(+
3)+
(3)
?(+
3)+
(0)
=?
(+3)
+ (
4)=
?(+
3)+
(1)
=?
(+3)
+ (
5)=
?
b) S
ubtr
actio
n(+
(+4)
=?
(-I-
6)(0
)=
?(+
6)(+
3)=
?(-
I-6)
(1)
= ?
(+(
-I-
2)=
?(-
I-6)
(2)
= ?
(+6)
(+1)
=?
(+6)
(3)
= ?
Thi
s sh
ould
lead
to th
e ge
nera
lizat
ion
that
tosu
btra
ct a
sig
ned
or d
irec
ted
num
ber,
we
chan
geth
e si
gn a
nd a
dd
c) M
ultip
licat
ionG
eom
etri
c Pr
ogre
ssio
nE
xam
ple:
5,
15, 4
5,?
4,8,
16,
?
44
44
44
X3
X2
X1
X0
x 1
x-2
128
40
18
d) D
ivis
ion
( 15
)(
3)=
n, m
eans
( 3
) n
=(
15)
and
that
n =
+ 5
n, m
eans
( 3
) n
=(+
15)
and
that
n5
= n
, mea
ns (
-1-
3) n
=(
15)
and
that
n =
5
= n
, mea
ns (
+ 3
) n
=(+
15)
and
that
n =
+ 5
43
G. G
raph
ing
H. M
ahem
atic
al p
atte
rns
in s
cien
ce
Gra
nhin
g is
incl
uded
in s
ome
othe
r un
its.
Ifin
tere
sted
in g
raph
ing,
ple
ase
refe
r L
o U
nits
Ian
d X
Patte
rns,
be
they
num
eric
al, g
eom
etri
cal,
or a
nyot
her
type
, hav
e in
tere
sted
man
thro
ugh
the
ages
. The
anc
ient
Gre
eks
and
Egy
ptia
ns, w
how
ere
esse
ntia
lly g
eom
eter
s,de
velo
ped
man
y ge
o-m
etri
c pa
ttern
s in
to th
eir
arch
itect
ure.
The
yal
so s
pent
con
side
rabl
e tim
e ar
rang
ing
dots
into
geom
etri
c sh
apes
suc
h as
tria
ngle
s an
d sq
uare
s,an
d th
en e
num
erat
ing
them
Tri
angl
e N
umbe
rs
Squa
re N
umbe
r,s
ee
Pent
agon
Num
bers
o
The
Ara
bs, H
indu
s, a
nd G
reek
s se
emed
tow
ork
mos
tly w
ith n
umbe
r pa
ttern
s
A n
umbe
r pa
ttern
or
seri
es is
the
build
ing
upof
a gr
oup
of n
umbe
rsby
usi
ng s
ome
sche
mat
icm
etho
d. M
athe
mat
ics,
bei
ng a
str
uctu
re o
rst
udy
of p
atte
rns
and
thei
r re
latio
nshi
ps, b
e-co
mes
vita
l and
inte
rest
ing
to a
ll of
us.
Qui
teof
ten
stud
ents
look
at m
athe
mat
ics
sim
ply
asa
cour
se w
ith m
any
rule
s an
d a
few
mag
ic tr
icks
dem
onst
rai.e
d by
the
teac
her.
The
dev
elop
men
tof
new
idea
s an
d te
chni
ques
are
oft
en b
roug
htab
out b
y th
e st
udyi
ng o
f pa
ttern
s, e
xper
imen
ts,
and
thei
r nu
mer
ical
res
ults
. It d
oes
seem
pla
us-
ible
for
us
to th
en in
clud
e in
our
mat
hem
atic
s44
1. E
xam
inin
g va
riou
s nu
mbe
r pa
t-te
rns
cour
se th
e st
udy
ofnu
mer
ical
rel
atio
nshi
psw
hich
dis
play
reg
ular
and
irre
g&ar
pat
tern
s
Thi
s w
ould
be
a go
od ti
me
to p
rese
nt to
the
stud
ents
man
y pr
oble
ms.
Hav
e th
em tr
y to
com
plet
e th
e nu
mbe
r pa
ttern
s. S
ome
exam
ples
whi
ch m
ight
be
used
are
the
follo
win
g:
a. L
ook
at th
e tr
iang
le a
nd s
eeif
you
cou
ld f
illin
the
blan
k sp
aces
with
the
prop
er n
umbe
rso
that
the
patte
rn w
hich
isin
dica
ted
is c
on-
tinue
d
Pasc
al's
Tri
angl
e of
Num
bers
(B
lais
e Pa
scal
Fren
ch M
athe
mat
icia
n, 1
623-
1662
)
b T
he f
ollo
win
g is
als
o co
vere
d in
Uni
t I. M
ul-
tiply
sev
eral
num
bers
by
9. T
hen
add
up th
edi
gits
of
the
prod
ucts
9 x
43 =
387
3+
8 +
7 =
?9
x 17
=15
31
+ 5
+ 3
=?
9 x
400
=36
003
+ 6
+ 0
+ 0
=?
9 X
521
63=
4694
674
+ 6
+ 9
+4
+ 6
+7
=?
Can
you
rec
ogni
ze a
ny p
atte
rn w
hich
see
ms
to b
e tr
ue in
the
case
s yo
u ha
ve e
xam
ined
?If
not
, try
sev
eral
mor
e
c. F
rom
this
info
rmat
ion,
coul
d yo
u te
ll if
the
follo
win
g nu
mbe
rs a
re d
ivis
ible
by
9 w
ithou
tac
tual
ly d
oing
the
divi
sion
?49
567
8624
5
d. T
ake
each
of
the
num
bers
fro
m 1
to 1
0 an
dsq
uar,
_, (
mea
ns ti
mes
them
selv
es: 5
X 5
, 3 X
3,
45
Gle
nn, W
illia
m H
., an
d Jo
hnso
n, D
onov
anA
. Inv
itatio
n to
Mat
hem
atic
s. E
xplo
ring
Mat
hem
atic
s on
You
r O
wn
Seri
es. S
t.L
ouis
. Web
ster
, 196
0
Gle
nn, W
illia
m H
., an
d Jo
hnso
n, D
onov
anA
. Num
ber
Patte
rns.
Exp
lori
ng M
athe
-m
atic
s on
You
r O
wn
Seri
es. S
t. L
ouis
.W
ebst
er, 1
960
4 x
4) th
em. C
ompa
re th
enu
mbe
r w
ith it
ssq
uare
. Do
you
reco
gniz
e an
y pa
ttern
?
e. C
uttin
gth
e C
ircl
eH
ow m
any
piec
es d
o yo
u ge
t if
you
cut a
circ
le a
ll th
e w
ay a
cros
s w
ithou
tcu
tting
thro
ugh
the
sam
e po
int m
ore
than
twic
e?
No.
No.
of
Incr
ease
In
of C
uts
Dra
win
gPl
oces
No.
of
Piec
es
f. A
rith
met
ic a
nd g
eom
etri
cpr
ogre
ssio
ns a
ndse
ries
pro
blem
s w
ould
fit
in n
icel
yhe
re 3
, 6,
9, 1
224
, 12,
6, 3
,___
_1,
3, 0
, 4, 1
, 6, 3
,
2. L
ette
rs s
tand
for
num
bers
the
Up
to th
is ti
me
the
stud
ents
hav
ebe
en u
sing
shor
than
d fo
r sc
ienc
e0,
A, 0
, and
so
on.
As
a pl
aceh
olde
r or
var
i-ab
le. T
his
wou
ld b
e an
opp
ortu
ne ti
me
toin
tro-
duce
the
idea
that
we
can
use
any
sym
bol t
hat
we
wan
t and
use
it as
a p
lace
hold
er. S
omet
imes
it is
con
veni
ent t
o us
e le
tters
of
our
alph
abet
tous
e as
pla
ceho
lder
s.T
hese
lette
rs s
houl
d be
thou
ght o
f as
num
bers
and
not
as
lette
rs. P
re-
sent
to th
e st
uden
ts m
any
of th
e sa
me
prob
lem
sth
at c
onta
ined
num
ber
fram
es b
ut n
ow a
rew
ritte
n in
term
s of
lette
rs
3. D
isco
veri
ng p
atte
rns
whi
ch c
anIt
now
bec
omes
tim
e fo
r th
e st
uden
t to
be p
re-
be e
xpre
ssed
as
form
ulas
sent
ed w
ith a
pro
blem
, with
all
esse
ntia
lin
for-
mat
ion.
The
stu
dent
sho
uld
then
put
dow
n th
isin
form
atio
n an
d lo
ok f
or th
e pa
ttern
. The
stu-
dent
's m
ain
task
wou
ld b
e to
exp
ress
the
patte
rnal
gebr
aica
lly. S
ever
al p
robl
ems
whi
chco
uld
oedo
ne in
this
man
ner
coul
d in
clud
e:
*a. T
ime
it ta
kes
an o
bjec
t to
fall
to th
egr
ound
fro
m s
ome
heig
ht*b
. How
far
doe
s a
car
go a
fter
the
brak
esar
e ap
plie
dbe
fore
it s
tops
?*c
. Pow
er in
com
pari
son
to w
eigh
tfo
r ai
r-pl
ane
engi
nes
d. C
ompa
ring
cen
tigra
de a
nd F
ahre
nhei
t
e. O
hm's
law
Am
ps =
Vol
tsO
hms
46
Saw
yer,
W. W
. Mat
h Pa
ttern
s in
Scie
nce.
Am
eric
an E
duca
tion
Publ
icat
ions
.C
o-lu
mbu
s, O
hio;
Wes
leya
n, U
nive
rsity
,19
60
f.E
lect
: ic
pow
erW
atts
= V
olts
X A
mps
g. S
ound
Vel
ocity
of
Soun
d =
Fre
quen
cyX
wav
e le
ngth
4. D
isco
veri
ng s
impl
e la
ws
by e
x-W
e fi
nally
com
e to
the
last
sta
ge o
f m
athe
mat
i-Sa
wye
r, W
. W. M
ath
Patte
rns
in S
cien
ceca
l pat
tern
s in
sci
ence
. The
stu
dent
s w
ill n
owbe
com
e re
spon
sibl
e fo
r se
tting
up
an e
xper
imen
t,ta
bula
ting
the
resu
lts, l
ooki
ng f
or a
pat
tern
, and
then
exp
ress
ing
this
pat
tern
in te
rms
of a
for
m-
ula.
Mos
t of
the
mat
eria
ls n
eede
d to
con
duct
thes
e si
mpl
e ex
peri
men
ts c
an b
e fo
und
in th
esc
ienc
e an
d in
dust
rial
art
s de
part
men
ts o
r at
hom
e. S
ever
al e
xam
ples
of
expe
rim
ents
whi
chm
ight
be
cond
ucte
d in
this
man
ner
are
incl
uded
belo
w
peri
men
tatio
n
*a.
plat
form
on
rock
ers
*b. p
ulle
ys*c
. hom
e-m
ade
wei
ghin
g m
achi
ne*d
. bou
ncin
g of
bal
ls, s
uch
as, t
enni
s ba
ll,go
lf b
all
e. H
ook'
s la
w o
n sp
ring
sf.
Incl
ined
pla
neM
echa
nica
l adv
anta
geL
engt
hH
eigh
tg.
Clo
ckw
ise
and
coun
terc
lock
wis
e m
omen
t*
Exa
mpl
es m
ay b
e fo
und
in W
. W. S
aw-
-For
ce x
Dis
tanc
e =
For
ce x
Dis
tanc
eye
r's p
amph
let,
Mat
h Pa
ttern
s in
Sci
ence
FACTS FROM FIGURESAND FIGURING FROM FACTS
UN
IT I
V
FAC
TS
FRO
M F
IGU
RE
S A
ND
FIG
UR
ING
FR
OM
FA
CT
S
Intr
oduc
tion
One
of
the
purp
oses
of
the
cour
se is
toha
ve th
e st
uden
ts le
arn
by d
oing
. Thi
sw
ill d
efin
itely
be
the
key
to y
our
succ
ess
with
this
unit.
The
uni
t mus
t be
mad
e pr
actic
al to
the
stud
ents
' int
eres
ts a
nd a
bilit
ies.
All
of th
e m
ater
ial p
rese
nted
with
in th
eun
it ca
n be
mad
e us
eful
to th
e st
uden
t. H
e m
ustf
irst
be
show
n w
hat s
tatis
tics
is a
nd h
ow it
is b
eing
use
d in
his
eve
ryda
y lif
e. H
e m
ust
be ta
ught
to u
nder
stan
d th
e m
athe
mat
ics
of s
tatis
tics
and
also
to le
arn
way
s of
anal
yzin
g th
e re
suits
. Muc
h pr
actic
al e
xper
ienc
ew
ill b
e ga
ined
by f
orm
al p
robl
ems
pres
ente
d by
you
, the
teac
her,
dev
elop
ed a
roun
d da
ta g
athe
red
from
man
y ar
eas
of li
fe. H
e w
ill a
lso
lear
n m
ucn
by a
pro
ject
pro
blem
des
igne
d an
d st
udie
dby
him
self
. Muc
h of
the
data
to b
e us
edin
the
unit
coul
d be
st b
e co
llect
edby
the
teac
her
befo
re b
egin
ning
the
unit.
Thi
s w
ill a
id y
ouin
the
expl
anat
ion
of d
ata
colle
ctio
n.O
ne m
etho
d of
pre
sent
atio
n th
at m
ight
be d
esir
able
wou
ld b
e to
teac
h th
e un
it as
pre
sent
ed in
the
outli
ne a
nd c
oncl
ude
with
indi
vidu
alpr
ojec
ts b
y th
e st
uden
ts w
hich
use
all
the
tech
niqu
es
stud
ied
in th
e un
it.
Obj
ectiv
es1.
To
lear
n th
e im
port
ance
and
val
ueof
sta
tistic
s
2. T
o le
arn
how
to c
olle
ct, t
abul
ate,
and
grap
h da
ta
:3. T
o le
arn
how
to s
umm
ariz
e da
ta b
yca
lcul
atin
g m
easu
res
of c
entr
al te
nden
cy a
ndra
nk
4. T
o le
arn
how
to c
alcu
late
mea
sure
sof
dis
pers
ion
and
to u
se th
ese
calc
ulat
ions
in p
redi
ctin
g ev
ents
5 T
o le
arn
how
to s
elec
t sam
ples
and
then
use
thes
e sa
mpl
es f
or q
ualit
y co
ntro
l
Con
tent
Proc
edur
es a
nd A
ctiv
ities
Tea
chin
g A
ids
A.
The
Rol
e of
Sta
tistic
s in
Dai
ly A
f-B
egin
by
show
ing
exam
ples
of
stat
istic
sfo
und
Huf
f, D
arre
ll. H
ow to
Lie
With
Sta
tistic
s
fair
sin
new
spap
ers,
mag
azin
es, b
ooks
,and
pam
phle
ts
1. W
hy is
the
anal
ysis
of
data
im-
port
ant?
Wha
t doe
s st
atis
tics
in-
volv
e?
2. H
ow is
sta
tistic
s us
ed in
bus
ines
s,ed
ucat
ion,
indu
stry
, sci
ence
, and
gove
rnm
ent?
Pres
ent a
sim
ple
set o
f da
ta, s
uch
az te
st s
core
sfo
r th
e cl
ass.
Use
the
data
to il
lust
rate
new
vo-
cabu
lary
, suc
h as
dat
a, s
tatis
tics,
tabl
es, r
awsc
ores
, dis
trib
utio
n.Sh
ow th
e ne
ed f
or o
rgan
i-za
tion,
arr
ange
men
t, an
d su
mm
arie
s of
data
Find
sou
rces
whe
re s
tatis
tics
have
bee
nm
isus
ed.
Rea
d th
ese
to th
e cl
ass
A g
ood
bulle
tin b
oard
idea
with
whi
ch to
begi
nth
e un
it m
ight
be
to w
ork
as m
uch
voca
bula
ryas
pos
sibl
e in
to a
char
t or
pict
ure
diag
ram
of
ast
atis
tical
pro
blem
II' h
at b
usin
ess
ente
rpri
ses
are
base
d on
stat
is-
tical
dat
a or
on
the
colle
ctio
n of
dat
a su
ch a
sin
-su
ranc
e, m
arke
ts f
or g
rain
and
lives
tock
, bus
ines
s
49
John
son,
D. A
., an
d G
lenn
, W. H
. The
Wor
ld o
f St
atis
tics
Wal
lis, W
ilson
A.,
and
Rob
erts
, Har
ry V
.T
he N
atur
e of
Sta
tistic
s. C
ollie
r, 1
962.
pa.
Cen
sus
Bur
eau
Rep
orts
Bul
letin
of
the
U.S
. Bur
eau
of S
tand
ards
Bul
letin
s of
the
U.S
. Dep
artm
ent o
fA
gric
ultu
reR
epor
ts o
f th
e U
.S. D
epar
tmen
t of
Lab
or
anal
ysts
? D
iscu
ss g
over
nmen
t ent
erpr
ises
that
colle
ct a
nd a
naly
ze s
tatis
tical
dat
a, s
uch
asth
eC
ensu
s B
urea
u, th
e W
eath
er B
urea
u, th
eB
urea
uof
Sta
ndar
ds, t
he D
epar
tmen
t of
Agr
icul
ture
Exp
lain
how
the
data
col
lect
ed is
use
d.Fo
r ex
-am
ple,
get
a s
peak
er f
rom
the
wea
ther
bure
auto
exp
lain
the
use
of s
tatis
tics
in h
isw
ork
3. W
hat a
re s
ourc
es o
f st
atis
tical
Go
into
mor
e de
tail
here
as
to s
peci
fic
sour
ces
data
for
cert
ain
type
s of
dat
a. I
llust
rate
the
vari
edty
pes
of s
tatis
tical
dat
a fo
und
in c
urre
nt n
ews-
pape
rs a
nd m
agaz
ines
.W
hat l
ibra
ry b
ooks
or
pam
phle
ts a
re d
evot
ed to
sta
tistic
al r
epor
ts
4. W
hat a
re th
e pe
rson
al u
ses
ofD
iscu
ss c
omm
unity
pro
blem
s w
hich
nee
d a
solu
-tio
n. D
iscu
ss s
choo
l dec
isio
ns b
ased
on
loca
lda
ta, t
est s
core
s, a
nd a
ccid
ents
, dis
cuss
the
role
of s
tatis
tics
in a
thle
tics
stat
istic
al d
ata?
B. L
earn
ing
to R
ead
and
to P
rese
ntD
ata
1. H
ow d
o w
e re
ad ta
bles
of
stat
is-
tical
dat
a?
Hav
e th
e st
uden
ts k
eep
a no
tebo
ok in
whi
chth
ey c
olle
ct n
ews,
adv
ertis
emen
ts, a
rtic
les
base
don
sta
tistic
s
Hav
e st
uden
ts c
olle
ct o
rigi
nal d
ata
by s
urve
ying
stud
ents
, tea
cher
s, p
aren
ts, r
egar
ding
act
iviti
es,
opin
ions
, or
mea
sure
s. T
hese
mea
sure
s m
ay b
eof
hei
ght,
wei
ght,
age,
sho
e si
ze, h
at s
ize,
hea
rtbe
at, f
or e
xam
ple.
Dat
a sh
ould
then
be
stud
ied
late
r to
det
erm
ine
mea
sure
s of
cen
tral
tend
enci
esan
d m
easu
res
of d
ispe
rsio
ns. A
wri
tten
anal
ysis
shou
ld b
e gi
ven
Illu
stra
te b
y ov
erhe
ad p
roje
ctor
or
disp
lays
or
dupl
icat
ions
, a v
arie
ty o
f ta
bles
lrif
dat
a fr
omcu
rren
t rep
orts
. Hav
e st
uden
ts c
olle
ctta
bles
of
data
and
wri
te in
terp
reta
tions
of
the
data
. Hav
eea
ch s
tude
nt f
ind
a ta
ble
of d
ata
and
pres
ent i
ton
a s
heet
of
note
book
pap
er. T
hen
wri
te s
ome
ques
tions
whi
ch c
an b
e an
swer
ed b
y an
alys
is o
fth
e ta
ble
Stre
ss th
e im
port
ance
of
wri
ting
inte
llige
nt q
ues-
tions
abo
ut th
e da
ta
2. H
ow d
o w
e co
nstr
uct a
tabl
e fo
rU
se d
ata
from
a c
lass
test
, sur
vey,
ath
letic
eve
nts,
repo
rtin
g da
ta?
or s
imila
r lo
cal s
ituat
ions
toco
nstr
uct a
tabl
e
50
Wor
ld A
lman
acE
ncyc
lope
dias
Bus
ines
s W
eek,
For
tune
, and
bus
ines
s pe
ri-
odic
als
Farm
jour
nals
, con
sum
er jo
urna
ls
Spor
ts d
ata
of th
e lo
cal o
r sc
hool
pap
er.
Dat
a fr
om th
e at
hlet
ic d
epar
tmen
t. D
ata
from
insu
ranc
e co
mpa
nies
, wea
ther
bure
au, o
ther
sou
rces
Dat
a fr
om n
ewsp
aper
s, m
agaz
ines
, bul
le-
tins,
ove
rhea
d pr
ojec
tor
and
over
lays
3. H
ow d
o w
e ta
bula
te s
core
sfr
eque
ncy
dist
ribu
tion?
C. P
rese
ntin
g D
ata
in P
ictu
res
Dis
cuss
the
title
, col
umn
head
ings
, and
row
labe
ls th
at w
ill m
ake
the
data
mea
ning
ful.
Hav
eea
ch s
tude
nt e
,olle
ct d
ata
by a
sur
vey,
an
expe
ri-
men
t, or
by
a se
arch
of
curr
ent n
ews
and
or-
gani
ze th
is d
ata
in a
tabl
e
Dev
elop
a s
erie
s of
fiv
e to
ten
sets
of
defi
ned
data
whi
ch c
ould
be
used
in th
e de
velo
pmen
tof
this
uni
t fro
m s
hort
, ove
r-si
mpl
ifie
d si
tua-
tions
, to
long
er, m
ore
invo
lved
situ
atio
ns
in a
Use
cla
ss d
ata
to d
iscu
ss in
terv
als,
inte
rval
limits
, and
dat
a ta
bula
tions
in a
fre
quen
cy d
is-
trib
utio
n. A
pply
thes
e pr
inci
ples
to a
var
iety
of
stat
istic
al r
epor
ts to
det
erm
ine
whe
re s
core
s ar
ere
cord
ed, w
hat a
re m
axim
um o
r m
inim
um s
core
sre
port
ed
1. D
ot g
raph
s of
fre
quen
cy d
istr
i;_tio
ns
2. B
ar g
raph
s an
d hi
stog
ram
s
3. C
ircl
e or
rec
tang
ular
dis
trib
utio
ngr
aphs
4. L
ine
grap
hs a
nd c
ompo
und
line
grap
hs
Use
gra
phs
from
cur
rent
new
s or
Con
fere
nce
Boa
rd R
epor
ts o
r al
man
acs
to il
lust
rate
type
sof
gra
phs
and
how
to r
ead
them
Hav
eA
phs
of s
ever
al ty
pes
prep
ared
so
that
stud
ents
will
hav
e ex
ampl
es o
f ea
ch ty
pe
Hav
e st
uden
ts c
olle
ct a
t lea
st o
ne e
xam
ple
ofea
ch ty
pe o
f gr
aph
to m
ount
on
note
book
pap
er.
