teach the best aspects of recent mathematics programs
TRANSCRIPT
-
DOCUMENT RESUME
ED 026 269s00000 9
By-Gelbaum, Bernard B. And Others[Orange County Science Education Improvement Project Syllabuses, K-6.3
'Orange County Public Schools, Calif.Spons Agency-National Science Foundation, Washington, D.C.Pub Date 66Note-819p.EDRS Price MF-$35 HC441.05Descriptnrs-*Arithmetic, *Cureiculum, *Elementary School Mathematics, Fundamental Concepts, *Instructional
Materials, Mathematics, Number Concepts, *Teaching Guides, Teaching Procedures
These syllabuses for K-6 Were written, evaluated, and revised by a team ofwriters from the Orange County Science Education Improvenient Project (OCSEIP).OCSEIP is a cooperative enterprise undertaken by the University of California (Irvine),California State College at Fullerton, the Orange County Schools Office, and localdistricts throughout Orange County. These syllabuses, were written to help teachersteach the best aspects of recent mathematics programs. Presented are somemethods of approach, intuitive examples, suggestions for additions arid deletions, andapplications in mathematics. The mathematical content for these syllabuses includesmaterials froM geometry, sets, numbers and numeration% order and relations,addition and subtraction, problem solving, and measurement. RP) .
-
e.,
acs.
EL
P. S
YL
LA
BU
S
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derg
arte
n
U.S
. DE
PA
RT
ME
NT
OF
HE
ALT
H, E
DU
CA
TIO
N &
WE
LFA
RE
OF
FIC
E O
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DU
CA
TIO
N
TH
IS D
OC
UM
EN
T H
AS
BE
EN
RE
PR
OD
UC
ED
EIA
CT
LY A
SR
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
Y R
EP
RE
SE
NT
OF
FIC
IAL
OF
FIC
EO
F E
DU
CA
TIO
9
PO
SIT
ION
OR
PO
LIC
Y.
"7-7
-7--
i-
-
AC
KN
OW
LE
DG
ME
NT
S
The
Ora
nge
Cou
nty
Scie
nce
Edu
catio
n Im
prov
emen
tPr
ogra
m (
0.C
.S.B
.I.P
.) is
spc
xu3o
red
by th
eN
atio
nal S
cien
ce F
ound
atio
n an
d ho
sted
. by
U.C
.Ir
vine
.It
is a
coo
pera
tive
entu
re u
nder
take
rby
the
Uni
vers
ity o
f C
alif
orni
a, I
rvin
eC
alif
orni
a St
ate
Col
lege
at F
ulle
rton
the
Ora
nge
Cou
nty
Scho
ols
Off
ice
an d
. loe
al s
choo
ldi
stri
cts
thro
ugho
ut O
rang
e C
ount
y.T
his
sylla
bus
was
wri
tten
by 0
.C.S
.E.I
.P. t
o he
lp te
ache
rs te
ach
the
best
asp
ects
of
rece
nt m
athe
mat
ics
prog
ram
a.It
is n
ot m
eant
to b
e an
othe
r te
xtbo
ok f
or a
'new
pro
gram
.In
stea
d., i
t is
mea
nt to
be
ash
arin
g an
d sv
nthe
sis
of e
ffec
tive
teac
hing
met
hods
.T
he o
utlin
e of
topi
cs is
a m
inir
amco
vera
ge v
hich
is c
am=
toal
l sch
ools
in O
rang
e C
ount
y.T
opic
s ad
equa
tely
cov
ered
in th
em
ajor
ity o
f te
xts
in u
se a
re g
iven
a m
inim
um-t
reat
men
t in
'the
sylla
bus.
The
fir
st d
raft
of
this
syl
labu
s va
s w
ritte
n du
ring
an
8 w
eek
sess
ion
at U
nive
rsity
of
Cal
ifor
nia,
Irv
ine
duri
ng th
e su
mer
of
1966
by:
Dr.
Will
iam
Wei
ser
- C
o-C
haim
=T
ed. B
robe
rgSu
san
Rop
er -
Co-
Cha
irm
anSy
lvia
Hor
neV
elm
a W
est -
Co-
Cha
irna
nR
. A. "
fork
The
fir
st d
raft
vat
s ev
alua
ted
8111
1 re
vise
dby
the
follo
win
g m
embe
rs o
f a
Uni
vers
ity O
fC
alif
orni
a, I
rvin
e E
xten
sion
cla
ss d
urin
g th
e .s
choo
l yea
r19
66-6
7:
Sylv
ia H
orne
- M
aste
r T
each
erG
eorg
ia B
rea'
Bar
bara
Cro
uch
gay
Savo
ieL
ee L
ou S
ell
Vir
gini
a Sn
vder
We
wis
h to
than
k au
the
part
icip
ants
in th
is p
rogr
am f
or th
eir
hard
wor
k an
d fi
neco
oper
atio
n.
Ber
nazd
B. G
elba
lmt,
Cha
ir=
Dep
artm
ent o
f M
a'4h
emat
ics,
Uni
vers
ity o
f C
alif
orni
a, I
rvin
eD
irec
tor,
0C
eS.E
.I.P
.
Rus
sell
V. B
elm
ar A
ssoc
iate
Pro
fess
orof
Mat
hem
atic
s, C
alif
orni
a St
ate
Col
lege
at P
ulle
rtat
Ass
ocia
te D
irec
tor,
0.C
.S.L
I.P.