Wri
te q
uest
ions
for
eac
h gr
aph
whi
ch c
an b
ean
swer
ed b
y re
adin
g th
e gr
aph
Dis
cuss
the
diff
eren
ce b
etw
een
a ba
r gr
aph
and
a hi
stog
ram
Rev
iew
per
cen
t in
term
s of
fin
ding
wha
t per
cent
a p
art i
s of
the
who
le
Hav
e st
uden
ts c
olle
ct d
ata
from
the
clas
s,sc
hool
, or
com
mun
ity. D
raw
gra
phs
of th
is d
ata.
Be
sure
the
righ
t typ
e of
gra
ph is
use
d an
d th
atea
ch g
raph
is p
rope
rly
labe
led
so th
at it
can
be
read
cor
rect
ly
Hav
e st
uden
ts d
raw
dif
fere
nt g
raph
s of
the
sam
e se
t of
data
to s
how
how
diff
eren
t asp
ects
51
The
Con
fere
nce
Boa
rd W
eekl
y gr
aphs
Film
s: T
he L
angu
age
of G
raph
s.St
atis
tical
Qua
lity
Con
trol
:A
ccep
tanc
e Sa
mpl
ing
Stat
istic
al Q
ualit
y C
ontr
ol:
Proc
ess
Con
trol
5. F
requ
ency
pol
ygon
s
6. R
eadi
ng a
nd in
terp
retin
g gr
aphs
corr
ectly
D. M
easu
res
of C
entr
al T
ende
ncy
and
Ran
k
1. M
ode
2. M
edia
ns a
nd r
ank
in a
gro
up
of th
e da
ta c
an b
e em
phas
ized
. Sho
w h
owgr
aphs
chan
ge w
hen
the
scal
e is
cha
nged
Als
o sh
ow e
xam
ples
of
how
to li
e w
ithst
atis
tics
Pres
ent b
y ov
erhe
ad p
roje
ctio
n, b
y ch
arts
, orb
ydu
plic
atio
n gr
aphs
whi
ch c
reat
e m
isco
ncep
tions
Use
sch
ool a
nd c
omm
unity
dat
a fo
rco
mpu
ting
mea
sure
s of
cen
tral
tend
ency
and
ran
k
Hav
e st
uden
ts f
ind
exam
ples
of
data
sum
mar
ies
for
thei
r no
tebo
ok
3. M
ean
Com
pare
the
diff
eren
t mea
sure
s of
cen
tral
tend
-en
cy f
or v
arie
ddi
stri
butio
ns
4. S
elec
ting
the
prop
er m
easu
re o
fD
iscu
ss h
ow m
easu
res
of c
entr
al te
nden
cy o
rce
ntra
l ten
denc
y fo
r a
set o
f da
tape
r ce
nts
may
be
used
tocr
eate
fal
se im
pres
-si
ons.
Str
ess
the
impo
rtan
ce o
f ac
cura
cy o
f co
m-
puta
tion
with
met
hods
of
chec
king
res
ults
E. M
easu
res
of D
ispP
rsio
n
1. R
ange
2. A
vera
ge d
istr
ibut
ion
3.;:,
t and
ard
devi
atio
n
4. W
hat i
s th
e pr
obab
ility
of
a sc
ore
or a
n ev
ent?
F. S
ampl
ing
and
Pred
ictio
ns
1. W
hy s
ampl
es a
re n
eces
sary
Use
loca
l dat
a fo
r co
mpu
ting
mea
sure
s of
dis
-pe
rsio
n
Use
a s
impl
iqed
tabl
e of
nor
mal
dis
pers
ion
topr
edic
t the
like
ly o
ccur
renc
e of
a s
core
or
anev
ent
Find
exa
mpl
es o
f da
ta w
hich
giv
e di
st r
ibut
ions
appr
oach
ing
a no
rmal
dis
trib
utio
n
1Tse
som
e of
the
sam
e pr
oble
ms
as w
orke
d ea
rlie
ron
ly e
xten
ding
thei
r us
e fo
r ea
ch n
ew c
once
pt.
Sele
ct o
ne o
r tw
o ne
w s
ets
of d
ata
and
carr
yth
roug
h th
e an
alys
is
Rev
iew
the
sect
ion
on r
ando
m s
ampl
ing
and
prob
abili
ty f
rom
Uni
t II 52
Huf
f. H
ow to
Lie
With
Sta
tistic
s
John
son
and
Gle
nn. T
he W
orld
of
Stat
is-
tics
Wal
lis a
nd R
ober
ts. T
he N
atur
e of
Sta
-tis
tics
Hav
e a
stat
istic
ian
or a
ctua
ry f
rom
a lo
cal
insu
ranc
e co
mpa
ny o
r in
dust
ry ta
lkab
out s
tatis
tics
in in
sura
nce,
med
icin
e,an
d bu
sine
ss
Tab
le o
f ra
ndom
num
bers
2. H
ow to
sel
ect r
ando
msa
mpl
es
3. H
ow s
ampl
ing
resu
lts a
reim
-pr
oved
by
incr
easi
ng th
e sa
mpl
esi
ze
4. H
ow s
ampl
es a
re u
sed
inin
dust
ryan
d go
vern
men
t
Perf
orm
exp
erim
ents
in s
ampl
ing
by d
raw
ing
card
s, m
arbl
es, r
ando
mnu
mbe
rs, c
ount
ing
let-
ters
, sur
veyi
ng s
tude
nts,
reco
rdin
g da
ily s
tatis
-tic
s. C
ompa
re th
e sa
mpl
ere
sults
with
the
char
-ac
teri
stic
s of
the
entir
epo
pula
tion.
Com
bine
sam
ples
to s
how
the
resu
lts,o
fin
crea
sing
sam
ple
size
by
sequ
entia
l sam
plin
g
Mea
sure
ave
rage
hei
ght f
or th
ecl
ass
by a
com
-pl
ete
cens
us a
nd b
y sa
mpl
ir7;
and
com
pare
re-
sults
Hav
e a
qual
ity c
ontr
ol e
xper
tfr
om in
dust
ry o
rgo
vern
men
t dis
cuss
sta
tistic
alm
etho
ds, s
am-
plin
g an
d qu
ality
, con
trol
A g
ood
qual
ity c
ontr
ol d
ispl
ayis
the
follo
win
g:H
ave
one
urn
with
90
per
cent
whi
te a
nd 1
0 pe
rce
nt b
lack
. Whi
te is
an
eff.
ctiv
e pa
rt a
nd b
lack
ade
fect
ive
part
. Thi
s re
pres
ents
the
proc
ess
inco
ntro
l. T
o re
pres
ent t
hem
achi
ne b
reak
dow
n,su
bstit
ute
an u
rn w
ith 5
0 pe
r ce
ntw
hite
and
50
per
cent
bla
ck.
How
man
y sa
mpl
es a
rene
eded
until
the
brea
kdow
n sh
ows
up?
Col
lect
ions
of
mar
bles
, car
ds,
pebb
les,
for
exam
ple
Qua
lity
cont
rol s
tatis
ticia
nfr
om in
dust
ry
Publ
ic O
pini
on P
oll w
orke
r
Film
s: I
ntro
duct
ion
to W
ork
Sam
plin
g
Dat
a fo
r In
vest
igat
ion
If y
ou w
ant t
o co
llect
dat
aab
out c
urre
nt e
vent
s, th
e lis
tbel
ow w
ill s
ugge
st s
ome
poss
ible
idea
s. Y
ou c
an th
en u
se th
ese
data
to a
pply
the
prin
cipl
es d
iscu
ssed
inth
is u
nit.
Lab
el c
ompl
etel
y th
eda
ta c
olle
cted
, des
crib
e th
em
etho
d of
col
lect
ion,
dra
w a
grap
h of
the
data
and
wri
te q
uest
ions
Lha
t can
be a
nsw
ered
fro
m th
e da
ta in
you
rno
tebo
oks.
1. S
cori
ng r
ecor
ds a
t ath
letic
even
ts2.
Wea
ther
rep
orts
tem
pe-a
ture
,rai
nfal
l, hu
mid
ity, s
torm
s3.
Tra
ffic
rec
ords
acci
dent
s, a
mou
nt o
f tr
affi
c, n
umbe
rof
veh
icle
s, n
umbe
r of
par
king
plac
es
4. S
choo
l abs
ence
s or
tard
ines
ses
5. S
choo
l cos
ts6.
Lun
chro
om o
r ca
ndy
sale
s7.
Vita
l sta
t:stic
birt
hs, d
eath
s, m
arri
ages
, une
mpl
oyth
ent
8. R
ecre
atio
nal a
ctiv
ities
of f
rien
dsra
dio,
mov
ies,
boo
ks, m
agaz
ines
,sp
orts
, hob
bies
9. M
arke
t quo
tatio
nsst
ocks
, pai
ns, c
attle
10. N
ewsp
aper
s -a
ds,p
ictu
res,
com
ics
11. B
usin
ess
cond
ition
ssa
les,
pri
ces,
ban
k de
posi
ts, i
nter
est
rate
s
12. T
ax r
aLe
and
expe
nditu
res
1:3.
Sch
ool g
rade
s an
d m
arks
53
14. M
oney
in c
ircu
latio
n, p
ublic
deb
t15
. Ele
ctro
card
iogr
ams,
tem
pera
ture
rea
ding
s, r
espi
ratio
n ra
tes
16. U
tility
bill
sgas
, ele
ctri
c, w
ater
17. U
se o
f le
tters
, wor
ds, n
umbe
rs p
er p
age
in a
boo
k18
. Clo
thes
inve
ntor
ycol
or, n
umbe
r, ty
pe19
. Hei
ghts
, wei
ghts
, sho
e si
zes
20. D
istr
ibut
ion
of b
irth
days
, and
the
prob
abili
ty o
f so
me
birt
hday
fal
ling
with
in a
cer
tain
tim
epe
riod
21. C
oin
date
s
Ref
eren
ces
Huf
f, D
arre
ll. H
ow to
Lie
With
Sta
tistic
s. N
orto
n an
d C
ompa
ny, 1
954
John
son,
D. A
., an
d G
lenn
, W. H
. The
TV
orld
of
Stat
istic
s. W
ebst
er P
ublis
hing
Com
pany
, 196
1M
oste
ller,
Rou
rke,
and
Tho
mas
. Pro
babi
lity.
Add
ison
Wes
ley
Publ
ishi
ng C
ompa
ny, 1
961
Nat
iona
l Con
fere
nce
Boa
rd W
eekl
y G
raph
s. F
ree.
460
Par
k A
venu
e, N
ew Y
ork
22, N
. Y.
Wal
lis, W
. A.,
and
Rob
erts
, H. V
. The
Nat
ure
of S
tatis
tics.
Col
lier,
196
2. p
a.
Film
s Intr
oduc
tion
to W
ork
Sam
plin
g. U
nive
rsity
of
Cal
ifor
nia
The
Lan
guag
e of
Gra
phs.
Cor
onet
Stat
istic
al Q
ualit
y C
ontr
ol: A
ccep
tanc
e Sa
mpl
ing.
Uni
ted
Wor
ld F
ilms
Stat
istic
al Q
ualit
y C
ontr
ol: P
roce
ss C
ontr
ol. U
nite
d W
orld
Film
s
.1111111111Wir
0E-40
,--,7-.1111111111111011101111.'
FACT OR FANCY
UN
IT V
FAC
T O
R F
AN
CY
Intr
oduc
tion
Fact
or
Panc
y is
a u
nit o
n lo
gic
desi
gncd
pri
mar
ily to
giv
e th
est
uden
ts a
n aw
aren
ess
that
ther
e ca
n be
orga
niza
tion
to o
ne's
thin
king
abo
ut d
aily
pro
blem
s di
vorc
edfr
om th
e st
uden
t's ty
pica
l em
otio
nal
invo
lvem
ent.
To
this
end
, the
exa
mpl
es, s
ituat
ions
,an
d
prob
lem
s in
volv
ed s
houl
d be
with
inth
e lif
e ex
peri
ence
of
the
stud
ents
and
mea
ning
ful t
o th
em. L
ivel
y cl
ass
part
icip
atio
n in
all
dis-
cuss
ions
sho
uld
be e
ncou
rage
d.
As
muc
h m
athe
mat
ics
as p
ossi
ble
can
be in
ters
pers
ed in
the
exam
ples
but
the
teac
her
shou
ld s
et r
ealis
tic g
oals
, for
ther
e is
a
dive
rgen
ce o
f op
inio
n as
to ju
st h
owm
uch
logi
c a
high
sch
ool s
tude
nt is
capa
ble
of c
ompr
ehen
ding
. A s
tage
may
be
reac
hed
whe
re
ther
e ar
e fa
ctor
s of
mot
ivat
ion,
mat
urat
ion,
and
inte
llige
nce
limits
whi
chre
stri
ct im
prov
emen
t bey
ond
this
poi
nt.
Sym
bolic
logi
c, f
or
exam
ple,
tend
s to
bec
ome
mor
e an
d m
ore
abst
ract
and
a c
ut-o
ff p
oint
may
be
reac
hed
whe
re n
egat
ive
mot
ivat
ion
is a
ppar
ent.
One
of
the
mos
t im
port
ant
obje
ctiv
es in
teac
hing
logi
c is
teac
hing
for
tran
sfer
of
lear
ning
. Gen
eral
izat
ions
shou
ld b
e fo
rmed
and
the
stud
ent t
augh
t to
reco
gniz
eth
e ap
plic
abili
ty o
f th
ese
rule
s an
dpr
inci
ples
to s
ituat
ions
and
pro
blem
s ot
her
than
thos
e us
ed in
the
teac
hing
of
thes
e ru
les.
The
re m
ustb
e no
mis
conc
eptio
n th
at lo
gic
isbe
ing
taug
ht f
or th
e so
-cal
led
"tra
inin
gof
the
min
d."
By
the
use
of m
eani
ngfu
l exa
mpl
es a
nd il
lust
ratio
nsth
e te
ache
r sh
ould
end
eavo
r to
teac
hth
e di
rect
val
ue o
f lo
gic
in m
any
fiel
dsso
cial
,aes
thet
ic,
and
utili
tari
an.
Obj
ectiv
es1.
To
lear
n to
app
ly th
e pr
inci
ples
of lo
gic
to m
athe
mat
ics
and
othe
rsp
ecif
ic s
choo
l sub
ject
s
0T
o fo
rm s
peci
fic
rule
s of
logi
cal
reas
onin
g so
that
gen
eral
izat
ions
eln
be f
orm
ed w
hich
can
be
appl
ied
to o
ther
situ
atio
ns, e
spec
ially
pers
onal
beh
avio
ral e
xper
ienc
es a
ndth
e pr
oble
ms
invo
lved
3. T
o le
arn
to r
ecog
nize
fal
laci
esin
rea
soni
ng
4. T
o be
abl
e to
dis
tingu
ish
the
diff
eren
ce b
-tw
een
trut
h an
d va
lidity
,de
duct
ive
and
indu
ctiv
e re
ason
ing
5. T
o st
udy
sym
bolic
logi
c,pa
rtic
ular
ly tr
uth
tabl
es
6. T
o le
arn
abou
t syl
logi
sms
and
thei
r us
e in
rea
soni
ng
7. T
o le
arn
abou
t the
appl
icat
ion
and
use
of s
ymbo
lic lo
gic
inco
mpu
ter
oper
atio
ns
Con
tent
Proc
edur
es a
nd A
ctiv
ities
The
teac
her
can
intr
oduc
e th
is u
nit
by g
ivin
gpr
oble
ms
who
se s
olut
ion
depe
nds
upon
logi
cal
reas
onin
g. F
or e
xam
ple:
the
mis
sion
ary
and
can-
niba
l pro
blem
55
Tea
chin
g A
ids
Seve
ral r
efer
ence
s w
ill b
e va
luab
le th
roug
h-ou
t the
ent
ire
unit:
John
son,
D. A
. bvi
e an
d R
.,aso
ning
inM
athe
mat
ics.
Web
ster
, 196
3
Supp
es a
nd H
ill. M
athe
mat
ical
Log
ic f
orth
e Sc
hool
v. S
tanf
ord
Uni
v. 1
961
Stan
dard
4-,
,aom
etry
text
s
A. L
ogic
al r
easo
ning
1. C
hara
cter
istic
s of
rea
soni
ng
Val
idity
3. T
ruth
Poin
t out
the
impo
rtan
ce o
f di
vorc
ing
emot
ions
from
logi
cal t
hink
ing
Cla
ss s
houl
d re
cogn
ize
step
s:1.
Dif
ficu
lty is
fel
t2.
Pro
blem
is c
lass
ifie
d an
d de
fine
d3.
Sea
rch
for
clue
s4.
Sug
gest
ions
for
sol
utio
n ap
pear
5. E
valu
atio
n of
sug
gest
ions
6. S
olut
ion
is a
ccep
ted
Cor
rect
pat
tern
of
reas
onin
g w
hich
giv
es v
alid
but n
ot n
eces
sari
ly tr
ue c
oncl
usio
nsG
ive
exam
ples
, suc
h as
:A
ll bo
ys a
re g
eniu
ses
John
is a
boy
John
is a
gen
ius
Bas
ic a
ssum
ptio
ns tr
ue a
nd r
easo
ning
is c
orre
ctG
ive
exam
ples
of
assu
mpt
ions
or
prem
ises
fro
mm
any
fiel
ds
Stre
ss d
iffe
renc
e be
twee
n va
lidity
and
trut
h
Giv
e de
tails
of
a m
urde
r an
d ha
ve s
tude
nts
try
to th
ink
ease
thro
ugh
and
conv
ince
cla
ssnm
tes
they
hav
e re
ache
d a
logi
cal c
oncl
usio
n
Hav
e st
uden
ts m
entio
n pr
oble
ms
in s
choo
l lif
e.ho
me
life,
in r
elat
ions
to m
embe
rs o
f op
posi
tese
x, a
nd d
iscu
ss s
teps
they
wou
ld h
ave
to ta
kein
thin
king
thro
ugh
to a
suc
cess
ful s
olut
ion.
Wat
ch f
or e
mot
iona
lly-t
oned
ana
lyse
s
Lis
t com
mon
ste
ps in
thes
e pr
,,,le
ms
and
rdat
eto
mat
hem
atic
al a
naly
sis
Stud
ents
may
be
led
to a
com
pari
eon
such
as:
In a
that
& m
atie
al p
robb
1. Y
ou m
ust g
et th
e N
% h
h p
robl
em c
lear
ly in
min
d2.
Fin
d ou
t \ %
h a
t sp
ecif
ic q
uest
ions
are
ask
ed:1
. Dec
ide
wha
t is
give
n in
the
prob
lem
Ntl
Fles
ch, R
udol
ph. T
hP A
rt o
f C
lear
Thi
nk-
ing.
Har
per,
195
1
If a
vaila
ble,
sho
w f
ilm: D
o Y
ou K
now
Tho
rto
Mak
e A
Sto
tt in
ent (
tf F
act?
Ass
ocia
tion
Film
s, I
nc.
Mid
wes
tern
Bra
nch
Off
ice
561
Hill
grov
e A
venu
eL
a G
rang
e, I
llino
is
Rip
ley,
H. A
. Miiw
tu _
TIy
.tttI
t s D
et e
ct °
-gr
ams)
Hou
ghto
n M
iffl
in, 1
932.
or
com
-pa
rabl
e m
ater
ial
If a
vaila
ble,
sho
w f
ilms:
How
to J
udge
Aut
hori
ties
How
to J
udge
Fae
tsH
ow to
Thi
nkC
oron
et F
ilms.
Inc
.C
oron
et B
ldg.
Chi
cago
1. I
ll.
B. D
efin
ition
s an
d as
sum
ptio
ns
4. D
ecid
e w
hat p
roce
sses
izop
erat
ions
of
mat
h-em
atic
s ar
e su
itabl
e to
use
5. E
stim
ate
the
resu
lt6.
Per
form
the
com
puta
tion
7. C
heek
the
resu
lt
In a
n eu
ryda
y lif
e si
tuat
ion:
1. Y
ou m
ust d
eter
min
e w
hat
the
cond
ition
.; su
r-ro
undi
ng th
e pr
oble
m a
re2.
Rec
ogni
ze th
e re
al p
robl
em.D
isre
gard
fac
tors
whi
ch h
ave
no b
e.)r
ing
3. C
onsi
der
wha
t you
kno
wan
d th
e as
sunr
.)-
tions
you
mus
t mak
e4.
Dec
ide
wha
t is
the
best
appr
oach
fro
m a
mon
gth
e m
any
that
mig
ht b
e co
nsid
ered
Ant
icip
ate
the
poss
ible
sol
utio
n6.
Arr
ive
at t1
1,2
resu
lt yo
ueo
nsid
er to
be
the
best
logi
cal e
oncl
usio
n or
dec
i,ion
.A
pply
the
solu
tion
App
ly p
rinc
iple
s di
scov
ered
to f
utur
epl
ans
ofm
etho
ds o
f at
tack
on
prob
lem
s
Thr
ough
out d
iscu
ssio
n em
phas
ize
impo
rtan
ce o
f...
NT
v ge
onie
tty te
xts
can
prov
ide
ex-
defi
nitio
n ill
any
arg
umen
t 'br
dis
cuss
ion
Ui.
es f
or th
is s
ectio
n
1. D
efin
ition
sa.
Und
efin
ed te
rms
Giv
e m
athe
mat
ical
exa
mpl
es, s
uch
as:
poin
t,le
ngth
, sur
face
, spa
ce
b. P
rope
rtie
s of
a d
efin
ition
Dis
cuss
oth
er e
xam
ples
. suc
h as
:.ecu
rity
, saf
edr
ivin
g, h
appi
ness
11y
havi
ng s
tude
nts
attA
npt t
ode
fine
com
mon
term
s. s
uch
as c
hair
.th
erm
omet
er, l
emon
, tri
-an
gle.
for
exa
mpl
e\ le
au u
p to
the
prop
ertie
s of
a go
od d
eiln
ithm
,usi
ng c
ompl
ete
sent
ence
s
1. N
ame
the
term
2. ll
ace
in n
eare
st c
lass
;ive
di,ti
ngtv
.thin
g ch
ante
teri
stic
s4.
Use
orl
y te
rms
prev
ious
lyde
fine
d5.
Tes
t tru
th o
f re
vers
e fo
rm
Hav
e st
udcn
ts w
rite
def
initi
ons,
suc
h as
"bal
l"as
use
d in
bas
ebal
l to
dist
ingu
ish
it fr
om "
stri
ke"
Cri
tic;z
e de
fini
tions
, suc
h as
: "Si
st(
's a
repe
ople
who
hav
e th
e sa
me
mot
her"
Dis
tingu
ish
from
impl
icat
ive
stat
emen
tsw
hich
requ
ire
proo
f. I
nter
sper
se a
s m
any
exam
ples
as
poss
ible
fro
m m
atht
mat
ics
(pos
tula
tes
and
ax-
iom
s)
Giv
e ex
ampl
es in
whi
ch th
e st
uden
tis
to g
ive
an a
ssum
ptio
n up
onw
hich
a s
tate
men
t is
prob
-ab
ly b
ased
, suc
h as
:A
bab
y sh
ould
not
eat
hot
dog
sT
o be
pop
ular
Joh
n m
ust b
uy a
ear
Stat
e as
sum
ptio
ns m
entio
ned
in th
eD
ecla
ratio
nof
Ind
epen
denc
e
Dis
cuss
rol
e of
pre
judi
ce in
acc
epta
nce
of a
s-su
mpt
ions
we
use
in li
fe s
ituat
ions
, suc
h as
raci
alin
tole
ranc
e
Dis
cuss
som
e as
sum
ptio
ns c
omm
only
acce
pted
by h
igh
scho
ol s
tude
nts
Exa
mpl
es:
A c
ar g
ives
one
inde
pend
ence
Hap
pine
ss is
dep
ende
nt u
pon
envi
ronm
ent
Onc
e ou
t of
scho
ol, y
ou'll
aut
omat
ical
lybe
happ
y
Cha
rac'
,eri
ze s
teps
:1.
Def
ined
term
s2.
Pre
mis
es3.
Gen
eral
izat
ions
bas
ed u
pon
expe
rim
enta
tion
4. C
oncl
usim
(te
nt a
t ive
)
( ay
es e
xit
ples
of
indu
ctiv
e m
etho
d as
a m
etho
din
mat
hem
atic
s w
hich
may
esta
blis
h a
high
pnth
abill
ty r
athe
r th
an a
fin
al p
roof
, suc
h as
:n
+ 4
1 is
a p
rim
enu
mbe
rO
Vhe
n n
----
41
it is
not
trtli
o
D. D
educ
tive
reas
onin
g
E. S
yllo
gism
s
Use
pra
ctic
al e
xam
ples
,su
ch a
s:H
ow in
duct
ive
reas
onin
gha
s in
flue
nced
sel
ec-
tion
of c
loth
es, p
rogr
ess
ofai
r tr
avel
, pop
u-la
rity
of
sayi
ngs
(Rai
nbe
fore
7, c
lear
bef
ore
11),
sup
erst
ition
s, f
orex
ampl
e
Thi
s to
pic
is a
n in
tuiti
vein
trod
uctio
n to
syl
-lo
gism
s
Cha
ract
eriz
e:1.
Val
id o
r in
valid
arg
umen
t2.
Tru
e or
fal
se c
oncl
usio
n
Giv
e ex
ampl
es th
at c
anbe
test
ed b
y st
uden
tsFl
esch
, Rud
olph
. The
Art
of C
kar
Thi
nk-
tor
valid
ity a
nd tr
uth,
such
as:
ing
All
squa
res
have
rig
htan
gles
All
rect
angl
es h
ave
righ
tang
les
The
refo
re, a
ll re
ctan
gles
P'V
sqva
res
To
beco
me
a po
pula
rst
uden
t is
desi
rabl
e
Bec
omin
g a
foot
ball
play
erw
ill m
ake
you
popu
lar
The
refo
re, b
ecom
ing
afo
otba
ll pl
ayer
is d
e-si
rabl
e
Hav
e st
uden
ts r
ewor
dex
ampl
es to
mak
e th
ear
gum
ent v
alid
Cha
ract
eriz
e pa
rts
of a
syl
logi
sm:
1. G
ener
al s
tate
men
t(M
ajor
Pre
mis
e)2.
Par
ticul
ar s
tate
men
t(M
inor
Pre
mis
e)
3. C
oncl
usio
n
Giv
e ex
ampl
es, s
uch
a.s:
Gen
eral
sta
tem
ent :
If
equa
ls a
read
ded
to e
qual
s,su
ms
arc
equa
l
Part
icul
ar s
tate
men
t: x
2t;
Con
clus
ion:
x8
Gen
eral
sta
tem
ent:
If I
leav
e:A
lool
at 3
:30.
Iar
rive
hom
e at
4:0
059
John
son,
D. A
., an
d G
lenn
,W
. H. S
ets.
Sent
ence
s, a
nd O
pera
tions
. AN
ebst
er, 1
960
F. S
ets
1. U
se in
def
initi
ons
2. U
se in
syl
logi
sms
G. E
quiv
alen
t tra
nsfo
rmat
ions
Tra
ns-
form
atio
ns s
houl
d be
con
side
red
optio
nal b
y th
e te
ache
r
Part
icul
ar s
tate
men
t: I
left
sch
ool a
t :1:
30C
oncl
usio
n: I
arr
ived
hom
e at
4:0
0
Hav
e st
uden
ts w
ork
exam
ples
in w
hich
one
of
the
thre
e st
atem
ents
is o
mitt
ed a
nd m
ust b
esu
pplie
d by
them
Hav
e st
uden
ts c
reat
e th
eir
own
syllo
gism
s to
be
solv
ed b
y ot
her
mem
bers
of
the
clas
s
Use
\ is
ual a
ids,
suc
h as
.ello
phan
e ci
rck,
s. to
clar
ify
Ven
n di
agra
ms
whi
ch a
e m
odel
s of
set
s
Syllo
gism
s us
ed in
pre
viou
s to
pic
coul
d n
diag
ram
med
be
Giv
e m
any
exam
ples
of
usi.
in d
educ
tive
reas
on-
ing.
Use
bot
h fa
lse
and
true
arg
umen
ts in
ex-
ampl
es a
nd h
ave
stud
ents
det
erm
ine
trut
h or
fals
ity b
y us
ing
circ
les
lf s
tude
nts
have
stu
died
set
s pr
evio
usly
, it m
aybe
fea
sibl
e to
go
into
the
topi
e m
ore
thor
ough
lyan
d in
volv
e so
me
Boo
lean
alg
ebra
. Kee
p ex
-am
ples
pra
etie
al a
nd w
ithin
stu
dent
s' e
xper
-ie
nce.
Inv
olve
som
e nm
them
atie
s in
exa
mpl
es
To
mot
ivat
e cl
ass,
cim
side
r an
exe
rcis
e in
logi
csu
ch a
s us
ed b
y L
ewis
Car
-oll:
1. N
o ki
t ten
that
hw
es a
sh is
unt
each
able
2. N
o kh
ten
wIt
hout
a ta
il w
ill p
lay
with
a
Kitt
ens
with
whi
sker
s al
way
s lo
ve f
ish
4 N
o te
acha
ble
kitte
n ha
s gr
eett
eyes
No
kit t
ens
hai
. e ta
ik,in
loss
the
hz,v
e w
hisk
ers
Whz
o is
dm
. on
e de
duct
ion
that
can
be
deaw
nfr
om th
is s
et o
f st
einm
.?
z-to
.,1..k
nts
couk
l be
enco
urag
ed to
bri
ng in
ex-
ampl
es o
f th
is .1
,7-p
e fo
r a
clas
s co
ntes
t
"Con
stru
ctin
g L
ogic
Puz
zles
" by
Hor
ace
Will
iam
s in
The
Mat
hem
atic
sT
each
er. N
ovem
ber,
196
1
Car
roll,
Lew
is. M
athe
mat
ical
Rec
reat
ions
qf
Lew
is C
arro
ll. C
ol. I
Sym
bolic
Log
ic a
wl
The
Gam
e of
Log
ic. D
over
, 195
8
Buc
halte
r, B
arba
ra. "
The
Log
ic o
f N
on-
sens
e."
Mat
hem
atic
s T
each
er. M
ay ,
1962
Wyl
ie, C
. IL
101
lles
in T
houg
ht ti
rdL
ogic
% D
over
, 195
7
1. S
tate
men
ts o
f th
e ty
pe"a
ll A
isU
s(to
cla
rify
B'
a. N
o A
isto
n-I.
b. N
o no
n-B
is A
C. A
ll no
n-B
is n
on-A
2. S
tate
men
ts o
f th
e ty
pe"N
o A
isB
"a.
No
B is
Ab.
All
B is
non
-Ac.
All
A is
non-
B
3. S
tate
men
ts c
onta
inin
gth
e w
ord
"som
e"a.
Som
e A
is B
Som
e B
is A
b. S
ome
A is
not
BSo
me
A is
B
FL N
on-e
quiv
alen
t tra
nsfo
rmat
ions
1. C
onve
rse
2. I
nver
se
"Aug
ustu
s de
Mor
gan
and
Mod
ern
Mat
mat
h's.
" M
athe
mat
ics
Teo
ch(r
. &In
..19
63
Exa
mpl
es s
hout
(' be
num
erou
s an
dw
ithin
the
real
m o
f th
e st
uden
t's e
xper
ienc
e.T
his
is th
ebe
ginn
ing
of th
e us
e of
the
tool
s of
sym
bolic
logi
c an
d sh
ould
be
anun
ders
tand
able
and
thou
,.tht
-pro
voki
ng e
xper
ienc
e
Hay
.. st
uden
ts w
rite
equ
ival
ent
tran
sfor
mat
ions
of e
ach
of th
e ty
pos
use
mul
tiple
cho
ice
exer
cist
's in
whi
ch th
eG
aHne
r, M
artin
. "L
ogic
Mac
hine
s."
stud
ent h
as to
sel
ect e
quiv
alen
t :t
n.en
tsSc
ient
ijic
Am
erka
n, M
arch
, 195
2
Mak
e a
prim
itive
logi
c m
achi
ne to
show
ver
yPf
eiff
er, J
ohn.
"Sy
mbo
lic L
ogic
."Sr
i( n
-
prac
tical
use
of
tran
sfor
mat
ions
tific
A m
uica
n. D
ec.,
1950
`f h
is to
pic
is in
clud
ed o
niy
for
com
piet
enes
s of
the
unit.
his
dou
btfu
l tha
t the
n_ w
oald
be m
any
clas
ses
inte
rest
ed o
r m
atur
e en
ough
/ofi
nd it
wor
th w
hile
. Om
ittin
g th
is m
ater
iai
will
not
affe
ct u
nile
rsla
ndin
g of
fut
ure
1.(1
)it s
Giv
e ex
ampl
es o
f ea
ch ty
pe
61
3: C
ontr
adic
tory
(ne
gatio
n)
4. C
ontr
apos
itive
a. I
ndir
ect
proo
f
I.Fa
llaci
es in
rea
soni
ng
1. H
idde
n as
sum
ptio
ns(D
icto
Sim
plic
iter)
2. I
rrel
evan
cy(N
on S
equi
tur)
3. C
ircu
lar
reas
onin
g
4. D
odgi
ng th
e Is
sue
(Ad
Popu
lum
)
5. H
asty
gen
eral
izat
ion
6. F
alse
ana
logy
7. P
arad
oxes
Indi
rect
pro
of c
ould
be
used
tore
ach
a co
nclu
-si
on f
or s
uch
stat
emen
ts a
s:It
did
not
rai
n la
st n
ight
The
bur
glar
did
not
ent
erby
this
win
dow
Thi
s fi
gure
is n
ot a
squ
are
Thi
s to
pic
coul
d se
rve,
at
this
poi
nt, a
s a
"bre
athe
r" f
or th
ose
who
may
have
bee
n lo
sing
inte
rest
in s
yllo
gism
s.K
eep
the
exam
ple
and
disc
ussi
on li
vely
Supp
ly m
issi
ng a
ssum
ptio
nsan
d w
rite
arg
u-m
ents
in s
yllo
gist
icfo
rm to
det
erm
ine
corr
ect-
ness
Poin
t out
the
wid
e us
e of
unst
ated
ass
umpt
ions
in a
dver
tisin
g
Stud
ezits
sho
uld
be a
ble
tobr
ing
in m
any
ex-
ampl
es o
f pr
actic
al v
alue
inth
eir
lives
The
use
of
the
Lat
in n
ames
of th
e fa
llaci
es m
ayin
tere
st s
omeo
ne in
mak
ing
anin
vest
igat
ion
ofth
e hi
stor
ical
bac
kgro
und
of tr
aditi
onal
lc;ie
A g
ood
exam
ple
of th
is is
the
purc
hase
of
a ea
r.A
job
is n
eede
d to
get
sup
port
for
the
car
that
is n
eede
d to
get
to th
ejo
b. T
his
coul
d be
ahe
ated
dis
cuss
ion
Poin
t out
.,om
mon
use
s by
stud
ents
Mak
e a
colle
ctio
n fr
omre
crea
tiona
l mat
hem
atic
sbo
oks
Hav
e st
uden
ts a
naly
zear
ticle
s to
see
if th
ey c
ande
tect
the
vari
ous
type
s of
falla
cies
62
Cha
se, S
tuar
t. G
uide
s to
Stra
ight
Thi
nkin
g.H
arpe
r, 1
956
Huf
f, D
arre
ll. H
ow to
Lk
With
Sta
tistic
s.N
orto
n, 1
954
Qui
ne, W
. V. '
Para
dox.
"Sc
ient
ific
Am
eri-
can.
Apr
il,19
62
de M
orga
n, A
. Bud
get o
fPar
adox
es. D
over
,19
54
J. T
ruth
tabl
estO
ptio
nal,
but s
tude
nts
find
this
topi
c fa
scin
atin
g. S
ectio
nsG
and
Har
e pr
ereq
uisi
tes
for
this
sec
tion)
Pres
ent a
ser
ies
of s
tate
men
tsin
whi
ch th
est
uden
t, on
the
basi
s of
a tr
uepr
emis
e, m
ust
deci
de if
the
follo
win
g st
atem
ents
are
true
, fal
se,
or u
ndet
erm
ined
.
Exa
mpl
e:
Prem
ise:
It i
s ra
inin
g, o
r it
is s
now
ing
A. I
t is
not s
now
ing,
and
it is
not
rai
ning
B. I
f it
is s
now
ing,
then
it is
not
rai
ning
C. I
f it
is n
ot s
now
ing,
then
it is
rai
ning
D. I
f it
is n
ot r
aini
ng,
then
it is
sra
- w
ing
E. I
t is
not t
he e
ase
that
it is
eith
er r
aini
ngno
r sn
owin
gF.
If
it is
rai
ning
, the
n it
is n
ot s
now
ing
G. I
t is
not r
aini
ng, o
r it
is n
ot s
now
ing
Prog
ress
in d
iffi
culty
to:
Prem
ise:
it is
impo
ssib
le f
or a
mal
lin to
hav
etw
o eq
ual j
akes
and
not
be a
bul
in
A. I
f a
mal
lin h
as tw
oeq
ual j
akes
, the
n it
isa
bulin
B. I
f a
mal
lin is
abu
lin, O
len
it ha
s tw
oeq
ual j
akes
C. I
f a
mal
lin is
not
abu
lin, t
hen
it do
es n
otha
ve tw
o eq
ual j
akes
D. T
here
exi
sts
a m
allin
whi
ch is
not
a b
ulin
but w
hich
has
two
equa
ljak
es
E. A
mal
lin is
a b
ulin
, or
it do
es n
ot h
ave
two
equa
l jak
esF.
A m
allin
has
two
equa
ljak
es, o
r it
is n
ota
bulin
Aft
er th
e ar
gum
ents
ens
uing
ove
rth
e co
rrec
tan
swer
s to
thes
eex
erci
ses
stud
ents
sho
uld
btre
ady
to a
ppre
ciat
e th
e va
lue
of t
h ta
bk.s
63
See
"Dev
elop
ing
a L
ogic
alC
once
pt in
Ele
-m
enta
ry M
athe
mat
ics"
by G
ertr
ude
Hen
drix
in T
he M
athe
mat
ics
Tea
cher
.D
ecem
ber,
195
5
1. T
erm
inol
ogy
Pres
ent i
n ta
bula
r fo
rm f
or f
utur
e re
fere
nce
Exa
mpl
e:
Kat
tsof
f, L
ouis
. "Sy
mbo
lic L
ogic
and
the
Stru
ctur
e of
Ele
men
tary
Mat
hem
atic
s"M
athe
mat
ics
Tea
cher
. Apr
il, 1
962
Spri
ng is
a s
easo
np
3 is
a p
rim
e nu
mbe
rPr
epos
ition
sq
Spri
ng is
a s
easo
nan
d 3
is a
pri
me
num
ber
Con
junc
tion
P A
A
Spri
ng is
a s
easo
nor
3 is
a p
rim
e nu
mbe
rD
isju
nctio
nI)
V q
If s
prin
g is
a s
easo
nth
en 3
is a
pri
me
num
ber
impl
icat
ion
P-->
qSp
ring
is a
sea
son
if a
ndon
ly if
3 is
a p
rim
e nu
mbe
rE
quiv
alen
cep<
*q
Spri
ng is
not
a s
easo
ne.
.. p
2. O
pera
tions
Prac
tice
stat
ing
prin
cipl
es o
f re
ason
ing
syi,b
ol-
ical
ly a
nd tr
ansl
atin
g sy
mbo
ls in
to w
ords
1C
onju
nctio
n
Pq
1P
VI
TT
TT
FI
FF
TF
FF
F
Dis
junc
tion
Pq
PVq
TT
TT
FT
FT
TF
FF
Equ
ival
ence
pq
p<--
--q
TT
TT
FF
FT
iF
FFI
T
Mak
e tr
uth
tabl
es -
Lr
exer
cise
s in
Par
t 1. M
ake
othe
r tr
uth
tabl
es b
ut w
atch
for
lai2
k oe
inte
rest
64
Impl
icat
ion
I)q
P>q
TT
TT
FF
FT
T*
FF
T*
*Stu
dent
s ha
ve d
iffi
culty
in b
elie
ving
this
Neg
atio
n1
-1:o
,1,
-, t,
3. A
pplic
atio
nsD
iscu
ss u
ses
of s
ymbo
lic lo
gic
in d
igita
l com
-pu
ter
calc
ulat
ions
Dep
endi
ng u
pon
abili
ties,
ther
e m
ay b
e in
tere
stin
goi
ng in
to a
mor
e de
taile
d di
scus
sion
of
the
prin
cipl
es u
nder
lyin
g th
e co
nstr
uctio
n of
com
-pu
ters
and
pro
gram
min
g 65
Gile
s, R
icha
rd. "
Bui
ldin
g an
Ele
ctri
cal
Dev
ice
for
Use
in T
each
ing
Log
ic."
The
Mat
hem
atic
s T
each
er. M
arch
, 196
2
"Sym
bolic
Log
ic a
nd L
ogic
al C
ircu
itry
inth
e H
igh
Scho
ol."
The
Mat
hem
atic
sT
each
er. J
an.,
1957
You
ng, F
rede
rick
. Dig
ital C
ompu
ters
and
Rel
ated
Mat
hem
atic
s. G
inn,
196
1
John
son,
D. A
., an
d G
lenn
, W. H
. Com
-pu
ting
Der
ices
. Web
ster
, 196
1
Gar
dner
, Mar
tin. L
ogic
Mac
hine
s an
d D
ia-
gram
s. M
cGra
w,
1958
MAKING A MILLIONOR SOMETHING LESS
UN
IT V
IM
AK
ING
A M
ILL
ION
OR
SO
ME
TH
ING
LE
SSIn
trod
uctio
nIn
vest
rden
ts h
ave
play
ed a
n im
port
ant r
ole
in th
e de
vAop
mei
n, c
f. th
eec
onom
y of
our
cou
ntry
. The
y ar
e fu
ndam
enta
l to
deve
lopi
ngin
the
Uni
ted
Stat
es th
e hi
ghes
t sta
ndar
d of
livi
ngev
er e
njoy
ed b
y a
peop
le in
all
of h
uman
his
tory
. Inv
estm
ents
hav
e he
lped
cre
ate
smal
l bus
ines
ses,
and
hav
e he
lped
thos
e sm
all b
usin
esse
sgr
ow in
to la
rge
corp
oi a
tions
. As
a re
sult,
new
pro
duct
s an
d im
prov
ed p
rodu
cts
resu
lting
fro
m in
dust
rial
res
earc
h ha
ve b
een
deve
lope
d w
hich
in tu
rn m
ake
life
mor
e co
mfo
rtab
le a
nd e
njoy
able
.
The
pur
pose
of
this
uni
t is
to a
cqua
int t
he s
tude
nt w
ith d
iffe
rent
type
s of
inve
stm
ents
,th
e pl
ace
of in
vest
men
ts in
an
indi
vidu
al's
savi
ngs
prog
ram
, the
inne
r w
orki
ngs
of th
e m
arke
t, an
d th
e in
terp
reta
tion
of f
inan
cial
repo
rts.
All
inve
stm
ent e
ntai
ls r
isk.
The
re is
no
sure
way
to w
ealth
. Mar
k T
wai
n's
Pudd
'nhe
ad o
nce
said
"A
pril
is a
par
ticul
arly
ris
kym
onth
ir w
hich
to s
pecu
late
in th
e st
ock
mar
ket.
The
oth
er m
onth
sar
e Fe
brua
ry, D
ecem
ber,
Mar
ch, N
ol. e
mbe
r, M
ay, O
ctob
er,
June
, Aug
ust,
July
, Sep
tem
ber,
and
Jan
uary
." I
t is
part
icul
arly
impo
rtan
t tha
t stu
dent
sre
cozn
ize
the
risk
invo
lved
in in
vest
men
ts.
Obj
ectiv
es:
1. T
o de
velo
p an
und
erst
andi
ng o
f st
ocks
, bon
ds, a
nd o
ther
inve
stm
ents
and
thei
rm
eani
ng a
nd p
urpo
se in
the
econ
omy
2. T
o de
velo
p an
und
erst
andi
ng o
f th
e pl
ace
of s
tock
s an
d bo
nds
inan
indi
vidu
al's
inve
stm
ent p
rogr
am
3. T
o de
velo
p un
ders
tand
ing,
acc
urac
y, a
nd s
peed
in th
e co
mpu
tatio
n sk
ills
invo
lved
Con
tent
Proc
edur
es a
nd A
ctiv
ities
Tea
chin
g A
ids
The
fol
low
ing
teac
hing
aid
s w
ill b
e m
ead
thro
ugho
ut th
e en
tire
unit.
As
ther
e is
som
e du
plic
atio
n in
thes
e m
ater
ials
, it
shou
ld n
ot b
e :n
eces
sary
to o
btai
n th
emal
l. 1Z
azt o
f th
e pa
mph
lets
are
ava
ilabl
efr
ee a
nd in
qua
ntity
.
67
Mak
e us
e of
Tex
tboo
ks o
n B
usin
ess
Mat
h-em
atic
s
Film
sB
ehin
d th
e T
icke
r T
ape.
UW
sd
colo
r 21
min
fre
e
Fair
Exc
hang
e. M
ovie
s U
SA 2
1 m
in f
ree
Opp
ortu
nity
U.S
.A. M
oder
n T
P 27
min
free
Our
Sha
reho
lder
s In
vest
in T
omor
row
. Gen
Mot
ors
17 m
in f
ree
Spec
ial R
epor
t to
Stoc
khol
ders
. Gen
Mill
s7
min
fre
e
'Wha
t Mak
es U
s T
ick?
Mod
ern
TP
12 m
infr
ee
Wor
i. of
the
Stoc
k E
xcha
nge.
Cor
on 1
15
min
Wor
king
Dol
lars
. Mod
ern
TP
13 m
in f
ree
Pam
phle
tsD
ivid
ends
Ove
r th
e Y
ears
. Pai
ne, W
ebbe
r,Ja
ckso
n, a
nd C
urtis
, Pill
sbur
y B
ldg.
,M
inne
apol
is, M
inne
sota
Und
erst
andi
ng th
e Y
ew Y
ork
Stoc
k E
x-ch
ange
. Pai
ne, W
ebbe
r, J
acks
on, a
ndC
urtis
, Pill
sbur
y B
ldg.
, Min
neap
olis
,M
inne
sota
How
to R
ead
a Fi
nanc
ial R
epor
t. M
erri
llL
ynch
Pie
rce
Fenn
er a
nd S
mith
. 240
Ran
d T
ower
, Min
neap
olis
2, M
inne
sota
Yew
Yoi
'k S
tock
Exc
hang
e 1,
'act
Boo
k. P
ub-
lic R
elat
ions
and
Mar
ket D
evel
opm
ent,
Publ
icat
ion
Div
isio
n N
ew Y
ork
Stoc
kE
xcha
nge,
11
Wal
l Str
eet,
New
Yor
k 5,
N. Y
.
Abo
ut T
his
Stoc
k an
d B
ond
Bus
ines
s.M
erri
ll L
ynch
Pie
rce
Fenn
er a
ndSm
ith, 2
40 R
and
Tuw
er, M
inne
apol
is 2
,M
inne
sota
Dir
iden
ds f
or M
ore
rzan
a D
ecad
e. P
ublic
Rel
atio
ns D
epar
tmen
t, A
mer
ican
Sto
ckE
xcha
nge,
86
Tri
nity
Pla
ce, N
ew Y
ork
6, N
. Y.