-
PRE
iAC
E
The
se u
nits
war
e w
ritte
n to
pro
vide
teac
ilers
with
inst
ruct
iona
l Mat
eria
lsth
at w
ould
impl
emen
t
mod
ern
appr
oach
es to
teac
hing
mat
hem
atic
s.St
ress
was
pla
ced
onde
velo
ping
in a
rtic
ulat
ed-
prog
ram
fro
mki
nder
gart
en th
roug
h co
llege
.
Inde
term
inin
g:th
e un
its to
be
deve
lope
d,th
e w
riid
ng te
em a
gree
d to
give
avi
"L
nwde
pth"
trea
tmen
t to
tit*
area
sco
nsid
ered
wea
k in
the
pres
ent
mat
hem
attc
n cu
rric
ula*
inO
rang
e 02
unty
-sch
ools
..
,T
he "
stra
nds"
of
mat
hem
atic
s, a
s pr
esen
ted
by th
e A
dvis
ory
Cas
taitt
ee o
nM
athe
mat
ics,
wer
e
used
as th
e ba
sis
for
eval
uatin
gthose topics in elementaryschool
mat
hem
atic
sto be
cons
ider
mi
for
incl
usio
n in
this
proj
ect.
Impl
eMen
ting
Mat
hem
atic
s Pr
omos
In C
alif
oitia
, AG
uide
K.8
was
use
d as
a.g
uide
in.th
e ex
amin
atio
nof
the
scop
e an
d, s
eque
nce
oral
-Win
Vir
nder
rira
ir"
curr
ent s
tate
-ado
pted
.tex
ts.
As
you
use
thes
em
ater
ials
, you
are
urg
ed to
be c
reat
ive
in y
our
teac
hing
and
to n
ot r
estr
ict
your
inst
ruct
ion
to th
e su
gges
tions
,ex
ampl
es, a
nd id
eas
give
nhe
re.
rn a
cme
area
s th
ere
are
-mor
e su
gges
tions
than
.you
will
nee
d or
can
use
effe
ctiv
ely.
Men
this
isth
e ca
se, s
elec
t'th
e m
ater
ial t
hat m
illbe
mos
t app
ropr
iate
for
you
rcl
ass.
.,
--T
he d
egre
e to
whi
chth
ese
or a
ny o
ther
mat
eria
lsw
ill im
proi
S m
athe
mat
ics
inst
ruct
ion
in y
our
-zoo
m d
epen
da o
n yo
uren
thus
iasm
and
des
ire
to p
rovi
dest
imul
atin
g m
ath
expe
rien
ces
fur
your
-
stud
ents
.
The
wri
ttng
tam
hop
es y
ouw
ill f
ind
thes
e m
ater
ials
help
ful a
id th
at, t
brom
gh th
edi
esea
very
.ap
proa
ch, y
our
stud
ents
will
be c
halle
nged
to d
roll*
tbei
r.m
etit
pote
ntia
l..
-
o
INMODUCTION
Kindergarten,
Mathematics in Kindergarten
should be informal and flexible,
but it must.also be
carefully planned to
capitalize upon the naturalcuriosity and eagerness for learningthat most kindergartners possess.
The
productive readiness period cannotbe left to chance, but must be
nurtured by a well informed tz.,cher.
Both planned and incidentalmath.lessons are necessary--neitheralone is adequate.
The length of the kindergartenday, the other than math curriculumand the natural short interest
span of a five yearold will determine the amount
of time spent each day on mathematics.
Game
and aids at a "Math" tablefor free Choice time provide intuitive
learning.
The content of the program is
presented in three natural phases.
ehase one:
Pre-number; phase
two:
matching sets, nuMber and numeration;
and phase three:
the operations.
Each phase is filled with experiences
involving manipulation of aids that
the child can see, touch,
move about, group, regroup
and discuss.
The ikillful teacher "gives" no answers
but motivates
questions from the children and answers
the questions with yet another questionmakin4 it possible for
,the children to use ,known facts todiscover the answers alone.
The teacher supplies the new language
allawben the Childjmeds it to
verbalize his thoughts.
It is strongly recommended thatindividual worksheets or workboOks do notbecome
part of the work
done by children in Kindergarten.
Experience provided by workbooks and
worksheets too often makes
little contribution for effective
learning.
The mechanics of finding andmarking the:correct place,
mhen eye and hand coordination may not
be adequately developed, interferes
with the mathematical ideas
being developed.
Furthermore, visual perception is often notdeveloped enough for some children to
understand the math concepts that pictures are
to convey.
The manipulation of objects makes a more
lasting contribution to deeper understanding
of the ideas under study and are obtainedwhen children
are active participantsand nat merely passive watchers and listeners.
-
TA
BLE
CV
CO
NT
EN
TS
PA
CE
AcK
NO
WIE
DG
ME
NT
S3
lb
GE
ME
TR
Y.
4,
SE
TS
RU
MM
ER
S A
ND
NU
/MA
=0N
****
***
Ill22
OR
DE
R A
ND
NIM
MO
NS
****
****
110
42
AD
DIT
ION
AN
D S
UB
TR
AC
TIO
NIll
II50
PR
OB
IEM
SO
LVIN
GO
OO
OO
OO
Olb
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ME
AS
UR
EN
ER
T62
AP
PE
ND
IX00
00II
70
BIB
LIO
GR
AP
HY
**
iv
EP-9840
-
GE
OM
ET
RY
KIN
DE
RG
AR
TE
N
00
0,
0
V%
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00
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0
3
215.
3
And finally with a few lines added,
the equations are
compressed into a nice, neat structure.
Completed
by the children, itlooks like the illustration.
Display two sets onrthe flannel board.
Have a child
identify the number of the set
and place the numeral
under till set.
Then explain to the children that wehave
a special way to
show that five is greater than three.
Place the> symbol between 5 and 3.
Read the statement
(5 > 3): "five is greaterthan three."
Compare other
numbers not greater than nine.
Explain to the
children t,2t
the open part of the sumbol is next tothe numeral for the
greater number.
The relation "less than" can be introduced
the same way.
Make sure the children understand
which symbol is which
before
.
going to more complexmath sentences.
Show the children a card with five numerals
in random order.
Ask a child to unscramble the numeralsand rewrite them on
the chalkboard in order of least to greatest.
The first
time do this with the class, ask which
numeral represents
the smallest number and proceed to place in
order.
Af,er
they are in order, place symbol
2 44:5 etc.
Have the
child
ren
read the symbols:
"nine is greater than two, two
is less than five," etc.
-
Open and Closed methematical sentences
in addition and subtraction.
Ope
n
0+1=
6.-
.0=
5+
3=
0
1+-0
=1
0-2
=3
CJo
se.c
i2+
3=
c
Open sentences are used to present
addition and subtraction.
Children should read them as sentencesbefore attempting
to solve them, reading "what" or"something" for the
placeholder.
When the correct numeral is in the
placeholder,
the sentence is closed.
This is difficult during the first
semester especially for average and
slower learners.
31
-
Use the tape recorder
tomake a tape that directs
the
children to do eachmath sentence on a ditto
sheet.
This ditto sheet-shouldbe constructed to
emphasize
the aifficult sentences
4.
suchchalic3- 2 = 3 and
= 4
.T
hen
give
eac
hin
you
r cl
ass
adi
tto a
ndso
met
hing
to c
ount
suc
h as
bead
s on
a w
ire
orbo
ttle
caps
.Next make the tapewhile the clas
does the ditto
(ndking
the tape while the
children are doing theditto helps the
teacher pace the tapecorrectly).
After the tape hasbeen made, it can beplayed again for
drill for those that
require it.
For example:
if th
emath sentence'isq-
2 = 5,
then
the
tape could say:
"This is a lunch
x story.
You ate 2 cookies
and then
you were nathungry any longer,
so y
outook five codkies
home.
Take 2 counters fbrthe 2 you ate andthen take $ more
for the 5 cookies youtodk home.
Now find out how many
counters you have.
Two
that
you
ate and 5
-tha
t you
took
home - How many coOkies
did Mother give you?
(pause)
Yes, 7.
w'h
atgoes in theplaceholder (mrsquare)? (pause)
Yes
, 7.
Draw tmo dots on oneside of the paper and draw one
dot on
the other side.
Fold the card inhalf so that only two
dots
show.
Show the card to the
class.
Have one child tellhow
many dots
be sees,
then
unfbId the card andhave the child
tell how many dots
"joined"
the
set.
Next adk a child to
wri
te th
e re
late
deq
uatio
n on
the
boar
d.2
+ 1
= 0
Sy reversing the
procedure, you can presentthe related
subtraction equation.
(mathematical sentence)
-
Open and closed mathematical sentences
addition and subtraction using two
addends.
in
Present story problems to the children by telling a
story
that can be written in an equation.
The teacher may say,
"Bobby, put one ornament on the Christmas tree and Susan,
put four ornaments on the tree.
Adk someone in the class
to come and write the math sentence orequation on the board.
The child may need the story told again, he then writes
the equation I + 4 =Don the board.
Then ask another
child to &One and, place the sum in the equation.
Then tell the children "there were five balls on
the tree
but the cat, Blackie, broke four of the balls,
how many
balls are left on the tree?
Then select a child to write
the equation on the board.
5 - 4 =0
Children at this level enjoy activities that appeal to
their
sense of humor.
Develop math sentences around riddle games.
To provide sentences that give the
children experience with the
placeholder in different positions in subtractionequations,
present the following riddles:
Say to the class, "/f I subtract one from
this nuMher, I get
two.
What is my nuMber?"
Call on a child to ',mite the equation(0- 1 = 2) on the
board.
Ask the class if the equation is correct.
Continue with 'this procedure using the
following riddles.
If I sUbtract two from
three, what number do
have?
-
"--
Open and closed
mathematical sentences
in addition andsubtraction
through 3 addends.
Ope
n
Clo
sed
31-2
4-47
-1
T
If you subtract onefrom
me, you willhave one.
.'What number am
I.
One part of a set
ofthree is a set of two.
I am the other part.
I
am a set
of h
owmany
objects.
-47T
VT
741f
Irer
MII
MI,
Use a balance scale
to illustrate
this concept.
Commercial
ones are
available.
This can also be
illustrated on a
bulletin board as perillustration.
Marks on cards must
balance
or b
eequal on each side
of the scale.
- 37 -
-
IMA
ID
ON
INIO
SK
amm
''Skalqcts'ol-.4
Aix!
er:r.St:r6EA
-
Intr
oduc
ing
Stor
yPr
oble
ms
10 f
eet
Ine-
iriT
h-3
8..Pr
oble
m s
olvi
ng s
eeks
to a
pply
the
abst
ract
ion
ofm
athe
mat
ics
to th
e w
orld
even
ts.
The
sto
rypr
oble
m h
as b
een
used
. at t
he e
nd o
fun
its a
s a
test
or e
xerc
ise
of th
e pu
vils
1 ab
ility
to a
pply
an
acqu
ired
. kno
wle
dge
of a
rith
met
ic.
Oft
en th
e st
ory
prob
lem
has
onl
ybe
en a
rea
ding
test
.If
a c
hild
coul
d re
ad, t
he q
uest
ions
are
triv
ial,
and
cons
titut
e
exam
ples
and
not p
robl
ems.