You
and
the
Inre
stm
ent W
orld
, a s
erie
s of
inve
stm
ent f
olde
rs a
vaila
ble
by w
ritin
gN
ew Y
ork
St o
ck E
xcha
nge,
11
Wal
lSt
reet
, New
Yor
k 5,
N. Y
.
I.In
vest
men
ts
A. S
tock
s1.
Int
rodu
ctio
n
a. N
eed
for
capi
tal
b. P
urpo
se o
f co
rpor
atio
n
c. I
ts le
gal a
spec
ts
2. K
inds
of
stoc
ks, c
omm
on a
ndpr
efer
red
a. D
ivid
end
diff
eren
ces
b. S
tabi
lity
of p
rice
s
e. P
rior
ity o
f pa
ymen
t
The
re a
re tw
o te
chni
ques
whi
ch w
ould
be q
uite
usef
ul in
the
intr
odue
tion
to th
e un
it. O
neap
proa
ch m
ight
be
from
tlw
poi
nt o
f vi
ewof
havi
ng m
oney
to in
vest
, the
oth
er b
eing
a n
eed
for
larg
e su
ms
of m
oney
for
dev
elop
ing
a bu
si-
ness
. The
fir
st a
ppro
ach
is u
sed
here
sin
ce it
ispo
ssib
le to
elic
it fr
om th
e st
uden
ts th
e va
riou
sty
pes
of p
ossi
ble
inve
stm
ents
.Sta
rt b
y as
sum
ing
that
one
or
v.h
mem
bers
of
the
clas
s ha
ve r
e-ce
ived
a la
fge
sum
of
mon
ey, s
ay $
5 O
K a
s an
inhe
rita
nce
from
a r
ich
uncl
e. A
ssum
e fu
rthe
rth
at th
e st
uden
ts h
ave
no im
med
iate
nee
d no
rus
e fo
r th
e m
oney
. Pos
eth
e qu
estio
n: "
Wha
tw
ould
you
do
with
it?"
Cla
ss d
iscu
ssio
n sh
ould
then
giv
e yo
u a
long
list
, inc
ludi
ng m
ost f
orm
sof
inve
stm
ent.
Furt
her
disc
ussi
on s
houl
dfo
llow
in a
n at
tem
pt to
ans
wer
the
ques
tion:
"Wha
tfa
ctor
s sh
ould
det
erm
ine
whe
re w
e pu
t the
mon
ey?"
It i
s no
w p
ossi
ble
to d
efin
e ea
ch f
orm
of in
vest
men
t and
inve
stig
ate
its c
hara
cter
istic
s.w
hich
is o
ne o
f th
e pu
rpos
es o
f th
e un
it
Dis
cuss
nee
d fo
r la
rge
sum
s of
mon
ey f
or d
e-ve
lopi
ng a
bus
ines
s, a
nd n
eces
sity
for
lega
lar
rang
emen
ts to
pro
tect
ow
ners
. Bri
ng n
ews-
pape
r st
ory
of s
ucce
ssfu
l cor
pora
teen
terp
rise
.lo
cal i
f po
ssib
le. S
tres
s po
ssib
ilitie
s fo
r til
l eco
.no
mic
gro
ups
to p
artic
ipat
e in
ow
ners
hip
ofco
rpor
atio
ns
The
se it
ems
are
clos
ely
rela
ted
and
can
bedi
s-cu
ssed
toge
ther
. Bro
kera
ge f
irm
s ha
ve w
ell-
pre-
pare
d pe
ople
ava
ilabl
e to
tell
the
stor
y of
sto
cks
to a
dult
clas
ses.
Loc
al r
epre
sent
ativ
es o
fth
ese
firm
s m
ay b
e in
vite
d to
exp
lain
thei
r fi
eld
to th
est
uden
ts. L
iste
d in
the
teac
hing
aid
s ar
e re
late
dfi
lms.
Cha
rts
are
avai
labl
e fr
om b
roke
rage
fir
ms
and
stoc
k ex
chan
ges
to s
how
com
pari
sons
of
type
s of
sto
cks,
mar
ket t
rend
s, a
nd r
ecor
dsof
mut
ual f
unds
Stud
ents
sho
uld
lear
n ho
w to
rea
d a
fina
ncia
lpa
ge e
arly
in th
e un
it.O
nce
com
mon
and
pre
-fe
rred
sto
cks
have
bee
n di
scus
sed,
-ne
wsp
aper
quot
es o
f cl
osin
g pr
ices
sho
uld
be u
sed
to a
cqua
int
69
The
film
Opp
ortu
nity
U.S
.A. a
s w
ell a
sse
vera
l oth
ers
liste
d, w
ould
ser
ve w
ell
as a
n in
trod
uctio
n.T
he p
amph
let T
heA
mer
ican
cor
pora
tion
in th
e Y
ou a
ndth
e In
rest
men
t Wor
ld s
erie
s w
ould
als
o be
good
Any
one
of
the
film
s lis
ted
rm
ost o
f th
epa
mph
lets
list
ed w
ill c
on. n
bute
gre
atly
to th
is s
ectio
n. I
n ad
ditio
n,lo
cal s
tock
brok
ers
can
supp
ly c
urre
nt b
roch
ures
on
typi
cal s
toek
s an
d m
utua
l fun
ds
If n
ewsp
aper
s ar
e no
t ava
ilabl
e fo
r ea
chst
uden
t, pa
rts
of th
e fi
nanc
ial p
age
can
be c
opie
d on
a tr
ansp
aren
cy a
nd u
sed
with
the
over
head
pro
ject
or
3. C
ompa
riso
n of
com
mon
sto
cks
a. C
onse
rvat
ive
b. S
pecu
lativ
e
c. G
row
th
d. C
yclic
al
e. W
hat a
re "
blue
chip
" st
ocks
?
4. M
arke
t tre
nds
a. B
ear
and
bull
mar
kets
b. C
orre
latio
n w
ith b
usin
ess
con-
ditio
ns
c. H
isto
ry o
f tr
ends
5. M
utua
l inv
estm
ent f
unds
a. D
iver
sifi
catio
n
b. S
tabi
lity
c. P
rofe
ssio
nal
plan
ning
d. R
ate
of c
omm
issi
on
6. O
pera
tion
of th
e m
arke
ts
a. S
tock
exc
hang
es
b. B
roke
rs' a
ctiv
ities
c. B
roke
rs' f
ees
the
stud
ents
with
the
stan
dard
met
hod
of r
e-po
rtin
g
Part
icul
ar e
mph
asis
sho
uld
be p
lace
d on
the
in-
terp
reta
tion
of th
e fr
actio
ns r
epor
ted
for
com
-m
on a
nd p
refe
rred
stoc
k, th
e nu
mbe
r of
sba
res
trad
ed, t
he D
ow-J
ones
indu
stri
al a
vera
ges
and
the
Stan
dard
-Poo
r in
dex
Inve
stig
ate
the
diff
eren
ces
and
sim
ilari
ties
be-
twee
n st
ocks
list
ed o
n th
e N
ew Y
ork
Exc
hang
e,A
mer
ican
Exc
hang
e, a
nd th
e L
ocal
Mar
ket.
Spec
ulat
ive
and
grow
th s
tock
s ar
e m
ore
com
-m
only
fou
nd in
the
loca
l mar
ket w
hile
con
serv
a-tiv
e an
d cy
clic
al s
tock
s ar
e fo
und
on th
e na
tiona
lm
arke
ts
The
kin
ds o
f st
ock
purc
hase
d ar
e us
ually
dete
r-m
ined
by
the
amou
nt o
f ri
sk o
ne is
will
ing
tcta
ke
70
Tak
e a
fiel
d tr
ip to
the
"big
boa
rd"
inM
inne
apol
is o
r St
. Pau
l or
othe
r lo
cal
stoc
k ex
chan
ges
7. C
ompu
tatio
n sk
ills
a. D
ivid
ends
on
pref
erre
d an
dco
mm
on s
tock
s
b. A
ctua
l rat
e of
ret
urn
on in
-ve
stm
ent
c. T
axes
on
divi
dend
san
d sa
le o
fst
ock
8. I
nves
tmen
t pra
ctic
e
B. B
onds
1. T
ypes
a. M
ortg
age
b. D
eben
ture
2. U
ses
a. C
orpo
ratio
n
b. U
nite
d St
ates
c. S
tate
Dem
onst
rate
and
pra
ctic
e sk
ills
invo
lved
. Thi
sw
ill in
clud
e th
e fo
ur f
unda
men
tal o
pera
tions
invo
lvin
g w
hole
num
bers
, fra
ctio
ns, d
ecim
als,
and
per
cent
s. I
f po
ssib
le, o
btai
n in
com
e ta
xfo
rms
and
prac
tice
com
putin
g th
e ta
x on
div
-id
ends
, and
buy
ing
and
selli
ng p
rofi
ts a
nd lo
sses
.E
xpla
in d
ivid
end
excl
usio
ns a
nd lo
ng te
rm in
-ve
stm
ent b
enef
its in
tax
com
puta
tion
Eac
h of
the
text
s lis
ted
at th
e en
d of
this
unit
has
som
e ex
ampl
es f
or c
ompu
tatio
nan
d dr
ill
Use
the
gift
of
$5,0
00 a
nd h
ave
each
stu
dent
A d
aily
new
spap
er w
ith a
sto
ck m
arke
tpi
ck h
is o
wn
stoc
k, k
eep
reco
rds
of p
rofi
ts a
ndqu
otat
ion
sect
ion
will
be
nece
ssar
ylo
sses
fig
ured
at s
tate
d in
terv
als
of ti
me.
You
mig
ht c
hoos
e to
lim
it on
e gr
oup
to p
refe
rred
stoc
k an
d on
e to
com
mon
sto
cks.
Stu
dy s
tock
sfr
om:
Past
rec
ord
of s
ale
pric
ePr
esen
t con
ditio
n an
d ea
rnin
g po
wer
Futu
re p
rosp
ects
Div
iden
d hi
stor
yD
ebat
e ty
pes
of s
tock
sass
ume
argu
men
ts
Pres
ent o
verv
iew
of
bond
s an
d th
eir
secu
rity
,di
ffer
entia
ting
betw
een
mor
tgag
e an
d de
ben-
ture
. Det
erm
ine
wha
t use
is m
ade
of b
onds
inth
e lo
cal c
omm
unity
. Com
pare
sec
urity
and
re-
turn
s to
that
of
stoc
ks
The
pur
chas
e of
cor
pora
te b
onds
sim
ply
mea
nsth
at y
ou a
re le
ndin
g m
oney
to th
at o
rgan
izat
ion.
You
are
the
cred
itor,
and
in r
etur
n fo
r th
e lo
an,
the
inst
itutio
n pl
edge
s to
pay
you
a s
peci
fied
amou
nt o
f in
tere
st o
n sp
ecif
ied
date
s
The
saf
est b
onds
in th
e w
orld
. The
y ar
e pa
yabl
efr
om th
e ta
x re
venu
e of
the
Fede
ral g
over
nmen
t.T
here
are
man
y ty
pes
of g
over
nmen
t bon
ds
The
deb
ts o
f th
e st
ates
res
t on
the
abili
ty to
mee
t the
ir o
blig
atio
ns w
hen
due.
The
y ca
nnot
71
d. M
unic
ipal
e. S
choo
l
3. C
ompu
tatio
ns
a. I
nter
est
b. A
ctua
l rat
e on
inve
stm
ent
c. B
roke
rage
fee
s
d. T
axes
on
retu
rns
C S
pecu
lativ
e In
vest
men
ts1.
Rea
l Est
ate
be c
ompe
lled
to p
ay th
eir
debt
s. B
ut s
tate
s ar
ede
siro
us o
f m
aint
aing
a s
ound
cred
it ra
ting,
ther
efor
e de
faul
t is
unlik
ely.
The
maj
or a
dvan
t-ag
es a
reex
empt
ion
from
fed
eral
inco
me
tax,
safe
ty, a
nd li
quid
ity. T
he r
ate
of in
tere
st is
low
.
Maj
or a
dvan
tage
: fed
eral
!mim
e ta
x de
(lue
-do
ns. D
isad
vant
ages
: low
yiel
d, p
ossi
bilit
y of
defa
ult
Dem
onst
rate
and
pra
ctic
e sk
ills
invo
lved
. Not
eta
x ex
empt
fea
ture
of
cert
ain
bond
s
Stud
y tI
nd d
iscu
ss th
ere
latio
nshi
ps b
etw
een
popu
latio
n gr
owth
, com
mer
cial
deve
lopm
ent,
and
infl
atio
nary
tren
ds, w
ith tF
eri
se a
nd f
all o
fre
al e
stat
e va
lues
. Dis
cuss
irm
obili
ty o
f th
isty
pe o
f in
vest
men
t. L
ocal
age
nt m
aybe
will
ing
to d
iscu
ss h
is w
ork
with
the
clas
s
A b
roch
ure
of th
e lo
cal s
choo
ldi
stri
ct p
er-
tain
ing
to a
rec
ent b
ond
issu
ew
ill b
e of
inte
rest
2. C
ontr
act b
uyin
g fo
r fu
ture
deliv
-E
xpla
in h
ow o
ne c
an c
ontr
act t
ofu
rnis
h at
aA
loca
l bro
ker
wou
ld p
roba
bly
Etv
e in
for-
ery
futu
re ti
me
a ce
rtai
n st
ock
orco
mm
odity
suc
hm
atio
n on
this
type
of
tran
sact
ion,
as
as g
rain
, at a
set
pric
e. T
hen
by p
layi
ng th
ew
ould
the
stoc
k ex
chan
ges
mar
ket o
ne m
ay b
e ab
le to
pur
chas
eth
e ite
mat
a p
rice
low
er th
an th
ede
liver
y pr
ice
guar
-an
teed
, thu
s re
aliz
ing
a pr
ofit.
Str
ess
the
high
lysp
ecul
ativ
e na
ture
of
this
type
ofb
usin
ess
tran
s-ac
tion.
A d
ebat
e m
ight
be
set u
p to
cons
ider
the
rela
tive
secu
rity
and
pos
sibi
litie
sfo
r pr
ofit
ofth
e va
riou
s ty
pes
of in
vest
men
tsst
udie
d
72
STRETCHING YOUR DOLLAR
UN
ITA
VII
STR
ET
CH
ING
YO
UR
DO
LL
AR
Intr
oduc
tion
In m
oder
n so
ciet
y m
any
pres
sure
s ar
e ex
erte
d up
on in
divi
dual
s an
d fa
mili
es; n
ot th
e le
ast o
f th
ese
is th
e co
mpe
titio
n fo
r th
efa
mily
pay
che
ck. M
any
prod
ucts
cle
sign
ed to
mak
e lif
e m
ore
satis
fyin
g ha
ve a
ppea
red.
It i
s ne
cess
ary
for
stud
ents
to u
nder
stan
dim
p( ,r
tant
con
cept
s ab
out m
oney
man
agem
ent i
n or
der
to c
ope
with
thes
e pr
essu
res.
Sou
nd in
form
atio
n ab
out t
he n
atur
e of
Lud
gets
and
the
cost
s of
inst
allm
ent b
uyin
g w
ill b
e an
ess
entia
l ele
men
t in
the
life
of th
e st
uden
t as
he p
lans
his
fut
ure.
Obj
ectiv
es1.
To
deve
lop
skill
s re
late
d to
bud
getin
g at
the
pers
onal
, fam
ily, b
usin
ess,
and
gov
ernm
enta
l lev
els
To
disp
el c
urre
nt m
isco
ncep
tions
abo
ut b
udge
ts a
nd th
eir
purp
oses
3. T
o un
ders
tand
the
plac
e of
con
sum
er c
redi
t in
the
fam
ily's
fin
anci
al p
lan
4. T
o re
cogn
ize
the
cost
s of
con
sum
er c
redi
t buy
ing
5. T
o un
ders
tand
the
adva
ntag
es a
nd d
isad
vant
ages
of
cons
umer
cre
dit b
uyin
g
6. T
o un
ders
tand
the
diff
eren
ce in
cos
t bet
wee
n cr
edit
purc
hase
and
the
cash
pri
ce
7. T
o re
aliz
e th
e im
port
ance
of
esta
blis
hing
inte
iPge
ni b
uyin
g ha
bits
8, T
o un
de-s
tand
that
con
sum
er c
redi
t has
a p
lace
in m
oder
n A
mer
ican
cul
ture
9. T
o be
abl
e to
com
pute
fin
ance
cha
rges
on
inst
allm
mt b
uyin
g, in
tere
st a
nd r
ate
of in
tere
st, t
he tr
ue r
ate
of in
tere
st
10. T
o be
abl
e to
sel
ect c
redi
t pla
ns w
hich
are
bes
t sui
ted
to th
e fa
mily
nee
ds
11. T
o re
cogn
ize
the
need
for
and
abi
lity
to e
stab
lish
a go
od c
redi
t rat
ing
Con
tent
I.B
udge
ting
Proc
edur
es a
nd A
ctiv
ities
Tea
t. :n
g A
ids
An
inte
rest
ing
tech
niqu
e fo
r in
trod
ucin
g th
eun
it, w
ould
be
ti1 ,?
adm
inis
trat
ion
of a
mon
eym
anag
emen
t I.Q
. tes
t. T
he f
ollo
win
g te
st w
aspu
blis
hed
in a
boo
k tit
led
Con
sum
er P
rold
einR
and
Pers
onal
Fin
ance
by
Arc
h A
V. T
'ls
trup
*T
his
ques
tionn
aire
, if
fille
d ou
t as
act.
,rat
ely
asyo
ur m
emor
y pe
rmits
, will
hel
p yo
u to
dis
cove
ryo
ur w
eakn
esse
s in
per
sona
l mon
ey m
anag
e-m
ent.
Eac
h "y
es"
answ
er r
ates
fiv
e po
ints
. Add
the
poin
ts to
fin
d yo
ur m
oney
man
agen
wnt
I.Q
.If
you
r sc
ore
isO
ver
75, c
onsi
der
your
self
a g
ood
mon
eym
anag
erB
etw
een
75 a
nd 5
5, c
onsi
der
your
self
ave
rage
Bet
wee
n 55
and
35,
you
are
bel
ow a
vera
geB
elow
35,
you
are
ver
y po
orr"
.1 t
Cop
yrig
ht 1
957.
McG
raw
-Hill
Com
pany
,In
c. U
sed
by p
erm
issi
on
" I.
Hav
e yG
u m
ade
aro
ugh
plan
for
you
rla
rge
expe
nses
for
thr
yr;I
r?
2. H
ave
you
kept
aw
ritte
n re
cord
of
your
ex-
pend
iture
s fo
r at
leas
t one
mon
th?
3. H
ave
you
exam
ined
you
rrec
ord
of e
xpen
di-
ture
s an
d m
ade
nece
ssar
ych
ange
s?
4. A
re y
ou s
eldo
m"b
roke
" be
fore
you
r ne
xtal
low
ance
or
inco
me
isre
ceiv
ed?
5. W
hen
"bro
ke,"
do y
ou g
ener
ally
get
alon
gas
bes
t you
can
until
you
r al
low
ance
is r
e-ce
ived
?
6. D
o yo
u av
oid
mak
ing
your
self
rili
sera
ble
and
unha
ppy
by f
retti
ngab
out s
omet
hing
you
wan
t but
can
nota
ffor
d?
7. A
re y
ou in
the
habi
t of
spen
ding
mod
erat
ely
on p
erso
nal
groo
min
g?
8. C
an y
ou g
ener
ally
be e
nter
tain
ed w
ithou
tsp
endi
ng r
r on
ey?
9. D
o yo
u 's
ually
res
istt
he s
pend
ing
pres
sure
sof
fri
ends
?
10. D
o yo
u re
sist
the
spen
ding
of
mon
ey a
ccor
d-in
g to
you
r w
him
with
out r
egar
d to
wha
tyo
u re
ally
need
?
11 W
hen
"bro
ke,"
do y
ou te
pd to
avo
idge
tting
an e
xtra
sum
from
you
r pa
rent
s or
guar
dian
?
12. I
f yo
u sa
w a
clot
hing
item
in a
sto
re w
ht r
eyo
u ha
ve a
char
ge a
ccou
nt, w
ould
you
belik
ely
to th
ink
abou
t how
to p
ay f
or it
bef
ore
you
boug
htit?
13. A
re y
ou c
aref
ulab
out n
ot le
avin
g ca
shin
your
roo
m o
rca
rryi
ng f
airl
y la
rge
sum
sof
mo-
- ey
on
your
pvi
mut
?
14. D
o yo
u us
uaPy
avoi
d bu
ying
clo
thes
that
you
may
wea
ron
ly a
few
tim
es?
74
A. C
omm
on m
isco
ncep
tions
1. A
bud
get i
s an
t boo
kkee
ping
15. D
o yo
u sp
end
a m
oder
ate
amou
nt o
f m
oney
for
food
bet
wee
n m
eals
?
16. D
o yo
u us
ually
sav
e ah
ead
for
som
ethi
ngyo
u w
ant v
ery
muc
h, s
uch
as a
new
dre
ssor
sui
t, a
gift
, a p
rom
?
17. D
o yo
u m
ake
it a
habi
t to
go to
mor
e th
anor
stor
e to
com
pare
pri
ce a
nd q
ualit
y be
-de
cidi
ng o
n a
big
purc
hase
?
18.
ald
you
say
that
abo
ut h
alf
your
pur
-ch
ases
are
pla
nned
in a
dvan
ced
and
are
not
mer
ely
"im
puls
e" b
uyin
g?
19. D
o yo
u kn
ow w
heth
er y
our
fam
ily c
arri
espe
rson
al b
elon
ging
s in
sura
nce,
pro
tect
ing
such
item
s as
you
r lu
ggag
e, c
loth
es, j
ewel
ry,
golf
and
tenn
is e
quip
men
t?
20. C
an y
ou r
esis
t buy
ing
barg
ains
just
bec
ause
they
are
adv
ertis
ed a
s ba
rgai
ns?
Stud
ents
and
thei
r pa
rent
s ha
ve m
any
mis
con-
cept
ions
abo
ut b
udge
ts. D
eter
min
e w
hat i
deas
the
stud
ents
hav
e ab
out t
he n
atur
e an
d pu
rpos
eof
bud
gets
Ane
cdot
al r
ecor
ds, s
uch
as th
e fo
llow
ing
can
beus
ed to
illu
stra
te th
ese
mis
conc
eptio
ns
Mrs
. Jam
es c
ame
to th
e ba
nk w
ith a
boo
k-ke
eper
's le
dger
con
tain
ing
the
year
's e
x-pe
nses
. "I'v
e ke
pt a
, rec
ord
of e
very
pen
ny,"
she
told
the
budg
et a
dvis
or. E
nter
ed in
the
ledg
er w
ere
such
item
s as
"po
stag
e, 5
c, p
hone
call,
10c
," a
nd s
o on
. "A
nd s
till w
e ca
n't s
ave
mon
ey"
she
excl
aim
ed. T
actf
ully
the
expe
rtto
ld h
er th
e tr
uth;
she
had
bee
n w
astin
g he
rtim
e. S
he h
ad m
ade
hers
elf
and
the
fam
ilym
iser
able
with
the
mis
take
n no
tion
that
budg
etin
g m
eans
kee
ping
a r
ecor
d of
eve
ry-
thin
g yo
u sp
end
2. A
bud
get i
s 'ro
t a s
yste
m o
f fi
xed
A C
hica
go h
ouse
wif
e to
ld a
bud
get e
xper
tpe
rcen
tage
sth
at s
he h
ad c
alcu
late
d ev
eryt
hing
sci
entif
i-
75
3. A
bud
get i
s no
t pin
chin
g pe
nnie
s
B. A
bud
get i
s a
plan
for
dis
trib
utin
gin
com
e in
ord
er to
giv
e ev
ery
mem
-be
r of
the
fam
ily th
e ut
mos
t sat
is-
fact
ion
from
the
mon
ey s
pent
C. T
he s
teps
in e
stab
lishi
ng a
bud
get
mig
ht b
e ge
nera
lly s
tate
d as
:
1. D
istr
ibut
ion
of y
our
know
n or
appr
oxim
ate
inco
me
2. C
ompa
re e
xpec
ted
expe
nditu
res
with
exp
ecte
d in
com
e3.