Exa
mpl
e: M
r.Jo
nes
boug
lxb
8 fe
nce
post
s. H
e pa
id$1
0 a
piec
e.M
iat w
as h
is to
tal e
vens
e?(I
f yo
u C
M r
ead,
this
con
stitu
tes
anex
ampl
e of
the
fact
that
8 x
10
=80
.)
Prob
lem
:14
r. J
ones
bou
ght
8 fe
nce
post
s. H
e se
tth
em o
ut in
a s
trai
ght l
ine,
10 f
eet
apar
t. W
hat
is th
e di
stan
ce f
ran
the
cent
er o
f on
e en
d po
st?
(If
you
can
reas
l, th
isis
a p
robl
em th
atre
quir
esca
refu
l con
side
ratio
nof
the
situ
atio
npr
esen
ted.
,w
ith p
erha
ps a
ske
tch
to s
how
seve
n sp
aces
.)
Hav
e st
uden
tsdr
amat
ize
the
stor
yof
the
4 :3
shep
herd
.who
coun
ted.
his
she
ep b
y m
atch
ing
each
shee
p w
ith a
peb
ble
asth
ey w
oe in
toth
e co
rral
.C
ount
the
stud
ents
this
way
as
they
leav
e th
e ro
omfo
r re
cess
and
com
eba
ck a
fter
rec
ess.
Ask
the
clas
s nH
ow d
o yo
u kn
owth
at e
very
one
retu
rned
?"T
he c
hild
ren
have
foun
d th
at s
ets
have
the
sam
enu
mbe
r of
mem
bers
with
out c
ount
ing
inth
e us
ual.
130:
1130
Mak
e pa
rty
plan
s a
part
of y
our
arith
met
iccl
ass.
Put a
ll nu
mbe
r pr
oble
ms
rela
ted
to th
e pa
rty
onsm
all p
iece
s of
pap
er.
Pick
out
var
ious
card
s an
dw
ork
out t
he p
robl
ems.
-
Dra
mat
izin
g pr
dblin
situ
atio
ns o
ccur
ring
incl
assr
oom
and
sta
ting
prdb
lem
s
Por
exam
ple,
if th
ree
child
ren
mak
e pl
ace
mat
s fo
rth
e gr
oup,
haw
man
y w
ill e
ach
have
to m
ake?
How
man
y di
mes
will
be
need
ed. t
o bu
y m
ilk?
The
teac
her
help
s th
e pu
pils
sol
ve a
pro
blem
by
havi
ng th
em d
ram
atiz
e or
mak
e dr
awin
gs,
toill
ustr
ate
it. M
en th
e pu
pils
wri
te th
e pr
oble
mas
an
equa
tion
with
a p
lace
hold
er in
it.W
hen
the
pupi
ls a
re a
ble
to s
olve
pro
blem
s by
them
selv
es, t
hey
may
do
so.
Exa
mpl
e: T
ed h
ad s
ix m
arbl
es.
Whe
n he
cam
e ba
ckfr
an th
e st
ore
with
mor
e m
arbl
es h
e ba
d10
alto
geth
er.
How
man
y di
d. h
e ge
t at
the store?
6 +
= 1
0
Mak
e ch
arts
with
*pic
ture
s of
obj
ects
you
mig
ht I
n.."
'in
a s
tore
and
. sho
w th
eir
pric
es.
Hav
e th
e cl
ass
take
turn
s in
dra
mat
izin
g st
orie
s of
buy
ing
and.
selli
ng.
A s
tore
may
be
set u
p w
ith g
ood,
s br
ough
tfrom home.
This
may
be
a to
ystore, grocery store, etc.
Choose
ach
ild to
serve as storekeeper.
Give the children
sets of coias to buy given items.
Have the store-
keeper give the correct amount of Change.
(Chi
ldre
n co
uld
mak
ethe money- to
beused.)
See the game, "Going Walking" in the
Add
ition
and
SUbtraction section,
-
Number line in problem solving
Ute the nuMber line
in problem solving,
The
nuMber line should be
situated, near a wall so
children would not lookat it upside dowa.
Have several pupils jump
along the line.
Suggest
that
they
jump like a, frOgs hoplike a bard, or a
rabbit.
Ask the class if they:know the =Ober of
Imps anyonehas taken.
et 3 %say a
Two hops plus two
hops is
four
hop
s.
Choose
chili
:tren
trepresent points or
"stations"
on the nudberline starting with one.
Asko
"Bow
rimmqchildren are there?
Row
can
you
tell
with
out
counting?"
Giv
e ch
ildre
n ta
gs ir
ith ta
llym
arks
. Sam
e th
enm
atch
them
selv
es to
the
num
ber
Ito.
.
-
To
solv
e th
ese
stor
ypr
oble
ms
the
teac
her
ceus
have
the
child
ren
refe
r to
the
clas
s m
anbe
r lin
e or
thei
rin
divi
duai
num
ber
lines
at
thei
r de
sks.
The
teac
her
may
say
,"T
here
are
28
child
ren
in m
rfi
rst g
rade
but
one
day
ther
e w
ere
25 c
hild
ren
inth
e ro
om.
How
man
y pu
pils
wer
eab
sent
? M
ech
ildre
n ca
n re
fer
to th
e lu
mbe
rlin
e to
rea
ch th
eco
rrec
t ans
wer
.Pu
pils
loca
te th
e nu
mbe
r28
and
then
cou
nted
. spa
ces
back
to25
to d
isco
ver
that
3 w
ere
abse
nt.
The
num
ber
line
can
also
be
used
to s
olve
the
follo
win
g pr
oble
m.