Sin
ce w
ants
pro
babl
y ex
ceed
in-
com
e, th
e es
tims
of e
ssen
tial
cally
. She
kne
w th
e fo
rmul
a w
hich
said
you
wer
e su
ppos
ed to
spen
d "x
" pe
r ce
nt o
ncl
othe
s, "
y" p
er c
ent o
n ho
usin
g, a
nd s
o on
.T
his
had
calle
d fo
r ch
angi
ng th
efa
mily
'sliv
ing
habi
ts, b
ecau
se th
ey w
ere
spen
ding
too
muc
h on
hou
sing
. The
y m
oved
,th
eir
rent
was
low
er b
utth
ey w
ere
no lo
nger
hap
py.
"Nob
ody
coul
d sa
y ho
w m
uch
you
'oug
ht' t
osp
end
on h
ousi
ng,"
the
expe
rt to
ldhe
r. T
here
is o
nly
wha
t you
hav
e to
spe
nd a
ndw
hat y
ouw
ant t
o sp
end.
Buy
one
thin
g,an
d yo
u ca
n't
buy
anot
her
Mrs
. Sm
ith w
as o
n th
e ve
rge
of a
ner
vous
brea
kdow
n w
hen
she
cam
e to
an
advi
ser.
"Whe
n w
e w
orke
d ou
t our
bud
get,"
she
said
,"f
igur
ed a
ll ki
nds
of w
ays
to s
ave
mon
ey.
Geo
rge
agre
ed to
cut
out
two
pack
s of
cig-
aret
tes
per
wee
k, T
om, o
ur s
on,
said
he
itoul
d ge
t alo
ng o
n ha
lf h
is a
llow
ance
and
soon
." T
hese
thin
gs s
aved
the
fam
ily $
5 pe
rw
eek
but e
very
one
was
unh
appy
. She
had
the
wro
ng n
otio
n of
a b
udge
t
Stud
ents
nee
d ex
peri
ence
s in
est
ablis
hing
budg
ets.
The
bes
t pla
ce to
beg
in w
ould
bese
tting
up
a pe
rson
al b
udge
t. T
his
is p
artic
u-la
rly
usef
ul f
or th
ose
pupi
ls w
ho h
ave
outo
fsc
hool
em
ploy
men
t
Rea
dy m
ade
acco
unt b
ooks
ava
ilabl
e fr
omin
sura
nce
com
pani
es, d
ime
stor
es, b
ooks
tore
san
d st
atio
nery
sho
ps c
an b
e us
ed b
ut a
re n
otpa
rtic
ular
ly u
sefu
l for
stu
dent
s. T
hey
are
ofte
n to
o co
mpl
icat
ed a
nd te
nd to
mak
eth
est
uden
t a b
ookk
eepe
r
A b
udge
t pla
n m
ight
fol
low
the
patte
rnbe
low
:
Tot
al in
com
eT
otal
exp
endi
ture
sN
et g
ain
or lo
ss
Tot
al b
udge
t,
76
Ann
ual
Mon
thly
Mon
ey M
anag
emen
t for
You
r Fa
mily
.C
ost
Cos
tB
ette
r H
omes
and
Gar
dens
. Mer
edith
Publ
ishi
ng C
o. D
es M
oine
s, I
a. 1
962
item
s sh
ould
be
cut,
part
icul
arly
Tax
esfo
od a
nd c
loth
ing
Fede
ral
Stat
eSo
cial
Sec
urity
4. R
earr
ange
the
rest
of
the
list i
nSa
ving
s an
dor
der
of p
refe
renc
ein
vest
men
ts
Tot
al C
onsu
mpt
ion
Item
sFo
odC
loth
ing
Hou
sing
Furn
ishi
ngs
Oth
er I
nsur
ance
Adv
ance
men
tC
hari
tyE
duca
tion
Chu
rch
Rec
reat
ion
Aut
omob
ile
Peri
odic
Exp
ense
s
Oth
er it
ems
Tot
al E
xpen
ditu
res
Sinc
e ev
ery
stud
ent d
oes
not h
ave
inco
me
with
whi
ch a
per
sona
l bud
get c
an b
e es
tabl
ishe
d,hy
poth
etic
al s
ituat
ions
can
be
esta
blis
hed
usin
ga
proj
ecte
d in
com
e fi
gure
fro
m h
is f
utur
e vo
ca-
tiona
l cho
ice
As
item
cos
ts a
re e
stim
ated
, an
inte
rest
ing
activ
ity is
the
com
pari
son
of p
rice
s of
pre
sent
-da
y pu
rcha
ses
with
thos
e a
cent
ury
ago.
At t
hat
time,
the
follo
win
g co
sts
are
typi
cal:
Milk
50 p
er q
t.R
aisi
ns 7
0 pe
r lb
.B
utte
r 12
0 pe
r lb
.B
eef
90 p
er lb
.C
hees
e90
per
lb.
Cof
fee
250
per
lb.
Boo
ts $
5 pe
r pa
irC
oats
$12
Ves
t $6
77
D. F
amily
bud
gets
Tw
o ca
nnot
live
as
chea
ply
as o
ne
It w
ould
als
o be
adv
isab
leto
com
pare
sal
arie
sas
wel
l. So
me
typi
cal s
alar
ies
wer
e:
Lab
orer
373.
40 p
er d
ayM
inis
ter
$700
per
yea
rFa
rm la
bore
r $
3 pe
r m
onth
,plu
s bo
ard
For
thos
e st
uden
ts w
hoha
ve d
iffi
culty
kno
win
gw
here
thei
r m
oney
isgo
ing,
kee
ping
cas
hac
coun
ts c
an b
e us
eful
in o
btai
ning
som
e id
eaas
to w
hat
item
s to
bud
get f
or. I
tsh
ould
be
clea
r th
at th
e ca
sh a
ccou
nts
shou
ld n
ot d
eter
-m
ine
the
budg
et. T
hebu
dget
sho
uld
be e
stab
-lis
hed
by th
e am
ount
of
inco
me
and
wha
t the
irdi
vidu
al w
ants
to b
uy w
ith it
The
fam
ily b
udge
ts th
eite
ms
to b
e in
clud
edw
ill d
iffe
r fr
om th
e pe
rson
albu
dget
. The
am
ount
sbu
dget
ed w
ill a
lso
diff
er
Stud
ents
mig
ht g
ain
a go
odde
al o
f un
der-
stan
ding
reg
ardi
ng m
oney
man
agem
ent b
y co
n-st
ruct
ing
fam
ily b
udge
ts f
orth
eir
own
fam
ily.
If m
arri
age
is in
the
plan
s of
the
pupi
ls, a
pro
-
The
mos
t com
mon
cau
seof
mar
ital
ject
ed f
amily
bud
get c
an b
ede
vise
d. I
t is
im-
disc
ord
is r
elat
ed d
irec
tly o
rin
-po
rtan
t to
brin
g ou
t the
fact
that
fam
ily b
udge
ts
dire
ctly
to m
oney
man
agem
ent
shou
ld b
e es
tabl
ishe
d on
lyby
join
t pla
nnin
g of
the
husb
and
and
wif
e
E. B
usin
ess
budg
ets
F. G
over
nmen
talb
udge
ts
Exp
lore
the
prob
lem
s of
budg
etin
g in
sm
all
U. S
. Bur
eau
of I
nter
nal
Rev
enue
, Was
h-
busi
ness
es, c
orpo
ratio
ns, c
hurc
hes,
PTA
's o
rin
gton
, D. C
. Tax
Kits
scho
ol g
roup
s in
whi
ch th
est
uden
ts h
old
mem
-be
rshi
p
The
item
s fo
r th
ese
budg
ets
are
diff
eren
t but
Invi
te s
upei
tend
ent t
o ta
lk o
nsc
hool
the
basi
c pr
oble
ms
rem
ain
the
sam
e. S
ampl
ebu
dget
sbu
dget
s fo
r so
me
of th
ese
grou
pssh
ould
be
dupl
icat
ed a
nd h
ande
d to
stud
ents
for
dis
cus-
sion
. Per
haps
cop
ies
of th
esc
hool
bud
get a
reav
aila
ble.
Alth
ough
mor
e re
mov
ed f
rom
the
daily
life
of
the
indi
vidu
al, s
uch
a st
udy
may
be v
alua
ble
ifno
t cov
ered
in th
e so
cial
stud
ies
clas
s
Wri
te y
our
cong
ress
man
for
info
rmat
ion.
Ask
city
off
icia
ls f
or in
form
atio
n
II. I
nsta
llmen
t buy
ing
A. I
ntro
duct
ion
Thi
s un
it sh
ould
mak
e us
e of
the
item
s lis
ted
inA
n id
eal,
free
boo
klet
whi
ch c
anbe
use
d
78
Thi
s un
it sh
ould
pro
vide
the
answ
ers
to k
ey q
uest
ions
, suc
has
:1.
Whe
re c
an w
e go
to b
orro
wm
oney
?2.
Whi
ch in
stitu
tions
are
bes
t?3.
Wha
t is
mea
nt b
y "t
rue"
in-
tere
st?
4. W
hen
shou
ld w
e bo
rinw
mon
eyto
pay
for
dur
able
s in
stea
d of
buyi
ng o
n th
e in
stal
lmen
t pla
n?5.
How
has
cre
dit h
elpe
d ex
pand
our
econ
omy
to th
e hi
ghes
tst
anda
rd o
f liv
ing
ever
enj
oyed
by a
nat
ion?
6. W
hy is
it im
port
ant f
or a
per
-so
n to
hav
e a
good
cre
dit
ratin
g?7.
Wha
t are
the
adva
ntag
es a
ndth
e di
sadv
anta
ges
of in
stal
l-m
ent b
uyin
g?8.
How
doe
s a
pers
on f
inan
ce th
epu
rcha
se o
f an
aut
omob
ile?
B. W
hen
does
one
use
cas
h, c
harg
eac
coun
ts, o
r in
stal
lmen
t buy
ing?
C. F
inan
cing
inst
allm
ent p
urch
ases
1. F
inan
cing
inst
itutio
nsa.
Sal
es F
inan
ce C
o.b.
Ban
ksc.
Sm
all l
oan
com
pany
d. L
oan
shar
kse.
Sav
ing
and
loan
ase
n.1.
Cre
dit u
nion
s
D. I
nsta
llmen
t Cha
rges
1. N
eces
sity
of
char
ges
a. in
tere
stb.
bad
deb
t and
cre
dit l
osse
sc.
adm
inis
trat
ive
cost
s
the
stud
ents
' bud
get w
hich
can
not b
e pu
rcha
sed
for
lack
of
read
y ca
sh. S
ome
stud
ents
may
hav
em
entio
ned
this
pos
sibi
lity
in th
eir
pers
onal
or
fam
ily b
udge
ts in
the
prev
ious
sec
tion
The
rem
aini
ng p
art o
f th
e ou
tline
fol
low
s cl
osel
yth
e ch
apte
rs in
the
pam
phle
t lis
ted
to th
e ri
ght
Hav
e th
e st
uden
ts li
st s
ome
of th
e ite
ms
in th
eir
own
hom
e w
hich
wer
e pu
rcha
sed
on th
e in
stal
l-m
ent p
lan
Hav
e th
e st
uden
ts d
iscu
ss th
e ne
cess
ity o
f ob
-ta
inin
g a
mor
tgag
e w
hen
purc
hasi
ng a
hom
e or
busi
ness
Ans
wer
s to
this
que
stio
n ca
n be
elic
ited
from
stud
ents
aft
er n
otin
g th
e ki
nds
of it
ems
whi
char
e pu
rcha
sed
by th
ese
met
hods
Hav
e th
e st
uden
ts in
vest
igat
e th
e ch
arac
teri
s-tic
s of
eac
h ty
pe o
f ag
ency
and
con
tras
t the
irop
erat
ion.
A b
rief
stu
dy o
f th
ese
man
y ag
enci
esle
ads
dire
ctly
into
a d
iscu
ssio
n of
how
one
choo
ses
a fi
nanc
ing
inst
itutio
n an
d th
us to
in-
stal
lmen
t cha
rges
Hav
e th
e st
uden
ts p
lay
the
role
of
a lo
an a
genc
yan
d as
k th
em to
exp
lore
wha
t cos
ts th
ey w
illha
ve w
hich
mus
t be
incl
uded
in th
e co
st to
the
borr
ower
79
as a
text
for
this
sec
tion
is "
Usi
ng I
nsta
l-m
ent C
redi
t," b
y C
lyde
W. P
help
s, p
ub-
lishe
d by
Com
mer
cial
Cre
dit C
o., E
duca
-tio
nal D
ivis
ion,
Bal
timor
e, M
d.
Cha
pter
I, "
Usi
ng I
nsta
lmen
t Cre
dit"
Cha
pter
II,
"U
sing
Ins
talm
ent C
redi
t"
Cha
pter
III
, "U
sing
Ins
talm
ent C
redi
t"
2. T
he d
olla
r co
stO
n an
inst
allm
ent p
urch
ase,
the
dolla
r co
st is
the
diff
eren
cebe
-tw
een
the
cash
pri
ce a
ndth
e"t
ime
pric
e."
3. T
he a
nnua
l int
eres
t rat
e on
pur
-ch
ases
. By
form
ula
the
rate
is:
't =
2 m
Iw
here
13(n
+ 1
)
R is
the
annu
al c
ost r
ate
m is
the
num
ber
of !
nsta
llmen
tpa
ymen
ts p
er y
ear
Iis
the
tota
l am
ount
of
the
fi-
nanc
ing
char
geP
is th
e ne
t am
ount
of
the
cred
itad
vanc
en
is th
enu
mbe
r of
inst
allm
ent
paym
ents
cal
led
for
inth
eco
ntra
ct
E. M
isco
ncep
tions
dea
ling
with
inst
all-
men
t per
cent
age
rate
s
F. C
ompa
ring
the
cost
sof
inst
allm
ent
purc
hase
s1.
rat
es2.
ser
vice
s
G. P
ossi
ble
bene
fits
of
inst
allm
ent p
ur-
chas
es
H. P
robl
ems
of e
xces
sive
debt
I. M
aint
enan
ce o
f a
good
cred
it ra
ting
J. T
he r
ole
of in
stal
lmen
tpu
rcha
ses
inth
e na
tion'
s ec
onom
y
Prov
ide
the
stud
ents
with
seve
ral p
robl
ems,
som
ewha
t inv
olve
d, w
here
the
dolla
r co
st is
tobe
fou
nd
An
exce
llent
dis
cuss
ion
ofth
is f
orm
ula
is f
ound
in "
Usi
ng I
nsta
lmen
tC
redi
t." T
he f
orm
ula
shou
ld n
ot b
e ta
ught
but
shou
ld b
e de
velo
ped
slow
ly a
s ou
tline
d in
the
pam
phle
t. N
umer
ous
prob
lem
s cl
n be
dev
ised
by
the
teac
her
whe
redo
llar
cost
s an
d an
nual
inte
rest
rat
es a
re c
om-
pute
d. I
t is
impo
rtan
t for
the
stud
ents
to r
ealiz
eth
at b
oth
dolla
r co
st a
ndan
nual
inte
rest
rat
ear
e im
port
ant
piec
es o
f in
form
atio
n in
mak
ing
deci
sion
s re
gard
ing
inst
allm
entp
urch
ases
Hav
e th
e st
uden
ts b
rim
tocl
ass
som
e ad
ver-
tisem
ents
fro
m n
ewsp
aper
s an
dm
agaz
ines
and
try
to s
ecur
e en
ough
in:w
mat
ion
from
them
toco
mpu
te d
olla
r co
st a
ndin
tere
st r
ate
The
se to
pics
cou
ld p
roba
bly
be h
andl
ed b
est b
ysm
all g
roup
dis
cuss
ion
peri
ods
whe
re th
e st
uden
tsha
ve a
n op
port
unity
to d
eelo
pth
e co
ncep
ts o
nth
eir
own.
Eve
ry e
ffor
t sho
uld
be m
ade
to k
eep
the
unit
activ
e an
d al
ive,
and
to p
rovi
de th
est
uden
ts w
ith m
uch
activ
ity
Hav
e a
stud
ent g
roup
stu
dyab
out a
nd o
r vi
sit
a cr
edit
bure
auan
d re
port
to th
e cl
ass
80
Cha
pter
IV
, "U
sing
Ins
talm
entC
redi
t"
"Usi
ng I
nsta
lmen
t Cre
dit,"
Cha
pter
s V
,V
I, I
X, X
Film
sB
anks
and
Cre
dit.
colo
rC
oron
et 1
0m
in. b
&w
Pam
phle
ts:
Usi
ng I
nsta
lmen
t Cre
dit,
Cly
de P
help
s.C
onsu
mer
Cre
dit C
o. B
altim
ore,
Md.
Mon
ey M
anag
emen
t Boo
klet
s.H
ouse
hold
Fina
nce
Cor
p. P
rude
ntia
lPl
aza,
Chi
cago
, Ill.
Con
sum
er C
redi
t Fac
ts f
orY
ou. B
urea
uof
Bus
ines
s R
esea
rch.
Wes
tern
Res
erve
Uni
vers
ity, C
leve
land
6, 0
.
FINANCING FREEDOM
UN
IT V
III
FIN
AN
CIN
G F
RE
ED
OM
Intr
oduc
tion
Wal
k up
and
dow
n M
ain
Stre
et, a
ttend
a c
ount
y fa
ir, m
eet
with
a g
roup
of
hous
ewiv
es, o
r vi
sit w
ith w
orke
rs in
indu
stry
.O
neto
pic
of c
onve
rsat
ion
you
will
und
oubt
edly
hea
r w
ill b
e ta
xes.
Alm
ost e
very
one
will
be
com
plai
ning
of
the
amou
nt o
f m
oney
appr
opri
-at
ed f
rom
pay
che
cks
by g
over
nmen
t loc
al, s
tate
,and
fed
eral
. Muc
h of
this
is p
ure
"gri
ping
," b
ased
upo
n ob
serv
atio
nan
d in
com
plet
eco
nsid
erat
ion
of f
acts
. Few
peo
ple
will
sug
gest
cut
ting
out s
peci
fic
area
sof
gov
ernm
enta
l act
iviti
es w
hich
thes
e ta
xes
fina
nce.
The
mai
n sp
irit
of th
is u
nit s
houl
d be
one
of
info
rmat
ion.
The
stu
dent
s sh
ould
bec
ome
fam
iliar
with
the
purp
oses
of
taxe
sth
eypa
y an
d w
ill p
ay. T
hesh
ould
be
taug
ht in
a m
anne
r th
at w
ill f
oste
r a
trul
y cr
itica
leva
luat
ion
of th
e en
tire
tax
stru
ctur
e. M
athe
-m
atic
al b
y-pr
oduc
ts o
f th
is s
tudy
will
be
num
erou
s. F
orex
ampl
e, p
erce
ntag
e co
mpu
tatio
n an
d th
e m
akin
g of
gra
phs
are
inte
gral
par
tsof
the
stud
y of
taxa
tion.
Bef
ore
star
ting
this
uni
t, it
wou
ld b
e w
ise
to d
iscu
ss it
with
the
teac
hers
of
soci
al s
tudi
es a
nd b
usin
ess
educ
atio
n in
you
r sc
hool
.T
his
will
poi
nt o
ut a
reas
of
conc
entr
atio
n an
d av
oid
unne
cess
ary
dupl
icat
ion
as w
ell a
s co
ntra
dict
ion.
The
usu
al p
erio
ds o
f dr
ill s
houl
d be
con
tinue
d. B
e su
re to
k ,p
thes
e sh
ort,
freq
uent
and
info
rmal
. Aga
in it
mus
t be
emph
asiz
edth
at th
is is
a g
uide
to a
ctiv
ities
, a s
et o
f se
lect
ed s
ugge
sted
thin
gs to
do. I
t is
not i
nten
ded
to b
e fo
llow
ed r
igor
ousl
y. Y
ou m
ayel
abor
ate
on s
ome
sect
ions
and
om
it ot
hers
.Y
ou a
re u
rged
to e
xam
ine
care
fully
the
colu
mn
head
ed "
Tea
chin
gA
ids"
wel
l bef
ore
star
ting
the
unit
so th
at h
elpf
ul m
ater
ials
can
be
obta
ined
for
cla
ss u
se a
tth
e pr
oper
tim
e.
Obj
ectiv
es1.
To
lear
n th
e im
port
ance
of
taxe
s in
the
life
of a
llin
divi
dual
s
2. T
o ac
quir
e sk
ills
whi
ch w
ill e
nabl
e th
em to
fill
out
,hei
r an
nual
inco
m, t
ax r
etur
ns c
orre
ctly
and
acc
urat
ely
3. T
o un
ders
tand
why
taxe
s ar
e ne
cess
ary
to m
aint
ain
the
Am
eric
an w
ay o
f lif
e
4. T
o be
tter
unde
rsta
nd th
e ta
x st
ruct
ure
in th
e U
nite
d St
ates
Con
tent
A. T
axes
1. D
efin
ition
s2.
His
tory
3. M
ajor
cla
ssif
icat
ions
a. in
com
e, p
riva
te, a
nd c
orpo
rate
b. e
xcis
e or
spe
cial
sal
es ta
xc.
pro
pert
y ta
xes,
rea
l, an
d pe
r-so
nal
d. g
ener
al s
ales
tax
4. M
eani
ngof
dire
ctan
d:n
dire
ct ta
xes
B. S
ervi
ces
paid
for
by
mon
ey r
ecei
ved
from
taxa
tion
Proc
edur
es a
nd A
ctiv
ities
Find
def
nitio
ns. W
rite
one
acc
epta
ble
to c
lass
.H
ave
a br
ief
repo
rt o
n hi
stor
y of
taxa
tion.
Dis
-cu
ss "
Tax
at io
n as
' a b
adge
of
free
dom
"
Con
side
r so
me
prod
ucts
suc
h as
an
auto
mob
ilean
d in
vest
igat
e in
dire
ct ta
xes
Ask
stu
dent
s to
list
as
man
y se
rvic
es a
s th
eyca
n. T
his
shou
ld p
roba
bly
be d
one
incl
ass
and
81
Tea
chin
g A
ids
Con
sult
Wor
ld A
lman
ac f
or c
urre
nt y
ear
for
stat
istic
s
U. S
. Bur
eau
of I
nter
nal R
even
ue, W
ash-
ingt
on, D
. C. T
ax K
its
C. G
over
nmen
ts to
whi
ch w
e pa
y ta
xes
1. F
eder
al2.
Sta
te3.
Cou
nty
4. L
ocal
D. F
eder
al T
axat
ion
1. M
ajor
sou
rces
of
reve
nue
2. M
ajor
exp
endi
ture
s of
the
fede
ral
gove
rnm
ent
3. R
dera
l tax
es y
ou w
ill b
e su
re to
Pay
a. I
ncom
e
b. E
xcis
e
E. S
tate
Tax
es1.