1.T
here
are
17
book
s on
the
tabl
e.W
hen
Mar
ypu
bs tw
o m
ore
book
s on
the
tabl
e ho
w m
any
dow
e th
en h
ave
onth
e ta
ble?
2.M
ark
hail
11 p
enci
ls. H
elo
ft 3
of
them
.H
ow
man
y do
es h
e ha
vele
ft?
3.Jo
luha
s lli
. mar
bles
.B
ill g
ave
him
11.
.H
ow m
any
mar
bles
doe
s Jo
hn h
ave
now
?
li..
Luc
y ha
d, n
ine
cray
ons
yest
erda
y.T
oday
she
has
5.H
ow m
any
cray
ons
did
she
lose
?
The
abo
ve p
robl
ems
are
disc
usse
d or
"ac
ted
out."
The
y ar
e no
t wri
tten
nor
are
they
mea
nt to
be
read
by
the
child
ren.
-
Inte
n3re
ting
prob
lem
sfr
om p
ictu
res
7.7r
.7^3
7r,"
"r,1
0.77
-'-rY
,r,7
77t7
T.7
7'7'
-:7
7k"I
'Mk
Her
e is
an
oppo
rtm
nity
for
the
child
ren
to s
eese
quen
ce in
adi
ffer
ent f
orm
.T
hey
mus
t sol
veth
epr
oble
m o
f nu
mbe
ror
der
follo
win
g a
path
.T
hech
ildre
n ar
e in
stru
cted
,to
follo
w th
e pa
th w
ritin
gth
e ne
xt la
rger
mm
iem
l.St
art a
t 111
."
ErN
EE
Firs
t gra
ders
can
be
help
edto
lear
n th
at m
any
com
bina
tions
hav
e th
e sa
me
sum
by
the
use
of la
rge
post
ers.
For
exam
ple:
pict
ures
of m
othe
r an
imal
s an
dth
eir
youn
g.
On
the
mot
her
vrite
the
sum
of
all
the
=W
AW
A. O
n ea
ch o
fth
e ba
by f
inin
iaS
the
grou
p ca
nw
rite
a c
ombi
natio
nth
at e
qual
s th
at s
um.
-
,A
ny, 0
,,77
0,1
Chi
ldre
n ca
n m
ake
thes
e ca
nbin
atio
ns w
ith th
e he
lpof
the
teac
her
Ithi
le f
aste
r ch
ildre
n ca
n m
ake
thes
eco
mbi
natio
ns b
y th
emse
lves
.
Tra
nsla
ting
num
ber
prob
lem
s in
to m
ath
sent
ence
sM
ake
a. a
pro
blem
Hav
e th
e pu
pils
wor
k in
gro
ups
and
take
turn
s be
ing
as d
escr
ibed
. bel
ow.
Let
one
pup
il do
the
timin
g.H
ave
him
exp
erim
ent w
ith p
erio
ds a
one
min
ute
to th
ree
min
utes
to d
eter
nine
bow
muc
h tim
esh
ould
be
allo
wed
..(I
t will
be
help
ful i
f yo
u ca
npr
ovid
e a
thre
e m
inut
e eg
g tim
er, w
atch
with
sec
ond
hand
., or
sto
n w
atch
.)T
he p
upil
'Who
is "
it" w
rite
son
the
boar
d. a
n ev
atio
n w
ith n
to h
old
a pl
ace
for
the
answ
er.
The
n he
poi
nts
to s
omeo
ne in
the
grou
pan
d. a
sks
him
to m
a,ke
up
a ve
rbal
pro
blem
to f
it th
eeq
uatio
n.If
the
chos
en p
upil
can
stat
e a
prob
lem
and.
giv
e th
e an
swer
bef
ore
the
time
is u
p, h
e be
com
esni
t."If
he
cann
ot th
ink
of a
pro
blem
, the
pup
ilw
ho is
"itn
ask
s so
meo
ne e
lse
to m
ake
a pr
oble
m.
Eve
ryon
e in
the
grou
p sh
ould
ver
ify
the
answ
ers.
Supe
x-vi
se th
e ac
tivity
to m
ake
sure
that
the
pupi
lsdo
not
go
beyo
nd th
e ra
nge
of p
roce
sses
stu
died
thus
far
.Pe
rmit
disc
ussi
on o
f al
l dis
pute
dpr
Obl
ems.
Cha
rade
s
1.on
e te
lls s
tory
with
a p
robl
em in
it2.
one
dem
onst
rate
s w
ith p
hysi
cal O
bjec
ts3.
a te
am s
olve
s th
e pr
oble
m
Tra
nsla
ting
mun
ber
prob
lem
s In
to m
athe
mat
ical
Setti
ng m
, add
ition
equ
atio
nsse
nten
ces
(ora
l and
'wri
tten)
11fl
Mai
m c
ards
with
num
eral
s on
them
. Hav
e st
uden
ts p
ick
out m
iner
als
and
oper
atio
nal s
ymbo
ls, t
o m
ake
prob
lem
s.H
ave
anat
her
stud
ent s
olve
the
prob
lem
.
-
+1.
3 +
32
+6
2 +
3 6
+ 0
3 +
26
+2
3 +
+ 2
+ 1
4 +
4 6
+3.
3 +
4+
5 +
04
+2
5 +
12
++
62
+5
5 +
32
+1
+5
4 +
3
The
teac
her
s1c-
-14
give
the
pupi
ls a
n op
port
unity
to c
ompo
se th
eir
own
prob
lem
s us
ing
the
basi
cop
erat
iona
l com
bina
tions
bei
ng ta
ught
so th
ey c
anle
arnt
he c
oMbi
natio
ns.