Sou
rces
of
inco
me
2. M
ajor
exp
endi
ture
s of
the
stat
e
put o
n th
e ch
alkb
oard
.St
uden
ts h
ave
stra
nge
idea
s ab
out w
ho f
oots
the
bills
Spen
d a
lot o
f tim
e on
this
sec
tion
tcla
y a
good
foun
datio
n
Hav
e st
uden
t lis
t the
ser
vice
sun
der
the
gove
rn-
men
t whi
ch th
ey th
ink
pays
for
thes
e se
rvic
es.
Som
e w
ill b
e un
der
mor
e th
an o
ne
Mak
e a
circ
le g
raph
sho
win
g so
urce
in p
er c
ent
Thi
s m
ight
mak
e a
good
cla
sspr
ojec
t
Thi
s m
ight
be
an in
divi
dual
orc
omm
ittee
pro
j-ec
t. M
ake
a ci
rcle
gra
phsh
owir
:pen
dita
rein
per
cen
ts. F
ind
wha
t the
annu
al m
idge
t is
for
the
pres
ent y
ear.
Wha
t is
the
aver
age
cost
per
pers
on?
How
do
inco
me
taxe
s in
the
U. S
. com
-pa
re w
ith th
ose
in o
ther
cou
ntri
es?
Wha
t is
the
pres
ent n
atio
nal d
ebt?
Set u
p a
hypo
thet
ical
fam
ilyin
com
e, s
ourc
e of
inco
me
Use
pro
blem
s in
tax
kit.
Lis
t pro
san
d co
ns o
fw
ithho
ldin
g
Fill
out a
tax
form
Exp
lain
car
eful
ly s
hort
and
long
for
m
Mak
e a
list o
f ex
cise
taxe
s pa
id b
y th
est
uden
t'sfa
mily
Wri
te f
or in
form
atio
n. M
ake
aci
rcle
gra
phsh
owin
g pe
rcen
tage
Lis
t in
orde
r of
cos
t the
maj
or e
xpen
ditu
res
Mak
e a
bar
grap
h of
thes
e ex
pend
iture
s
Com
pare
sta
te g
raph
s w
ith f
eder
algr
aphs
82
U. S
. Bur
eau
of I
nter
nal R
even
ue, W
ash-
ingt
on, D
. C. T
ax K
its
"Rep
ort t
o G
over
nor
and
Leg
isla
ture
,"C
omm
issi
oner
of
Tax
atio
n, S
t. Pa
ul
Res
earc
h D
ivis
ion,
Min
neso
ta T
axpa
yers
Ass
'n.,
812
Min
neso
ta B
ldg.
, St.
Paul
Cou
nty
Ass
esso
r or
Cou
nty
Supe
rvis
or
3. S
tate
taxe
s th
e st
uden
t will
belik
ely
to p
aya.
inco
me
b. s
peci
al s
ales
F. L
ocal
Tax
es1.
Maj
or s
ourc
ea.
pro
pert
y ta
x(1
) re
al e
stat
e(2
) pe
rson
al p
rope
rty
b. li
cens
es2.
Maj
or E
xpen
ditu
res
a. s
choo
lsb.
roa
dsc.
pub
lic s
ervi
ce(1
) ho
spita
l(2
) lib
rary
Fill
in S
tate
Inc
ome
Tax
For
mfo
r a
hypo
thet
i-St
ate
Tax
For
ms
cal i
ndiv
idua
l and
:or
fam
ily
Find
dif
fere
nces
bet
weE
n st
ate
and
fede
ral f
orm
s
Get
sug
gest
ions
for
mak
ing
tax
colle
ctio
ns m
ore
eftie
ipnt
Find
gas
olin
e ta
x st
ate
and
fede
ral
Wha
t per
cen
t of
tota
l cos
t is
the
gas
tax?
Wha
t per
cen
t of
tota
l cos
t o!
ciga
rette
s is
the
tax?
Wha
t per
cen
t of
tota
l pro
pert
yill
the
com
-m
unity
goe
s to
the
stat
e?
Wha
t per
cen
t of
stat
es h
ave
a sa
les
tax?
Pan
eldi
scus
sion
or
deba
te--
"How
muc
hw
ould
sal
esta
xes
be o
n va
riou
s am
ount
s?"
"Wha
t are
arg
umen
ts f
or a
nd a
gain
st a
sale
sta
x?"
It h
as b
een
said
that
Min
neso
taha
s an
unf
avor
-ab
le ta
x cl
imat
e fo
r br
ingi
ng in
new
indu
stry
.H
ow d
oes
Min
neso
ta r
ank
as a
taxp
ayin
g st
ate?
How
doe
s M
inne
sota
ran
k am
ong
the
stat
es in
taxe
s co
mpa
red
to in
com
e?
Can
you
fin
d an
ythi
ng c
once
rnin
gco
rpor
atio
nta
xes?
Lea
gue
of W
omen
Vot
ers,
84
So.S
ixth
St.,
Min
neap
olis
14
Hav
e th
e co
unty
ass
esso
r or
supe
rvis
or c
ome
toC
ount
y as
sess
ors
have
muc
hin
form
atio
n
the
scho
ol. T
hese
que
stio
ns m
aybe
ask
ed:
on p
rope
rty
taxe
s
Wha
t is
mea
nt b
y m
ill r
ate?
How
is lo
cal m
ill r
ate
dete
rmin
ed?
How
are
ass
esso
rs c
hose
n?W
hat t
rain
ing
is n
eces
sary
?H
ow is
ful
l and
true
val
ue d
eter
min
ed?
Is th
is v
alua
tion
cons
iste
nt th
roug
hout
the
stat
e?W
hat i
s m
eant
by
hom
este
ad a
ndno
n-ho
me-
stea
d la
nd?
83
d. p
rote
ctio
n(1
) po
lice
(2)
fire
e. w
elfa
ref.
rec
reat
ion
G. C
ount
y T
axes
1. M
ajor
sou
rce
a. p
rope
rty
2. M
ajor
exp
ense
s
H. F
inan
cing
'.._
choo
ls
Wha
t is
mea
nt b
y ag
ricu
ltura
l lan
d?H
ow d
oes
it di
ffer
in th
e m
anne
r of
ass
ignm
ent?
Who
par
s pe
rson
al p
rope
rty
taxe
s?W
hat p
er c
ent o
f th
e pr
oper
ty ta
x is
per
sona
lpr
oper
ty ta
x?So
me
peop
le c
onsi
der
this
an
unfa
ir ta
x w
hy?
How
cou
ld m
oney
be
rais
ed in
its
plac
e?W
ork
prob
lem
sgiv
en m
arke
t val
ue, f
ull a
ndtr
ue v
alue
and
loca
l mill
rat
e. G
iven
tax
and
mill
rat
e, w
hat i
s th
e as
sess
ed v
alua
tion?
Lis
t maj
or e
xpen
ses
Lis
t the
fac
ts th
at h
ave
impr
esse
d yo
u m
ost o
nM
inne
sota
taxe
s
If y
ou f
eel t
axes
ale
i.00
hig
h, w
hat s
ervi
ces
wou
ld y
ou r
educ
e?
Are
ther
e an
y se
rvic
es th
at c
ould
be
prov
ided
mor
e ch
eapl
y by
pri
vate
indu
stry
than
by lo
cal,
coun
ty, o
r st
ate
gove
rnm
ent?
Get
a p
rope
rty
tax
form
fro
m th
e co
unty
ass
es-
sor
or c
ount
y au
dito
r
Find
wha
t per
cen
t of
prop
erty
taxe
s go
to th
eci
ty, t
he c
ount
y, th
e st
ate,
and
the
scho
ols
Mak
e a
grap
h
How
muc
h ta
x m
oney
is s
pent
for
edu
catio
n?(F
eder
al, s
tate
, and
loca
l)
From
you
r sc
hool
sup
erin
tend
ent f
ind
the
cost
per
pupi
l
How
is th
e co
st d
ivid
ed b
etw
een
stat
e an
d lo
cal
gove
rnm
ent?
Doe
s th
e sc
hool
rec
eive
any
fed
eral
aid
?
For
wha
t pur
pose
s is
fed
eral
aid
rec
eive
d?
84
Nat
iona
l Edu
catio
n A
ocia
tion
has
muc
hin
form
atio
n on
this
topi
c
Film
sC
itize
n D
are
Dou
glas
27
min
b &
w M
oder
nT
alki
ng P
ictu
res
Serv
ice,
Inc
., 3
E. 5
4th
St.,
New
Yor
k 22
, fre
e lo
an
......
.....1
1.11
1111
1111
1111
11F
Mak
e gr
aphs
of
sour
ces
of s
choo
lai
dsFe
dera
l Tax
atio
n. C
oron
et 1
0 m
in b
& w
1948
col
or
Hel
ping
the
Tax
paye
r-11
7hat
Hap
pens
Whe
n th
e G
orer
nmen
t Que
stio
ns Y
our
Tax
Ret
urn
1955
141
4 m
in s
d b
& w
, Ass
o-ci
atio
n Fi
lms,
Inc
., 34
7 M
adis
on A
ve.,
New
Yor
k 17
, fre
e lo
an
Prop
erty
Tax
atio
n. E
ncyc
lopa
edia
Bri
tan-
nica
Film
s
Sena
tor's
Dau
ghte
r 19
57 3
0 m
in s
dco
lor
Jam
Han
dy O
rgan
izat
ion,
282
1 E
.G
rand
Blv
d., D
etro
it 11
, fre
e lo
an
Tax
Pol
icyW
hat A
bout
it?
1954
15
min
sd b
& w
, 516
Fif
th A
venu
e, N
ew Y
ork
36 A
m F
ilm F
orum
Ser
.
tAn IEY AND YOUR MANOR
UN
IT I
XM
ON
EY
AN
D Y
OU
RM
AN
OR
Intr
oduc
tion
Mon
ey a
nd Y
our
Man
oris
a u
nit d
esig
ned
togi
ve th
e st
uden
t a b
ette
run
ders
tand
ing
of th
e ro
le o
fm
athe
mat
ics
in th
e liv
es o
f
the
hom
e ow
ner
and
hom
e bu
ilder
. As
anin
tegr
atin
g ac
tivity
, eac
hst
uden
t cou
ld b
e co
ncer
ned
with
des
igni
ng h
is o
wn
hom
efro
m
the
sele
ctio
n of
the
lot t
o th
e fu
rnis
hing
of
the
hom
e. T
o in
volv
ean
d in
tere
st e
ach
stud
enti
mm
edia
tely
, at t
he b
egin
ning
of th
e un
it
stud
ents
cou
ld v
olun
teer
to s
erve
as
cont
ract
ors,
pain
ters
, or
supp
ly d
eale
rs.T
hey
coul
d fo
rm a
nd n
ame
thei
r ow
n co
mpa
nies
tosu
pply
mat
eria
ls a
nd w
ork
onea
ch o
ther
's h
ouse
s.
As
the
unit
prog
ress
esth
e op
port
uniti
es f
orco
mpu
tatio
nal p
ract
ices
,m
ensu
ratio
n pr
oble
ms,
appr
oxim
ate
num
ber
calc
ulat
ions
and
othe
r ap
plie
dpr
oble
ms
are
exte
nsiv
e. T
oke
ep th
e cl
ass
enth
usia
smhi
gh it
may
be
nece
ssar
y to
curt
ail t
he e
xten
t of
such
prac
tice
if a
sat
urat
ion
poin
4.
is f
elt t
o ha
ve b
een
reac
hed.
The
re a
re m
any
mor
eac
tiviti
es li
sted
than
wou
ld b
e ap
plic
able
to th
e av
erag
ecl
ass
and
the
teac
her
shou
ldbe
ale
rt to
and
cap
italiz
e
upon
spe
cifi
c ar
eas
of in
tere
st d
ue to
the
com
mun
ity o
r ho
me
envi
ronm
ent.
Obj
ectiv
es1.
To
lear
n to
appr
ecia
te th
e ro
le o
fm
athe
mat
ics
as a
pplie
d to
the
build
ing
trad
es
2. T
o de
velo
p th
esk
ill n
eces
sary
to m
ake
and
use
the
mea
sure
men
tsre
quir
ed in
hom
e co
nstr
uctio
nan
d m
aint
enan
ce
3. T
o le
arn
toap
prec
iate
the
nece
ssity
for
bala
nce
of q
ualit
y,qu
antit
y, a
nd c
ost i
n ho
me
cons
truc
tion
4. T
o le
arn
to d
o so
me
of th
e co
mpu
tatio
nsre
quir
ed in
hom
e bu
ildin
gan
d m
aint
enan
ce
5. T
o le
arn
tr-t
imat
e an
d us
e ap
prox
imat
enu
mbe
rs
Con
tent
Proc
edur
es a
nd A
ctiv
ities
Tea
chin
g A
ids
A. T
he L
ot1.
Sel
ectio
n of
neig
hbor
hood
Invi
te lo
cal r
eal e
stat
eag
ent t
o sp
eak
on th
efa
ctor
s go
vern
ing
lote
valu
atio
n
2. U
se o
f su
rvey
ing
tool
sIn
vite
sur
veyo
r to
dem
onst
rate
the
use
of in
-Sh
uste
r, C
. N.,
and
Bed
ford
,F.
L. F
ield
Wor
k in
Mat
hem
atic
s.Y
oder
Ins
tru-
men
t Co.
, 193
5st
rum
ents
a. T
rans
itan
d po
les
Mak
e a
tran
sit o
r a
hyps
omet
erS7
Boy
Sco
uts
of A
mer
ica.
Mer
it B
adge
Boo
k-le
t #33
27 S
urve
ying
If a
vaila
ble,
sho
w f
ilm:
Fund
amen
tals
of
Lan
d Su
rvey
ing.
Uni
vers
ityof
Cal
ifor
nia
Reg
iona
l Sou
rce:
Sel
ecte
dFi
lms,
Inc
.,Fi
lm R
enta
l Dep
t., 1
018
So. W
abas
hA
ve.,
Chi
cago
5, I
ll.
Mak
e a
trav
erse
on
the
scho
ol g
roun
ds. T
his
coul
d ta
ke th
e fo
rm o
f a
trea
sure
hun
t
Mak
e an
anc
ient
sur
veyi
ng in
stru
men
t
(1)
Lot
div
isio
n an
d lo
t de-
Invi
te lo
cal b
uild
ing
insp
ecto
r to
tell
abou
tsc
ript
ion
(2)
Prob
lem
s of
leve
ling
B. T
he H
ouse
Pla
n1.
Rea
ding
blu
epri
nts
and
hous
epl
ans
2. U
se o
f sc
ales
in d
raw
ing
hous
epl
ans
rest
rict
ions
and
law
s
Mak
e a
larg
e w
all m
ap o
f a
subd
ivis
ion
with
diff
eren
t sha
ped
lots
, giv
e it
a na
me
and
lette
rst
uden
ts' n
ames
on
purc
hase
d lo
ts. L
ater
, the
yca
n cu
t a s
ilhou
ette
of
thei
r ho
use
plan
and
past
e it
on th
e lo
t in
the
prop
er p
ositi
o.i.
One
of
the
stud
ents
cou
ld a
ct a
s th
e ag
ent s
ellin
g th
elo
ts
Stud
y pr
inci
ples
of
scal
e dr
awin
g an
d m
ake
ala
rge
area
map
Hav
e st
uden
ts d
raw
indi
vidu
al p
lats
of
a su
b-di
visi
on a
nd b
uy a
nd r
ecor
d lo
ts
Map
an
area
usi
ng a
lidad
e an
d pl
ane
tabl
e
Cal
cula
te w
ater
and
sew
er li
nes
in s
ubdi
visi
on
Mak
e a
leve
l
Plot
ele
vatio
ns o
n lo
ts
Col
lect
pla
ns f
rom
loca
l lum
ber
deal
ers
and
mag
azin
es
Rev
iew
and
em
phas
ize
the
prin
cipl
es a
nd u
ses
of r
atio
and
pro
port
ion
Dra
w h
ouse
pla
ns o
f ow
n de
sign
to b
e bu
ilt o
npu
rcha
stA
lots
88
Fede
ral H
ousi
ng A
dmin
istr
atio
n, W
ash-
ingt
on, D
. C. "
Prin
cipl
es o
f Pl
anni
ngSm
all H
ouse
s,"
Tec
hnic
al B
ulle
tin N
o. 4
pp. 4
0-44
If a
vaila
ble,
sho
w f
ilm: C
augh
t Map
ping
.G
ener
al M
otor
s C
orp.
, Film
Lib
rary
,D
etro
it 2,
Mic
h. F
ree
If a
vaila
ble,
sho
w f
ilm: L
evel
Pro
trac
tor.
Uni
ted
Wor
ld F
ilms,
144
5 Pa
rk A
ve.,
New
Yor
k 29
, N. Y
.
Fede
ral H
ousi
ng A
dmin
istr
atio
n. W
ash-
ingt
on, D
. C. "
Prin
cipl
es o
f Pl
anni
ngSm
all H
ouse
s,"
Tee
lmic
al B
ulle
tin N
o..1
If:a
vaila
ble,
sho
w f
ilm: P
rinc
iple
s of
Sca
leD
raw
ing.
Cor
onet
Film
s, C
oron
et B
ldg.
.C
hica
go_l
,
3. E
stim
atio
n of
cos
t of
hous
e
C. T
he C
onst
ruct
ion
of th
e H
ouse
1. C
osts
of
mat
eria
ls
Dis
cuss
pro
blem
s of
alig
nmen
t of
hous
e on
lot
Che
ck a
rea
and
plac
emen
t of
hous
e to
con
form
to lo
cal b
uild
ing
rest
rict
ions
Cal
cula
te a
reas
of
colle
cted
hou
se p
lans
Cal
cula
te h
ouse
cos
ts b
y qu
oted
pri
ces
per
squa
refo
ot
Con
side
r bu
dget
allo
wan
ce in
det
erm
inin
g si
zeof
hom
e to
be
built
Invi
te lo
cal l
umbe
r de
aler
s to
dis
cuss
cos
ts a
ndty
pes
of lu
mbe
r
Col
lect
info
rmat
ion
and
sam
ples
fro
m d
eale
rs,
mag
azin
es, a
nd n
ewsp
aper
s on
cos
ts a
nd ty
pes
of m
ater
ials
Mak
e sc
rapb
ook
of it
ems
desi
red
in o
wn
hom
e
Est
imat
e co
sts
by le
arni
ng to
com
pute
with
appr
oxim
ate
num
bers
89
If a
vaila
ble,
sho
w f
ilm: A
reas
. Kno
wle
dge
Bui
lder
s, V
isua
l Edu
c. B
ldg.
, Flo
ral
Park
, N. Y
.
Tow
nsen
d, G
ilber
t, D
alze
ll, J
., an
d M
c-K
inne
y, J
. How
to E
stim
ate
For
the
Bui
ldin
g T
rade
s. A
mer
ican
Tec
hnic
alSo
ciet
y, 1
955
or s
imila
r so
urce
If a
vaila
ble,
sho
w f
ilm: H
omes
Unl
imite
d.Fr
ee. M
oder
n T
alki
ng P
ict.
Serv
ice,
3 E
.45
th S
tree
t, N
ew Y
ork
19, N
. Y.
If a
vaila
ble,
sho
w f
ilm: M
easu
rem
ent.
Cor
-on
et F
ilms,
Cor
onet
Bld
g., C
hica
go 1
,
Van
Leu
van,
Edw
in P
. Gen
eral
Tra
deM
athe
mat
ics.
McG
raw
, 195
2or
Oth
er c
ompa
rabl
e tr
ade
mat
hem
atic
s te
xt-
book
His
tory
of
Mea
sure
men
t Cha
rts.
Fre
e. F
ord
Mot
or C
ompa
ny, E
duca
tiona
l Rel
atio
nsD
ept.,
Dea
rbor
n, M
ichi
gan
Jolm
son,
D. A
., an
d G
lenn
, W. H
. The
wor
ld o
f M
easu
rem
ent.
Web
ster
, 196
1
The
Am
azin
g St
ory
of M
eesu
rem
ent.
Luf
-ki
n C
o., S
agin
aw, M
ichi
gan
a. E
xcav
atin
g
b. C
emen
t wor
k
c. L
umbe
r
d. R
oofi
ng m
ater
ials
e. G
lass
, tile
, bri
ck
f. P
aint
and
pap
er
2. C
onst
ruct
ion
deta
ils
3. C
ost o
f ho
me
Hav
e st
uden
ts f
orm
ow
n co
mpa
nies
to b
id o
nan
d su
pply
mat
eria
ls f
or in
divi
dual
hou
se c
on-
stru
ctio
ns
Kee
p re
cord
of
cost
s fo
r ea
ch h
ouse
Use
vol
ume
form
ulas
Cal
cula
te c
osts
of
bloc
k w
ork,
ste
ps, f
loor
s,w
alks
, for
exa
mpk
Cal
cula
te b
oard
fee
t and
cos
t of
lum
ber
Cal
cula
te s
hing
le a
nd o
ther
type
s of
roo
fing
cost
s
Cal
cula
te n
umbe
r of
eac
h ne
eded
Inve
stig
ate
tess
ella
tions
by
use
of ti
le s
ampl
es
Cal
cula
te a
reas
and
am
ount
s re
quir
ed
Cal
cula
tions
of
the
abov
e ty
pes
coul
d ea
sily
be
over
done
. Wat
ch f
or a
poi
nt o
f di
min
ishi
ng r
e-tu
rns
and
lack
of
inte
rest
Inve
stig
ate
the
use
of g
eom
etri
c pr
inci
ples
by
John
son,
D. A
. and
Gle
nn, W
. H. T
hem
eans
of
the
tool
s of
the
carp
ente
r, p
artic
ular
lyPy
thag
orea
n T
heor
em. W
ebst
er, 1
961
the
carp
ente
r's s
quar
e
Mak
e a
mod
el o
f a
hous
e of
som
e of
the
con-
stru
ctio
n de
tails
Mak
e a
fiel
d tr
ip to
a h
ouse
und
er c
onst
ruct
ion
Cal
cula
te to
tal c
ost
Inve
stig
ate
vari
ous
trad
es a
nd b
uild
ing
voca
-tio
ns to
lear
n ab
out t
rahi
ng n
eede
d, m
athe
-m
atic
s us
ed, w
ages
, and
oth
er f
acto
rs. S
tude
nts
may
be
able
to g
ive
mor
e co
mpl
ete
repo
rts
onas
sum
ed jo
b du
ring
pro
ject
Dis
cuss
inst
allm
ent p
aym
ents
. Rel
ate
to p
re-
viou
s st
udy
on in
stal
lmen
t buy
ing
in U
nit V
II
90
D. L
ands
capi
ng
1. C
ost o
f pr
epar
ing
soil
2. C
ost o
f up
keep
E. M
aint
enan
ce o
f H
ome
1. G
as, w
ater
, ele
ctri
city
2. I
nsur
ance
3. H
eatin
g
4. P
rope
rty
taxe
s
5. T
otal
mai
nten
ance
cos
t per
yea
r
F. F
urni
shin
g th
e ho
me
1. C
arpe
ting
2. A
pplia
nces
3. H
ome
furn
ishi
ngs
4. D
rape
ries
G. I
mpr
ovem
ent o
f ho
me
1. D
o-it-
your
self
pro
ject
s
Cal
cula
te s
od, s
eed,
shr
ubbe
ry c
osts
Cal
cula
te w
ater
, fer
tiliz
er, m
owin
g co
sts
Lea
rn to
rea
d m
eter
and
cal
cula
te c
ost,
Dis
cuss
var
ious
type
s an
d co
sts.
Rel
ate
to p
re-
viou
s st
udy
of in
sura
nce
Dis
cuss
var
ious
type
s an
d co
sts
Find
how
the
prop
erty
tax
rate
is c
alcu
late
d.R
efer
to p
revi
ous
stud
y of
taxe
s
Lea
rn w
hat i
s m
eant
by
"spe
cial
" as
sess
mcn
tsan
d "h
omes
tead
" ex
empt
ion
Dis
cuss
hom
e ow
ners
hip
vs. r
entin
g
Det
erm
ine
upke
ep c
osts
suc
h as
: pai
ntin
g, r
e-pa
peri
ng, r
epai
ring
Cal
cula
te a
reas
and
cos
ts
Dis
cuss
buy
ing
"on
time"
vs.
cas
h. R
elat
e to
prev
ious
stu
dy o
f in
stal
lmen
t buy
ing
Mak
e a
colle
ctio
n o;
pic
ture
s ill
ustr
atin
g th
e us
eof
geo
met
ric
desi
gn in
hom
e fu
rnis
hing
s
Cal
cula
te d
rape
ry m
ater
ial n
eede
d fo
r va
riou
spl
eatin
g de
pths
. Thi
s m
ay b
e a
"gir
ls o
nly"
proj
ect
Enc
oura
ge s
tude
nts
to r
epor
t on
proj
ects
und
er-
take
n in
thei
r ow
n ho
mes
em
phas
izin
g th
e m
ath-
emat
ics
invo
lved
91
Hav
e st
uden
ts p
rese
nt in
divi
dual
rep
orts
on
mat
hem
atic
s us
ed, m
ater
ials
nee
ded,
and
cos
tsof
pro
ject
s, s
uch
as f
enci
ng, e
ar p
ort,
gara
ge,
swim
min
g po
ol, r
etai
ning
wal
ls, r
ecre
atio
n ro
ompa
nelli
ng
BLAST OFF TO THE FUTURE
UN
IT X
BL
AST
OFF
TO
TH
E F
UT
UR
EIn
trod
uctio
nSp
ace
flig
ht is
her
e to
sta
y "
Man
's in
satia
ble
curi
nz.it
y, c
oupl
edw
ith h
is in
secu
re r
elat
ion
with
peo
ple
of o
ther
nat
iona
lor
igin
sco
mpe
ls h
im o
utw
ard
tow
ard
the
star
s. T
his
isan
era
of
astr
onau
ts a
nd s
putn
iks,
and
we
mus
t lea
rn a
bout
them
to li
ve in
our
soci
ety.
Nee
dles
s to
say
, spa
ce f
light
req
uire
sa
vast
kno
wle
dge
of m
athe
mat
ics.
In
this
uni
t, th
e m
athe
mat
ics
need
ed f
ora
basi
c un
der-
stan
ding
of
spac
e tr
avel
is in
trod
uced
. It s
tart
s w
ith th
esi
mpl
est o
f co
ordi
nate
geo
met
ries
, the
geo
met
ry o
f th
e lin
e.Fr
om th
is it
prog
ress
es to
two-
dim
ensi
onal
and
thre
e-di
men
sion
al c
oord
inat
e sy
stem
s, c
.tart
ing
with
bas
icco
ncep
ts a
nd p
roce
edin
g in
duct
ivel
y.
Indu
ctiv
e th
inki
ng is
the
key
to th
e pr
esen
tatio
n of
this
unit.
Pro
ofs
are
not g
iven
nor
expe
cted
.
The
uni
t als
o pr
ovid
es a
n in
trod
uctio
n to
the
coni
c se
ctio
ns, p
arab
olas
, elli
pses
, and
hyp
erbo
las,
an u
nder
stan
ding
of
whi
ch is
a ne
cess
ary
prer
equi
site
to th
e co
mpr
ehen
sion
of
spac
e fl
ight
.
Sign
ed n
umbe
rs a
nd a
bsol
ute
valu
ear
e re
intr
oduc
ed a
nd a
n ac
quai
ntan
ce w
ith th
e Py
thag
orea
n T
heor
em is
pres
uppo
sed.
The
rear
e m
any
teac
hing
aid
s an
d re
fere
nces
ava
ilabl
e, a
nd th
ese
need
s sh
ould
be
antic
ipat
ed b
efor
e st
artin
g th
e un
it.T
he te
ache
r sh
ould
fee
l fre
e to
use
any
part
of
this
uni
t whi
ch w
ill b
e ac
cept
able
to th
e pa
rtic
ular
cla
ss h
e is
teac
hing
. He
shou
ldm
ake
ever
y ef
fort
to h
ave
the
stud
ents
dis
cove
r the
rel
atio
nshi
ps p
rese
nted
rat
her
than
lect
ure
on e
ach
topi
c. T
he u
sual
sho
rt, f
requ
ent,
and
info
rmal
dri
ll pe
riod
s sh
ould
be
cont
inue
dw
hile
stu
dyin
g th
is u
nit.
Obj
ectiv
es1.
To
impr
ess
upon
the
stud
ent t
he im
port
ance
of
mat
hem
atic
s in
the
stru
ggle
for
the
conq
uest
of
spac
e2.
To
deve
lop
an u
nder
stan
ding
of
som
e el
emen
tary
mat
hem
atic
al c
once
pts
of s
pace
trav
el
3. T
o br
oade
n th
e st
uden
ts u
nder
stan
ding
of
the
conc
epts
of
alqe
bra
and
geom
etry
with
in th
e co
ntex
t of
the
spac
e ag
e4.
To
prov
ide
the
stud
ents
with
inte
l,,st
ing
by-w
ays
and
area
s of
exp
lora
tion
rela
ted
to d
ista
nces
, dir
ectio
n,an
d di
men
sion
Con
tent
A. C
oord
inat
e G
eom
etry
1. L
ocat
ion
of p
oint
son
a li
ne
Proc
edur
es a
nd A
ctiv
ities
Tea
chin
g A
ids
In a
dditi
on to
the
refe
renc
es f
ound
in e
ach
Sinc
e a
line
has
no b
egin
ning
and
no e
nd, t
hese
ctio
n of
the
outli
ne, t
here
are
seve
ral
stud
ents
can
sel
ect a
n ar
bitr
ary
zero
poi
nt. H
ave
impo
rtan
t sou
rces
of
refe
renc
es a
nd a
ids
the
stud
ents
loca
te p
oint
s co
rres
pond
ing
to th
ew
hich
the
teac
her
shou
ld c
onsu
lt in
pla
n-po
sitiv
e in
tege
rs f
irst
. The
n lo
cate
the
posi
tive
ning
the
wor
k of
the
unit.
Ber
ger
and
frac
tions
and
dec
imal
fra
ctio
nsJo
hnso
n, "
A G
uide
to th
e Pr
ocur
emen
t of
Tea
chin
g A
ids
for
l'kla
them
atic
s,"
Nat
iona
lC
ounc
il of
Tea
cher
s of
Mat
hem
atic
s, A
pril,
1959
. Will
iam
L. S
chaa
f, "
Rec
reat
iola
lM
athe
mat
ics,
" N
CT
M, 1
958.
Tea
cher
ssh
ould
als
o re
fer
to th
e ye
atly
indi
ces
prin
ted
in th
e D
ecem
ber
issu
es o
f T
heM
athe
mat
ics
Tea
cher
93
a. S
ettin
g up
aon
e-to
-one
cor
res-
pond
ence
bet
wee
n nu
mbe
rsan
d po
ints
on
the
line
b. O
ne a
nd o
nly
one
coor
dina
te is
nece
ssar
y to
loca
te th
epo
si-
tion
of a
ny p
oint
on
a lin
e
2 D
eter
min
ing
dist
ance
s la
, '-w
een
poin
t on
a nu
mbe
r lin
e
a. B
y co
untin
gun
its b
etw
een
the
two
loca
tions
b. D
evel
op a
dis
tanc
e fo
rmul
a fo
rth
e nu
mbe
r lin
e to
the
gene
rali-
zatio
n (X
2X
1) =
d
At t
his
poin
t the
teac
her
mig
ht li
ke to
digr
ess
abi
t and
hav
e th
e pu
pils
exp
lore
the
conc
ept
of"d
ense
ness
." H
ow m
any
poin
ts a
re f
ound
be-
twee
n 1
and
2? B
etw
een
.1 a
nd .2
?B
etw
een
.01
and
.02?
Thi
s di
scus
sion
cou
ld a
dd to
an
unde
r-st
andi
ng o
f on
e-to
-one
cor
resp
onde
nce
Pupi
ls s
houl
d no
w b
e ex
pose
d to
the
loca
tion
ofne
gativ
e nu
mbe
rs. I
t mig
ht b
e de
sira
ble
to u
seth
e tim
e-ho
nore
d th
erm
omet
er a
s an
illus
trat
ion
but m
ost o
f th
ese
pupi
ls h
ave
som
ekn
owle
r'ge
of th
e ex
iste
nce
of s
uch
num
bers
Her
e th
e pu
pils
may
be
aske
d to
det
erm
ine
the
dist
ance
bet
wee
n m
any
sets
of
poin
tsby
fir
stlo
catin
g th
e po
ints
, the
n co
untin
g th
eun
its b
e-tw
ee.'
them
. A c
hart
suc
h as
the
one
belo
w w
illhe
lp th
em to
dis
cove
r th
e pa
ttern
and
thus
the
gene
raliz
atio
n. I
t is
impo
rtan
t to
begi
n w
ithth
epo
sitiv
e in
tege
rs. T
his
will
ena
ble
the
stud
ents
to s
ee th
e pa
ttern
dev
elop
mor
ecl
earl
y
x,X
2D
ista
nce
52
101
86
77
25
19
v , 223
/2
P.1
42
73
05
1/21
.i
6.25
4.75
Pupi
ls s
houl
d do
as
man
y of
thes
e as
nec
essa
ryto
see
the
patte
rn. W
hen
the
gene
raliz
atio
n ha
sbe
en f
orm
ed, p
upils
sho
uld
dete
rmin
edi
stan
ces
94
*
3. L
ocat
ion
of p
oint
s on
a p
lane
a. T
wo
and
only
two
coor
dina
tes
are
nece
ssar
y to
loca
te o
r to
spec
ify
any
poin
t in
a pl
ane
b. A
n or
deri
ng o
f th
e tw
o co
ordi
-na
tes
is n
eces
sary
to s
peci
fy a
part
icul
ar p
oint
c. D
evel
op a
dis
tanc
e fo
rmul
a fo
ran
y tw
o po
ints
in a
pla
ne
.1; (
x2x1
)2 +
(y2
yi)2
betw
een
poin
ts w
ithou
t gra
phin
g th
e po
ints
and
coun
ting
Her
e is
a p
lace
whe
re th
e st
uden
t can
be
expe
cted
to d
evel
op th
e fo
rmul
a by
him
self
, and
dis
play
the
enth
usia
sm w
hich
acc
ompa
nies
this
dis
-co
very
The
con
cept
of
usin
g tw
o co
ordi
nate
s to
spe
cify
Mat
hem
atic
s fo
r th
e Ju
nior
Hig
h Sc
hool
.a
poin
t can
be
intr
oduc
ed n
icel
y w
ith a
map
Vol
ume
IL Y
ale
Cri
vers
ity P
ress
, 196
1.us
ing
lines
of
long
itude
and
latit
ude
for
posi
-C
hapt
er I
tioni
ng c
ities
. Aft
er s
ome
disc
ussi
on o
f th
is te
ch-
niqu
e of
map
ping
, mos
t stu
dent
s sh
ould
be,
reao
y to
abs
trac
t the
line
s of
latit
ude
and
long
i-tu
de to
the
Car
tesi
an g
rid.
If
the
need
for
an
orde
ring
of
the
coor
dina
tes
has
not y
et b
een
rais
ed b
y th
e pu
pils
, the
poi
nts
such
as
(2,5
)an
d (5
,2)
shou
ld b
e pl
otte
d an
d th
e st
uden
tssh
ould
be
aske
d fo
r su
gges
tions
whi
ch w
ould
spec
ify
the
exac
t poi
nt to
be
disc
usse
d. T
hete
ache
r m
ight
like
to r
etur
n to
the
map
and
loca
te tw
o ci
ties
who
se la
titud
e an
d lo
ngitu
dear
e re
vers
ed
By
this
tim
z., p
upils
sho
uld
reco
gniz
e th
e tw
oco
nditt
on,re
quir
ed f
or s
peci
fyin
g an
y po
int o
na
plan
e; n
amel
y, tw
o co
ordi
nate
s an
d an
ord
erin
gof
the
coor
dina
tes.
The
not
atio
n fo
r an
ord
ered
pair
of
num
bers
sho
uld
be in
trod
uced
and
the
pupi
ls s
houl
d re
ceiv
e ex
tens
ive
prac
tice
inpl
ot ti
ng P
oint
s. T
his
prac
tice
can
be m
ade
mor
ein
trig
uing
to p
upils
if th
e co
ordi
nate
s, w
hen
conn
ecte
d, tr
ace
out i
nter
estin
g pi
ctur
es
In o
rder
to d
evel
op th
e di
stan
ce f
orm
ula,
it w
illbe
nec
essa
ry to
rev
iew
with
the
stud
ents
the
Pyth
agor
ean
theo
rem
. All
of th
e pu
pils
will
hav
eha
d so
me
expo
sure
to th
e pr
inci
ple
befo
re th
isco
urse
is ta
ken
but t
he e
xten
t to
whi
ch th
e co
n-ce
pt m
ust b
e re
view
ed w
ill d
epen
d up
on th
ena
ture
of
the
clas
s. A
goo
d be
ginn
ing
wou
ld b
eto
pla
ce tw
o po
ints
on
the
grid
, one
of
the
x tA
isan
d on
e on
the
y ax
is; (
0,3)
and
(4,
0). T
om
easu
re th
e di
stan
ce b
etw
een
the
poin
ts th
est
uden
ts m
ight
sug
gest
spr
eadi
ng th
e co
mpa
ss
95
Jack
son,
R. L
. "T
he D
isco
very
of
a C
on-
cept
--A
Dem
onst
ratio
n L
esso
n."
Mat
he-
mat
ics
Tea
cher
. Mar
ch, 1
961
4. L
ocat
ion
of p
oint
s in
spa
ce
a. T
hree
and
only
thre
e co
ordi
-na
tes
are
nece
ssar
y to
loca
teor
to s
peci
fy a
nypo
int i
n th
ree
dim
ensi
onal
spa
ce
b. A
n or
deri
ng o
f th
e th
ree
co-
ordi
nate
s is
nec
essa
ry to
spe
cify
a pa
rtic
ular
poin
t
betw
een
the
poin
ts, t
hen
layi
ng o
ffth
e di
stan
ceon
the
x ax
is a
ndco
untin
g th
e sq
uare
s. I
n th
isca
se th
e m
etho
dw
ould
be
appr
opri
ate
prov
ided
the
mea
sure
men
t is
accu
rate
.O
f co
urse
the
tech
niqu
e br
eaks
dow
n w
hen
the
dist
ance
be-
com
es a
n ir
ratio
nal
num
ber.
Thi
s pr
oble
m c
anbe
poi
nted
out
by
sele
ctin
g tw
oot
her
coor
di-
nate
s. T
hus
the
need
for
ate
chni
que
(the
Py-
thag
orea
n th
eore
m in
this
cas
e) is
crea
ted
The
Pyt
hago
rean
rel
atio
nshi
p ca
nea
sily
be
tran
slat
ed to
the
dist
ance
for
mul
a by
usin
g th
eco
ncep
t dev
elop
ed in
IB
2,us
ing
the
abov
e co
-or
dina
tes:
d =
1,X
2-1-
Y2
Sinc
e X
= (
30)
and
Y =
(4
(I3
- 0)
2 4-
-0)2
Oth
er c
oord
inat
es c
an n
ow b
e us
ed.P
upils
mus
tbe
led
slow
ly to
this
con
cept
with
cons
tant
refe
renc
e to
the
Car
tesi
an c
oord
inat
egr
id
In s
elec
ting
prob
lem
s fo
r th
epu
pils
, poi
nts
shou
ld b
e se
lect
ed o
nly
in q
uadr
ant
I fi
rst,
follo
wed
by
poin
ts s
elec
ted
from
othe
r qu
ad-
]. a
nts.
It m
aybe
nec
essa
ry to
rev
iew
with
the
stud
ents
at l
east
an
intu
itive
idea
of a
bsol
ute
valu
e
Pupi
ls w
ill e
xper
ienc
e so
me
diff
icul
tyin
vis
ual-
izin
g th
ree
dim
ensi
onal
spa
ce o
n a
two
dim
en-
sion
al s
urfa
ce. H
owev
er f
rom
the
prec
edin
g de
-ve
lopm
ent,
man
y w
ill b
e ab
le to
pred
ict t
henu
mbe
r of
coo
rdin
ates
nee
ded
to lo
cate
apo
int
in s
pace
and
als
o to
rec
ogni
ze th
ene
ed f
oror
dere
d tr
iple
ts o
f nu
mbe
rs
It m
ight
be
adva
ntag
eous
to s
pend
abr
ief
peri
odin
per
spec
tive
draw
ing.
No
doub
t mos
tte
ache
rsha
ve p
var
iety
of
tech
niqu
e.1
whi
chai
d in
ove
r-co
min
g th
is p
robl
em
96
Som
ethi
ng li
ke D
-Stix
set
s, a
vaila
ble
com
-m
erci
ally
, may
be
of h
elp
here
Nat
iona
l Cou
ncil
of T
each
ers
ofM
athe
-m
atic
s, M
ulti-
Sens
ory
Aid
s in
thP
Tea
ch-
ing
of M
atlu
mat
ies.
18t
h ye
arbo
ok,
1945
c. D
evel
op a
dis
tanc
e fo
rmul
afo
ran
y tw
o po
ints
in s
pace
5. E
xplo
re th
e po
ssib
ilitie
s of
afo
urth
dim
ensi
on
B. V
eloc
ity, a
ccel
erat
ion,
and
gra
vita
-tio
n
1. T
he m
eani
ng o
f sp
eed
and
velo
city
a. S
peed
ind:
ate
s ho
w f
ast a
nob
ject
is m
ovin
g
b. V
eloc
ity in
dica
tes
the
rate
of
mot
ion
in a
par
ticul
ar d
irec
-tio
n
2. T
he m
eani
ng o
f ac
cele
ratio
na.
Acc
eler
atio
n is
the
rate
of
chan
ge o
f ve
loci
ty
Som
e th
ree
dim
ensi
onal
gra
phin
g an
d pl
ottin
gof
poi
nts
shou
ld o
ccur
at t
his
time.
Pup
ils m
aybe
inte
rest
ed in
con
stru
ctin
g m
odel
s to
illu
stra
teth
e lo
catio
n of
poi
nts
in s
pace
. Som
e cu
rve
stitc
hing
act
iviti
es c
an a
dd r
eal i
nter
est a
t thi
spo
int.
Pupi
ls m
ay w
ish
to c
onst
ruct
thre
e di
men
-si
onal
str
ing
mod
els
out o
f fl
uore
scen
t ela
stic
thre
ad. P
erha
ps a
kit
such
as
"Spa
ce G
eom
etri
es"
wou
ld b
e he
lpfu
l her
e
As
the
pupi
ls c
ontr
ast t
he f
orm
ulas
for
dis
tanc
eson
one
and
two
dim
ensi
onal
surf
aces
, sev
eral
may
com
e up
with
the
gene
raliz
atio
n fo
r th
ree
dim
ensi
onal
dis
tanc
es. A
LW
AY
S M
AK
E U
SEO
F ST
UD
EN
T H
UN
CH
ES
Gen
eral
izat
ions
of
the
abov
e co
ncep
ts m
ay b
em
ade
to a
fou
rth
dim
ensi
on. S
ome
stud
ents
may
be in
tere
sted
in e
xplo
ring
this
pos
sibi
lity;
eve
na
cons
truc
tion
of a
fou
rth
dim
ensi
onal
mod
el(t
he te
sser
act)
. The
ext
ent t
o w
hich
a te
ache
rca
n dw
ell o
n th
is to
pic
mus
tbe
det
erm
ined
by
the
inte
rest
and
abi
lity
of th
e cl
ass
Abb
ott's
boo
k, "
Flat
land
" of
fers
an
inte
rest
ing
setti
ng f
or a
live
ly d
iscu
ssio
n of
life
as
it m
ight
be li
ved
in z
ero,
one
, tw
o, a
nd f
our
dim
ensi
ons.
STU
DE
NT
AC
TIV
ITY
AN
D I
NV
OL
VE
-M
EN
T I
S T
HE
KE
Y S
UC
CE
SS W
ITH
TH
ISM
AT
ER
IAL
Sinc
e m
uch
of th
e w
ork
in th
is u
nit i
s de
pend
ent
upon
an
unde
rsta
ndin
g of
the
conc
epto
f ve
loci
tyan
d ac
cele
ratio
n, p
artic
ular
pai
ns s
houl
d be
take
n in
pre
sent
ing
thes
e co
ncep
ts. M
ost h
igh
scho
ol p
hysi
cs b
ooks
pro
vide
a n
umbe
r of
exe
r-ci
ses
and
illus
trat
ions
dea
ling
with
vel
ocity
and
acce
lera
tion.
The
se e
xerc
ises
sho
uld
incl
ude
dis-
tanc
e, v
eloc
ity, a
nd ti
me
rela
tions
hips
usi
ng th
eex
pres
sion
d =
v t
Exp
lore
acc
eler
atio
n of
a c
ar. H
ave
som
e st
uden
tsga
ther
dat
a in
clud
ing
time
and
spee
d at
dif
feri
ngac
cele
ratio
n ra
tes.
Gra
phin
g sp
eed
agai
nst t
ime
will
sho
w a
ccel
erat
ion.
Thi
s w
ould
be
a go
odcl
ass
proj
ect
97
Abb
ott,
Edw
in A
. Fla
tland
: A R
oman
ce o
fM
any
Dim
ensi
ons.
Dov
er, 1
952
Gam
ow, G
eorg
e. O
ne T
wo
Thr
eein
fini
ty.
New
Am
. Lib
rary
, 195
4
Klin
e, M
orri
s. M
athe
mat
ics
in W
este
rnC
ultu
re. O
xfor
d U
niv.
Pre
ss, 1
953
Mill
s, R
ober
t. "T
he M
an in
Mot
ion.
" T
heM
athe
mat
ics
Tea
cher
. Feb
., 19
61. p
p10
7-10
9
Rus
h, J
. H. "
The
Spe
ed o
f L
ight
." S
cien
tif-
ic A
mer
ican
. Aug
., 19
55. p
. 62
Wei
dem
ann,
Cha
rles
C. "
Ani
mal
and
Bir
dSp
eeds
." T
he M
athe
mat
ics
Tea
cher
.Fe
b., 1
951.
pp
142
-143
Lar
son,
Har
old
D. F
aste
r an
d Fa
ster
. Row
Pete
rson
, 195
6
Mey
er, S
held
on. T
he M
athe
mat
ics
Tea
cher
.
3. N
ewto
n's
firs
t and
sec
ond
law
s of
mot
ion
a. A
n ob
ject
in m
otio
nw
ill n
otch
ange
its
spee
d or
dir
ectio
nun
less
act
ed u
pon
by a
for
ce
b. F
orce
is e
qual
to th
e pr
oduc
tof
the
mas
s an
d its
acc
eler
atio
n
4. A
ccel
erat
ion
due
to g
ravi
ty
C. C
urve
s of
fal
ling
obje
ets-
-the
par
a-bo
la1.
Dro
ppin
g an
obj
ect
a. d
= 1
6 t2
b. T
he e
ffec
t of
the
atm
osph
ere
on a
fre
ely
falli
ng o
bjec
t
2. D
ropp
ing
relie
f pa
ckag
es o
r bo
mbs
from
a m
ovin
g ob
ject
a. T
wo
forc
es a
rein
volv
ed in
this
situ
atio
n; th
at o
f gr
avity
act
-in
g on
the
obje
ct a
nd th
at o
fth
e m
ovin
g ob
ject
itse
lf
b. T
he c
urve
of
desc
ent o
f su
chan
obj
ect i
s pa
rabo
lic in
nat
ure
c. d
= v
t16
t2
3. J
avel
ins,
rif
les,
art
iller
y, a
ndIC
BM
'sa.