The
teac
her
also
ass
ists
the
pupi
lsin
rea
ding
and
.un
ders
tand
ing
the
prob
lem
s in
thei
r te
xt. M
e he
lps
them
to d
eter
min
e th
e qu
estio
ns a
sked
.en
d. to
obs
erve
the
mat
hem
atic
al te
rns
that
are
use
d.
Hav
e ch
ildre
n dr
emat
ize
num
ber
com
bina
tions
(ad
ditio
nor
sub
trac
tion)
thro
ugh
10.
Mak
e tw
o se
ts o
f ca
rds
with
num
eral
s 0-
9.Pa
ss o
ut th
e ca
rds,
one
to a
child
..O
ne c
hild
sta
nds
in f
ront
of
the
grou
pan
dsa
ys I
am
2.
Who
can
hel
p m
ake
me
6?T
he c
hil.d
with
num
ber
11. c
anes
up
and
he in
turn
say
s,"I
am
IL.
Who
can
hel
p no
be
8V1
If th
e te
ache
r w
ishe
s on
e ch
ild c
anbe
cal
led
tow
rite
the
stor
y in
equ
atio
n fo
im o
nth
e ch
alkb
oard
.
The
mils
raw
enj
oy p
layi
ng a
bal
l gam
e.D
raw
api
ctur
e of
a b
all d
iam
ond
onth
e bo
ard
and
wri
te a
num
eral
for
eac
h ba
se (
5,6,
1 en
d8)
.
To
scor
e, a
pup
il m
ust g
ive
anot
her
nam
efo
r ea
chnu
mbe
r th
at is
nam
ed o
n th
e ba
ses.
wri
te th
e fo
llow
ing
num
ber
=es
on
the
boar
d an
d ha
ve th
e pu
pils
sele
ctan
anr
opri
ate
one
for
each
num
ber
nam
e sh
own
on th
eba
il di
amon
d. D
ivid
e in
to tw
o te
ams
and
have
the
mils
kee
p sc
ore.
-
dram
atiz
e.T
he p
upils
may
als
o U
M: o
ojec
usin
the
clas
sroo
m o
r cu
tout
sat
the
flan
nel
boar
d. to
dem
onat
rate
the
actio
n in
a p
robl
em.
Sugg
este
d. p
robl
ems:
Thr
ee b
oys
wer
e pl
ayin
g m
arbl
es.
Mre
e bo
ys jo
ined
.th
em.
The
n ho
w m
any
boys
wer
e th
ere?
Soon
thre
e of
the
six
boys
join
ed.
anot
her
gam
. How
mm
y bo
ys a
re n
owpl
ayin
g m
arbl
es?
An
45 .
E3
=
-
Mat
chin
g m
ath
sent
ence
sw
ith p
ictu
res
8. C.11
1110
1IM
MIK
IIII
,
3
r.
7 0
3
.111
1111
11M
0111
Mil
It
EI
1
81 a
m".9
dfts
I1A
116G
ame
"Wha
t's M
y R
ulo?
"
The
sec
ret r
ule
pres
ente
dhe
re is
that
17:5
8th
e su
m o
f ea
ch p
air
of n
umbe
rs s
ugge
sts
the
thir
d nu
mbe
r.
Hav
e th
e cl
ass
look
at
line
"A".
Whe
t is
the
num
ber
of b
lue
bars
?W
hat i
s th
e :u
mbe
r of
blu
edo
ts?
Wha
t is
the
num
ber
ofbl
ue ta
3.1y
mar
ks?
In li
ne"B
"the
sec
retrule "1" and "3" suggests
"4.77EF 1:" and "5" suggest "7,"
and
num
ber
"1"
and.
13"
sug
gest
"4"
on li
new
hat n
umbe
r do
"3"
and.
"7"
sugg
est?
Pupi
ls s
houl
d. s
eeif
they
can
app
ly th
e ru
leto
all e
xam
ples
.
E a
nd. G
ham
a d
iffe
rent
rul
e.W
hat i
s th
eir
rule
?
Allow students the opportunityto make up their own
rule
s an
d. tr
y th
em o
ut o
nea
ch o
ther
,
-
Anything reasonable
should be considered
usable.
For example,
after the game hasbeen played a
number of times,
the rule might beto coMbine two
numbers so tbat
"1" and "3" would suggest
"13" and
"3"
and "7" would suggest
"37".
Or perhaps
"12"
and "49" might
suggest "1492" aml
"28" and. "16"
might suggest
"2168."
The rule in each case
should
be clear to everyone.
Children need. notvez4balize theirunderstanding of
the rules.
However, they-can eameismtheir under-
standing by offering auexample of their ownto
show that they have
found the
"secret."
Put a set of twofelt cutout airplanes onthe flannel
board.
Have the number ofthe set identified.
Tell
the children to
close their eyes.
Remove one of the
airplanes.
Llrect the childrento opaa their eyes
and tell how manyairplanes flew away.
Have them
tell hourmanyairplanes they still see.
Have some-
one tellthe set story andgive the relatednumber
sentence.
This procedure can
also be used. in the
introduction of othercombinations.
-
r'''''
GE
OM
ET
RY
Gra
de1
-
,,,17
7777
7777
r,M
777
Grade 1
Concepts from Kindergarten
Characteristics of circle, square,
rectangle and triangle through
measuring.
AeagefeeLebIlle
models
AA
Amatching uhite shapes
for student to color
red, green, and blue.
GEOMETRY
Don't assume tbat pupils have anybackground in
geometry.
Due to the lack of a basic textand
a variety of content
offerings in kindergarten,
first grade teachers would be wise to
start with
concepts introduced at
kindergarten level.
Cut shapes from pieces of coloredconstruction
18:
paper.