The
mot
ion
of o
bjec
ts to
bepr
opel
led
hori
zont
ally
and
up-
war
d is
dep
ende
nt u
pon
the
vert
ical
and
hor
izon
tal v
eloc
ityan
d th
e ve
rtic
al a
nd h
oriz
onta
ldi
stan
ce to
be
cove
red
Use
a g
yros
cope
to in
dica
te in
ertia
Thi
s co
ncep
t sho
uld
be m
entio
ned
brie
fly
inpa
ssin
g
The
coi
n-fe
athe
r tu
be c
omm
only
fou
nd in
phy
sics
labo
rato
ries
off
ers
anot
her
inte
rest
ing
dem
on-
stra
tion.
The
res
ult o
f th
is e
xper
imen
t mig
htle
ad th
e cl
ass
into
an
inte
rest
ing
disc
ussi
on o
fgl
ider
s an
d th
e ef
fect
of
the
atm
osph
ere
on th
eir
flig
ht
Thi
s is
the
firs
t app
licat
ion
in th
is u
nit o
f N
ew-
ton'
s fi
rst l
aw o
f m
otio
n. A
gra
phic
al o
r pi
ctor
ial
pres
enta
tion
of th
e fa
lling
obj
ect w
ould
pro
babl
ybe
the
mos
t eff
ectiv
e m
etho
d of
illu
stra
tion.
Thi
s pi
ctur
e le
ads
nice
ly in
to th
e co
ncep
t of
apa
rabo
lic c
urve
. Som
e di
scus
sion
will
no
doub
tde
velo
p re
gard
ing
the
prob
lem
of
how
abo
mba
rdie
r ca
n de
term
ine
how
far
ahe
ad o
f th
eta
rget
the
bom
b sh
ould
be
rele
ased
. Bom
b si
ghts
are
ofte
n av
aila
ble
from
war
surp
lus
stor
es a
ndca
n ad
d in
tere
st to
clas
s di
scus
sion
The
stu
dent
sho
uld
be a
war
e of
the
fact
that
this
traj
ecto
ry is
a c
urve
sim
ilar
to th
at d
iscu
ssed
in th
e se
ctio
n on
dro
ppin
g re
lief
pack
ages
and
bom
bs. T
he d
iscu
ssio
n of
the
traj
ecto
ry o
f a
shel
l sho
uld
brin
g ou
t the
impo
rtan
ce o
f th
ean
gle
of e
leva
tion
of th
e la
unch
er in
det
erm
inin
gth
e di
stan
ce to
be
trav
eled
by
the
mis
sile
98
App
licat
ion
Sect
ion,
Oct
ober
, 195
2. p
p45
3-45
4
Klin
e, M
orri
s. M
athe
mat
ics
and
the
Phys
i-ca
l Wor
ld. C
row
ell,
1959
Bak
st, A
aron
. Mat
hem
atic
s, I
ts M
agic
and
Mas
tery
. Van
Nos
tran
d, 1
952
y =
(u/
v) x
+16
x2
whe
re x =
hor
izon
tal
dist
ance
y =
ver
tical
dist
ance
u =
hor
izon
tal
velo
city
v =
ver
tical
velo
city
41
4. T
he n
atur
e of
the
para
bola
a. A
par
abol
ais
the
locu
s of
apo
int w
hich
mov
es in
a p
lane
so th
at it
s di
stan
cefr
om a
fix
edpo
int i
s al
way
s eq
ual t
o its
dis
-ta
nce
from
a f
ixed
line
b. T
he a
xis,
clir
ectr
ix a
nd f
ocus
ofth
e pa
rabo
la
c. T
he e
quat
ion
of th
e pa
rabo
la:
ay2
= b
xc
d. G
raph
ing
the
para
bola
5. T
he c
onst
ruct
ion
of th
e pa
rabo
la
a. P
aper
fol
ding
b. T
he p
arab
ola
as a
con
ic s
ectio
n
c. T
he u
se o
f a
com
pass
and
stra
ight
edg
e
d. C
urve
stit
chin
g
A d
iscu
ssio
n of
this
con
cept
can
be f
ound
inA
aron
Bak
st's
boo
k, "
Mat
hem
atic
s,It
s M
agic
and
Mas
tery
, "C
hapt
er 3
6, e
ntitl
ed. "
The
Fir
ing
Squa
d an
d M
athe
mat
ics"
If th
e cl
ass
is la
rgel
y sp
orts
orie
nted
, the
sam
eco
ncep
t can
be
deve
lope
d fr
omva
riou
s at
hlet
icev
ents
suc
h as
the
jave
lin th
row
intr
ack,
kic
king
,an
d pa
ssin
g in
foo
tbal
l, ba
tting
in b
aseb
all a
ndsh
ootin
g in
bas
ketb
all
Stud
ents
can
take
tim
e ex
posu
res
of s
omeo
nesh
ootin
g a
bask
etba
ll w
ith d
iffe
ring
arc
s.W
ith a
stro
ng li
ght,
the
refl
ectio
n w
ill tr
ace
apa
rabo
la
A n
umbe
r of
gra
phin
g ex
erci
ses
invo
lvin
g th
epa
rabo
la o
ffer
the
stud
ent a
n op
port
unity
toin
vest
igat
e th
e re
latio
nshi
ps b
etw
een
chan
ges
in th
e ge
nera
l for
mul
a an
d th
esh
ape
of th
ecu
rve.
Exa
mpl
esm
ight
be
the
addi
tion
of c
o-ef
fici
ents
oth
er th
an 1
for
eac
h of
the
vari
able
s,in
trod
ucin
g ne
gativ
e co
effi
cien
t,an
d in
ter-
chan
ging
the
vari
able
s of
the
equa
tion
Seve
ral i
nter
estin
g la
bora
tory
ses
sion
s ca
ngi
veth
e pu
pils
a g
ood
unde
rsta
ndin
g of
met
hods
of
cons
truc
ting
the
para
bola
. Pap
er f
oldi
ng e
xer-
cise
s ca
n be
fou
nd in
the
pam
phle
t, Pa
per
Fold
-in
g in
the
Mat
hem
atic
s C
lass
room
by
D. A
. Joh
n-so
n.
Cur
ve s
titch
ing
exer
cise
s ar
e ex
plai
ned
and
dis-
cuss
ed in
the
pam
phle
t, A
Rhy
thm
icA
ppro
ach
to M
athe
»zat
ics
by E
dith
Som
erve
ll. K
its a
reav
aila
ble
com
mer
cial
ly w
hich
con
tain
mat
eria
lsfo
r co
nstr
uctin
g m
any
inte
rest
ing
geom
etri
c pa
t-te
rns
with
ela
stic
thre
ad99
Stru
yk, A
dria
n. "
The
me
Pape
r, a
Rul
eran
d th
e C
entr
al C
onic
s."
The
Mat
he-
mat
ics
Tea
cher
. Mar
ch, 1
954.
pp.
189
-193
Olm
sted
, Hug
h. "
A P
arab
ola
Dev
ice.
"T
heM
athe
mat
ics
Tea
cher
. Oct
., 19
55. p
p.40
7-40
8
John
son,
D. A
. Pap
er F
oldi
ng in
the
Mat
he-
mat
ics
Cla
ssro
om. N
atio
nal C
ounc
il of
Tea
cher
s of
Mat
hem
atic
s, W
ashi
ngto
n,D
. C.
Som
erve
ll, E
dith
. A R
hyth
mic
App
roac
hto
Mat
hem
atic
s
6. U
ses
of th
e pa
rabo
lic c
urve
a. T
he d
evel
opm
ent
of th
e pa
ra-
bolo
id b
y th
e ro
tatio
n of
the
para
bola
on
its a
xis
b. I
n re
flec
tion
of li
ght
c. I
n ra
dio
rece
ptio
nan
d tr
ans-
mis
sion
d. A
s a
nom
ogra
ph in
com
putin
gsq
uare
s an
d sq
uare
roo
ts
7. O
verc
omin
g th
e gr
avita
tiona
lat
trac
tion
of th
e ea
rth
a. T
he e
scap
eve
loci
ty o
f th
eea
rth
is a
ppro
xim
atel
y 7
mile
spe
r se
cond
b. H
uman
tole
ranc
e of
hig
hac
cele
ratio
n ra
tes
c. A
nti-
grav
itym
achi
nes
D. T
he C
urve
of
Spac
e T
rave
lthe
ellip
se
1. U
nder
stan
ding
the
mot
ions
of
the
eart
h
a. T
he e
arth
rot
ates
on
its a
xis
ata
spee
d of
app
roxi
mat
ely
1,00
0m
ph a
t the
equ
ator
b. T
he e
arth
's r
evol
utio
n ab
out
the
sun:
65,
000
mph
c. T
he e
arth
's m
ovem
entt
hrou
ghth
e ga
laxy
: 60,
000
mph
d. T
he g
alax
y's
mov
emen
t thr
ough
the
univ
erse
is o
f un
know
nsp
eed
(3,5
00,
00 0
mph
estim
ate)
Use
s an
d oc
curr
ence
s of
the
para
bola
can
befo
und
in th
e pa
mph
let c
alle
d G
ums
inSp
ace
byD
. A. J
ohns
on. T
his
sect
ion
of th
e un
it ca
nal
sost
imul
ate
a go
od d
eal o
f cl
ass
disc
ussi
on.
Flas
h-lig
ht b
eam
s, a
utom
obile
hea
dlig
hts,
refl
ectin
gte
lesc
opes
, for
exa
mpl
e, o
ffer
man
y po
ssib
ilitie
sfo
r th
ese
disc
ussi
ons
Som
e of
the
hist
oric
al b
ackg
roun
d of
rock
etw
ork
whi
ch w
ould
be
inte
rest
ing
to th
est
uden
tsis
fou
nd in
Will
y L
ey's
boo
k on
Roc
kets
,Mis
sile
san
d Sp
ace
Tra
rel.
The
con
stru
ctio
n of
rock
ets,
the
deve
lopm
ent o
f fu
els
and
rece
ntin
nova
tions
to a
id in
the
deve
lopm
ent o
f ne
eded
thru
st a
redi
scus
sed
in s
impl
e la
ngua
ge a
nd in
gre
atde
tail
An
unde
rsta
ndin
g of
the
com
plex
mot
ions
of
the
eart
h an
d its
nei
ghbo
rs w
ill h
elp
the
stud
ents
see
the
com
plex
ities
of la
unch
ing
spac
e ve
hicl
esan
d ha
ving
them
arr
ive
in s
pace
at t
heri
ght
plac
e an
d th
e ri
ght t
ime.
Thi
s di
ffic
ulty
will
be
expl
ored
fur
ther
in s
ectio
n E
l(H
)
John
son,
D. A
. Cvr
ees
in S
pace
.Web
ster
,19
63
Saup
e, E
thel
. "Si
mpl
e Pa
per
Mod
els
of th
eC
onic
Sec
tions
." T
he M
athe
mat
ics
Tea
cher
. Jan
., 19
55. p
p. 4
2-44
See
Uni
t I o
n C
ompu
ting
Mac
hine
s
Ley
, Will
y. R
ocke
ts, M
issi
les
and
Spac
eT
rare
l. V
ikin
g, 1
958
Tel
ler,
Edw
ard.
"T
he G
eom
etry
of
Spac
ean
d T
ime.
" T
he M
athe
mat
ics
Tea
cher
.N
ov.,
1961
. pp.
505
-514
2. T
he n
atur
e of
the
ellip
se
a. A
n el
lipse
is th
e lo
cus
of a
poi
ntw
hich
mov
es s
o th
at th
e su
mof
its
dist
ance
s fr
om tw
o fi
xed
poin
ts is
con
stan
t
(1)
The
two
fixe
d po
ints
are
calle
d fo
ci
(2)
The
elli
pse
has
two
axes
b. T
he e
quat
ion
of th
e el
lipse
:
+ 3
2 b2
c. G
raph
ing
the
ellip
se
3. T
he c
onst
ruc4
don
of th
e el
lipse
a. B
y th
e us
e of
str
inz
and
pins
b. C
onst
ruct
ion
with
com
pass
esan
d st
raig
ht e
dge
c. P
aper
fol
ding
d. C
urve
stit
chin
g
e. T
he e
llips
e as
a c
onic
sec
tion
4. U
ses
of th
e el
liptic
al c
urve
a. T
he g
ener
atio
n of
an
ellip
soid
by r
otat
ion
of th
e el
lipse
on
eith
er th
e m
ajor
or
min
or a
xis
b. W
hisp
erin
g ga
lleri
es
c. R
efle
ctio
n of
sou
nd, l
ight
and
wav
es
d. G
ears
See
sect
ion
B-5
on
the
para
b la
The
labo
rato
ry a
ppro
ach
can
be q
uite
suc
cess
ful
in th
is s
ectio
n. T
he p
upils
mig
ht li
ke to
con
-st
ruct
item
s lik
e an
elli
ptic
al b
illia
rd ta
ble
orsw
imm
ing
pool
whi
ch il
lust
rate
s be
autif
ully
the
conc
entr
atio
n of
wav
es a
nd s
hots
em
anat
ing
from
one
foc
us a
nd c
onve
rgin
g on
the
othe
rfo
cus
Stud
ents
may
v, i
sh to
exp
erim
ent w
ith e
llipt
ical
Haz
ard,
Will
iam
. -C
urve
s of
Con
stan
tge
ars
and
perh
aps
follo
w th
roug
h w
ith s
ome
Bre
adth
." T
he M
athe
mat
ics
Tea
cher
.st
udy
of c
urve
s of
con
stan
t dia
met
erD
ec.,
1955
. pp.
89-
90
101
5. O
rbiti
ng o
f pl
anet
s an
d sa
telli
tes
a. K
eple
r's g
reat
disc
over
ythe
ellip
tical
orb
it of
the
plan
ets
Whe
re T
is th
e pe
riod
of
revo
-lu
tion
of th
e bo
dy, D
is th
eav
erag
e di
stan
ce f
rom
the
sun
and
K is
a c
onst
ant f
or a
llbo
dies
b. S
peci
fic
fact
ors
invo
lved
in th
eor
bitin
g of
sat
ellit
es
(1)
Prev
entin
g th
e sa
telli
tes
from
esc
apin
g th
e ea
rth'
sgr
avita
tiona
l for
ce c
om-
plet
ely
(2)
Prev
entin
g th
e sa
telli
tes
from
ret
urni
ng to
the
eart
h's
atm
osph
ere
and
burn
ing
up
(3)
Min
atur
izat
ion
of in
stru
-m
ents
allo
win
g sm
all p
aylo
ads
to g
athe
r im
port
ant
data
E. S
pace
land
ings
and
inte
rpla
neta
rytr
avel
1. T
he c
ompl
exity
of
eart
h m
otio
nsan
d ta
rget
mot
ions
in s
ched
ulin
gm
eetin
gs o
f th
e sp
acec
raft
and
the
I ar
get
a. L
unar
pro
bes
2. P
ossi
bilit
ies
of in
terp
lane
tary
trav
el a
nd tr
avel
out
side
our
sol
arsy
stem
The
his
tori
cal d
evel
opm
ent o
fK
eple
r's w
orks
can
prov
e to
be
very
help
ful h
ere.
Mor
ris
Klin
e's
book
, Mat
he»z
atic
s an
d th
e Ph
ysic
al W
orld
,an
dIi
akst
's b
ook,
Mat
hem
atic
s, I
ts M
agic
and
Mas
t-er
y w
ould
be
mos
the
lpfu
l
Thi
s se
ctio
n w
ill h
elp
the
pupi
l und
erst
and
why
ther
e is
mor
e in
volv
ed in
"hi
tting
the
moo
n"th
an m
erel
y po
intin
g a
rock
et in
that
dire
ctio
n.A
ddin
g th
e m
otio
n of
the
moo
n to
the
com
plex
mov
emen
ts o
f th
e ea
rth
mak
es th
epr
oble
m e
ven
mor
e in
volv
ed. S
imila
rpr
oble
ms
will
exi
st w
hen
atte
mpt
ing
a re
ndez
vous
with
a p
lane
t. Si
nce
luna
r pr
obes
are
pre
sent
ly u
nder
way
, the
yw
ould
rye
as a
goo
d st
artin
gpo
int f
or th
isdi
scus
sion
Usi
ng a
s a
basi
s th
e op
timum
spe
ed p
ossi
ble
atth
is ti
me
in o
uter
spa
ce tr
avel
, hav
e th
e pu
pils
com
pute
the
leng
th o
f tim
e ne
eded
tom
ake
the
102
Whi
pple
, F. L
., an
d H
ynek
, J. A
."O
bser
-va
tions
of
Sate
lEte
I."
Sci
entif
icA
mer
i-ca
n. D
ec.,
1957
. p.
37
Men
gel,
J. T
., an
d H
erge
t, Pa
ul."
Tra
ckin
gSa
telli
tes
by R
adio
," S
cien
tific
Am
eri-
can.
Jan
., 19
58. p
.
Lan
don,
M. V
. "M
echa
nics
of
Orb
iting
,"T
he M
athe
mat
ics
Tea
cher
. May
,19
59.
pp. 3
61-3
61
Ley
, Will
y. R
ocke
ts, M
issi
les,
and
Spac
eT
rare
l. V
ikin
g, 1
958
a. T
he ti
me
fact
or
F. T
he c
urve
of
loca
tion:
the
hype
rbol
a
1. T
he n
atur
e of
the
hype
rbol
a
a. it
s gr
aph
and
equa
tion
x2y2
b. a
sym
ptot
es
2. U
ses
of th
e hy
perb
ola
a. T
he p
aths
of
som
e co
met
str
avel
in h
yper
bolic
cur
ves
b. U
ses
in n
avig
atio
n in
det
er-
min
ing
loca
tions
on
the
eart
h's
surf
ace
c. T
he s
onic
boo
m
3. C
onst
ruct
ion
of th
e hy
perb
olic
curv
e
a. U
sing
the
com
pass
and
str
aigh
ted
ge
b. P
aper
fol
ding
c. T
he h
yper
bola
as
a C
onic
see
-do
n
jour
ney
to o
ther
pla
nc s
as
wel
l as
to th
e ne
ares
tst
ar. T
he p
upils
will
be
inte
rest
ed in
dis
cuss
ing
the
max
imum
pos
sibl
e sp
eed
of o
ur f
utur
e sp
ace
ship
s
Publ
icat
ions
pre
pare
d by
NA
SA (
Nat
iona
lA
eron
autic
s an
d Sp
ace
Adm
inis
trat
ion)
are
mos
the
lpfu
l in
prov
idin
g up
to d
ate
info
rmat
ion
onth
ese
prob
lem
s. A
rec
ent b
ook,
Pro
ject
s: S
pace
,by
Jud
ith V
iors
t was
wri
tten
spec
ific
ally
on
NA
SA p
roje
cts
Stre
ss th
e do
uble
fig
ure
inhe
rent
in th
e gr
aph
of th
e hy
perb
ola
Men
tion
that
the
hype
rbol
a is
a s
tart
ing
poin
tfo
r th
e an
alyt
ic s
tudy
of
loga
rith
ms.
See
any
mod
ern
adva
nced
alg
ebra
text
for
a d
iscu
ssio
n
10;3
Vjo
rst,
Judi
th. P
roje
cts:
Spac
e. W
ashi
ngto
nSq
uare
Pre
ss I
nc.,
1962
See
refe
rene
es o
n th
e pa
rabo
la, s
ectio
n C
-5
Stru
yk, A
dria
n. "
The
me
Pape
r, a
Rul
eran
d th
e H
yper
bola
." T
he M
athe
mat
ics
Tea
cher
. Jan
., 19
54. p
p. 2
9-30
G. O
ther
topi
cs to
inve
stig
ate
1. O
ther
cur
ves
and
thei
r ap
plic
a-tio
ns s
uch
as th
e ca
tena
ry, c
ardi
-oi
d, c
yclo
id, s
pira
l
2. O
ther
cur
ved
surf
aces
suc
h as
the
toru
s, m
oebi
ti, s
trip
, and
pre
tzel
3. L
inka
ges
as m
echa
nica
l dev
ices
for
cons
truc
ting
curv
es
4. V
ecto
rs
Men
tion
that
the
cate
nary
is th
e "c
urve
of
quic
kest
des
cent
." S
ee a
text
on
Cal
culu
s of
Var
iatio
ns.
Ask
the
clas
s if
an
inne
r tu
be (
toru
s) c
an b
etu
rned
insi
de o
ut
N.C
.T.M
., 18
th y
earb
ook
1945
. "M
ulti-
Sens
ory
Aid
s."
BIB
LIO
GR
APH
Ya.
BIB
LIO
GR
APH
Y
Abb
ott,
Edw
in 4
. Fla
tland
: A R
oman
ce o
f M
any
Dim
ensi
ons.
Dov
er, 1
952
pap.
als
oA
dler
, Irv
ing.
Thi
nkin
g M
achi
nes.
Joh
n D
ay, 1
951
Asi
nov,
Isa
ac. T
he R
ealm
of
Num
bers
. Hou
ghto
n, 1
959
Bak
st, A
aron
. Mat
hem
atic
s, I
ts M
agic
and
Mas
tery
. Van
Nos
tran
d, 1
952
Bak
st, A
aron
. Mat
hem
atic
al P
uzzl
es a
nd P
astim
es. V
an N
ostr
and,
195
4B
all,
W. W
. Rou
se, a
nd C
oxet
er, H
. S. M
. Mat
hem
atic
al R
ecre
atio
ns a
nd E
ssay
s. M
acm
illan
, 195
7B
erke
ley
and
Wai
nwri
ght.
Com
pute
rs, T
heir
Ope
ratio
ns a
nd A
pplic
atio
ns. R
einh
old,
195
6B
oy S
cout
s of
Am
eric
a. M
erit
Bad
ge B
ookl
et #
3327
. Sur
veyi
ngB
rant
and
Kee
dy. E
lem
enta
ry L
ogic
for
Sec
onda
ry S
choo
ls. H
olt,
1962
Car
roll,
Lew
is. M
athe
mat
ical
Rec
reat
ions
of
Lew
is C
arro
ll. D
over
, 195
8C
hase
, Stu
art.
Gui
des
to S
trai
ght T
hink
ing.
Har
per,
195
6C
ulbe
rtso
n, J
ames
T. M
athe
mat
ics
and
Log
ic f
or D
igita
l Dev
ices
. Van
Nos
tran
d, 1
958
Dan
tzig
, Tob
ias.
Num
ber,
the
Lan
guag
e of
Sci
ence
. Dou
bled
ay, 1
056
DeG
razi
a, J
osep
h. M
ath
is F
un. E
mer
son
Boo
ks, 1
954
De
Mor
gan,
Aug
ustu
s. B
udge
t of
Para
doxe
s. D
over
, 195
4 2
vols
Eck
ert,
W. J
., an
d Jo
nes,
R. F
aste
r, F
aste
r. M
cGra
w, 1
956
o.p.
Fede
ral H
ousi
ng A
dmin
istr
atio
n, W
ashi
ngto
n, D
. C. "
Prin
cipl
es o
f Pl
anni
ng S
mal
l Hou
ses.
" T
echn
ical
Bul
letin
No.
4Fl
esch
, Rud
olph
. The
Art
of
Cle
ar T
hink
ing.
Har
per,
195
1, a
nd C
ollie
r pa
p.Fo
rd M
otor
Co.
, Edu
catio
nal R
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