Cut matching shapes from white paper.
Ask students to color the white
shapes to match
the color of given shapes of the same size.
Give students at least 3 different
sized circles,
squares, etc., so they can see
the similarities
of circles.
Ask, "How many sides has a rectangle?
What
can we say aboutthe lengths of the sides of
a rectangle?"
(Opposite sides have same length).
Measure using crayon lengths, widths
of finger,
etc.
"How many corners does a rectanglehave?"
Is a square a rectangle?
(Yes, a square is a
special rectangle.) What is special or
different about the square?" (four sides are
equal in length)
11:
17,29,
48,68,
70,80,
94,111,
124,128
8,13,51,
80.98-99,
152
-
r,'V
r.
-7}T
WA
,kr.
eA
-1.1
,W77
.177
71,7
"How many sides has a triangle?
Is a triangle
a rectangle?
Use what is known about the
re.-tangle
to test the triangle."
(4 corners? 4 sides? opposite
sides the same length?)
"What do circles locik like?
Do they have corners?
Straight edges? What is the longestwalk across
the inside of a circle?"
(along a diameter, but
don't use the term now)
Demonstrate the diameter
concept by having students joinhands and form a
circle.
Adk a student to show the longeststraight
path across the inside.
Prove by measuinik various
paths with a string.
Have students sit in a
rectangular pattern, a square pattern, a
triangular
pattern.
"How many children make up a
side of a
tectangle if the entire class
forms the pattern?
(Varies)
The square?" (Class members .:-4)
When children have learned to recognize
geometric
shapes, they will enjoy going throughold magazines
and newspapers looking forpictures illustrating
these shapes.
They will be surprised to find so
many.
Have the children cut outpictures of
these shapes and paste them on large
charts.
The
individual charts may show objectsof one geometric
shape or may be a combination ofshapes.
Let the children make geometricshapes from colored
paper.
Tell the children to create
animals from
the shapes.
The animals may be made fromtriangles,
circles, squares or triangles.
Have them describe
these animals to the class.
What shapes did they
use to create
the animals?
.49-
-
Spatial relationships
between
Objects (distances)
t3e
0,4
Name two points inthe classroom.
Ask students
to walk from one
point to another.
Discuss the
various paths taken.
Ask, "What is theshortest
path between the twopoints?" (line
segment)
Use models ofrectangles on floor orhave students
to talk the
shortest path fromeach side to
its opposite side.
"What two figures are
formed
if you walk the
shortest path between
opposite
corners?" (triangles)
"What can we sayabout
the lengths of theopposite sides of a
rectangle?"
(equal)
Try the same activitywith squares.
Ask students
to generalize
about the lengths
of the four sides
of any square.
Let children decide
how many
students should sit ineach line to form a
square.
(Separate class members intofour sets.)
Develop vocabularyand awareness.
Draw line on
chalkboard.
Draw triangles, squares,
circles and
rectangles above, on,and below the line.
Label
19:5-8
each figure with a
capital letter.
Adk, "Is
triangle A above, below, or onthe line?
How
many circles arebelow the line?" etc.
-SO
.
-
Spatial relationships -interior
and exterior points.
Draw a vertical line onthe chalkboard and place
points to the left, on,
and to the right of the
line.
Label the points using the
capital letters
of students' first names.
Ask, "Is point P to
the left, on, or tothe right of the line?
Name
two points to the
right of the line.
Is point
W on the line?"
On a table, place twoobjects about 15 inches
6
apart.
Have the pupils take
turns in arranging
a string to
show different ways to get
from one
object to the next.
Develop the idea thatthe
shortest distance is shownby holding the string
tight.
Go back to the drawingof the houses.
Develop the idea that
the shortest distance or
path between houses ismeasured along a straight
chalk mark.
Tell the pupils thatsuch a mark is a pictureof
a line or segment or
path.
Draw pairs of intersecting
triangles, squares,
circles, and rectangles.
Draw stars or points
inside the figures.
Ask, "How many stars are
inside the square?
(five).
How many stars are
inside the square ontle
(three)
A variety of
-Sl
-
-
inside square
inside square orgy
inside circle
inside circle only
inside square and circle
outside square
outside circle
outside square and circle
Comparing shapes and sizes of Objects.
shapes and Objects could be used.
Assign letters
to points and*ask the children to name the points
outside the square,Anside the circle, etc.
Draw-two intersecting lines as shown in the example.
Draw geometric figures, numbers, or letters in the
quadrants.
Have students ask questions of each
other.
1.
How many circles are to the left of the line?
2.
Name the figures above the line.
3.
Name the figures above and to the left of
the lines.
4.
Point to a circlit to the right and
below the.lines.
Showr pairs of objects alike in color and general
shape but different in size. (books, balls,
pencils, containers)
"How can we tell the objects
of each* pair apart?" (size)
Show two and then
three circles of different size.
"How can we
-
tell the two circles apart?
Three circles?"
Develop vocabularyof larger than, less than,
largest, smallest,
and in-between.
Show squares,
rectangles and triangles inthe same way.
Place the name of a shape, (triangle) onthe
Chalkboard.
Ask the children "Who wants to
draw a tria4le on the chalkboard?
Another
triange? A lova thin triangle?
A small
triangle? A trianglewith a very sharp point?"
Continue tids activity with other shapes.
Then
tell the children they are a jury.
Their Job is
to study the shapes - do they all belong?
Are
all the squares really squares?
Are they triangles?
Discuss sLapes that doet belong.
"Where does
it belong?"
Place rows of circles of various sizes
and colors
along the chalk tray.
Rave students duplicate
the patterns displayed.
Encourage discussion
and use of vocabulary terms.
1.
Arrange figures fromsmallest to largest
size (left to right).
2.
Arrange from largest to
smallest size,
3.
Vary positions.
-53-
-
Symmetry
4111
111.
MW
elm
Place a pattern of circles, squares, rectangles
and triangles on the chalk try.
Have individual
students match the pattern or allow groups to
build a pattern together.
TO vary this activity, have a student arrange a
model pattern on the chalk tray.
Then let class
members see it briefly, cover the model and aik
the class to reconstruct the pattern by themselves
from memory.
Provide students with the fbllowing shapes:
circle, rectangle, square, and an isosceles
(two sides equal) and equilateral (three sides
equal) triangle.
Importantmake a17. copies of
one shape of the same color, ie
Li eircies red,
all squares blue, etc.
Or, mae all shapes of white
paper.
This reduces color confusion and allaws
student to dwell on the symmetry of the shapes.
Students then fold shapes in half.
Special help
and discussion should develop out of the folding
of the triangle shape.
Students cut along the
folding line and paste the halves on sheets of
colored paper.
"Are the two halves alike?
Are they exactly alike
(congruent)?
How can we prove that they are?"
(By placing one over the other, by mapping all
points of one half on the second half.)
-
Have students fold A" x 11" sheets of typing
paper in half.
Let them experiment, cutting
various shapes, being sure to leave segments
along the folding line intact.
Paste the
designs on colored construction paper.
Discuss how familiar shapes such as hearts or
circles can be cut out by thinking about the
picture of one half of a heart or one half of
a circle.
Let them experiment to find out how
a circle shape mustbe cut on the fold.
Compare
shapes by placing th ?.. side by side on the
chalk
tray.
-
AIM
Recognition ofthree-dimensional
objectssphere and cUbe.
right circular
cylinder
right rectangular
prism
Collect a set ofthree-dimensional objects
such as marbles, balls,boxes, dice, sugar
6:117-125
cubes, ater tumblers, tin cans,
and mailing
tubes.
Give the children theobjects to feel and
inspect.
Encourage them to see
edges, (line
segments), faces (surfaces), and corners
(right angle7)87
Lead.them to discover specific
facts about each
object, e.g., hold up a-can.
"Does it have
.corners? (No)
Does it have edges?
(Yes, Have
aligarTs run fingers aroung the two
circular
rims.)
Does it have faces?"
(Yes, three)
To develop vocabularand concepts sequentially,
it is wise to beginwith the cube and right
rectangular prism (box)before introducing the
cylinder.
There is sometimes
confusion over
the edge-face distinction.
Draw a funny face
on the
face.of a box and point out
that we could
-.56-
-
side view of cube
not draw the same picture on an edge.
TO show the face of a cylinder, cover the
surface of a tim can with a rectangular piece
of paper and then remove the can.
Open the
resulting paper cylinder to show the rectangle
shape.
Ask students to hold objects behind their
backs while describing the object to the class
in terms of edges, faces, shapes, and corners.
The class member Who guesses the object correctly
then becomes the leader.
-Hold objects so that the children see the two-
dimensional shapes found in them, i.e., the
circle and rectangle shapes in the cylinder,
the square shape in the cube, and the rectangle
shapes in the right rectangular prism.
Darken the room and use a light source such as
a film projector to projectthe shadows of three-
dimensional objects on the wall or screen.
See
if students can guess the object by studying its
shadow projections, e.g., a cUbe, a rectangular
prism, and a cylinder may all projeCt a square,
liemme a second projection would be necessary to
distinguish between these objects.
-57-
-
ri
- -S" -.s' -
-
,
GR
AD
E 1
ME
ASU
RM
EN
T
Mea
sure
nnnt
non
-sta
ndar
d un
its to
stan
d.ar
d un
its.
Mas
a:te
m...
a* r
elat
ed to
dis
tanc
ear
ound
a c
ircl
e.
Lea
d th
e ch
ildre
n to
see
wha
t is
invo
lved
, in
choo
sing
a s
tand
ard
unit
of m
easu
re, r
athe
rth
an te
achi
ng r
ate
use
of th
e ru
ler
or y
ard-
stic
k.13
:334
1
Ask
the
child
ren
if th
ey a
re b
igge
rth
an a
soda
pop
bot
tle.
Hav
e th
em e
stim
ate
thei
rhe
ight
in"b
attle
" un
its.
Stan
d on
e or
two
child
ren
agai
nst a
ver
bica
l pie
ceof
add
ing
mac
hine
tape
.R
ecor
d an
d co
mpa
re th
eir
heig
ht in
bat
tles
toth
eir
estim
ate.
Onc
ea
canp
aris
on is
mad
e be
twee
n on
e or
two
child
ren
and
the
bottl
e, a
skth
e re
st o
f th
ecl
ass
to r
evis
e th
eir
estim
ates
.T
hen
the
child
ren,
wri
ting
inte
ams
of 5
or
6, c
anca
rry
out t
he e
zper
imen
t the
mse
lves
o
Mak
e fr
eque
nb u
se o
f te
rms
rela
ted
to h
eigh
t.
Ann
is s
hort
, Sal
ly is
sho
rter
and.
Rod
issh
orte
st.
(Tal
l, ta
ller,
talle
st,
etc.
)5:
50-5
1
Dra
w tw
o co
ncen
tric
cir
cles
in c
halk
on
the
play
-gr
ound
.R
ave
a pa
ir o
f ch
ildre
n, h
oldi
ng h
and.
s,go
aro
und
bath
cir
cles
whi
ln y
ou a
sk q
uest
ions
abou
t till
: dis
tanc
e co
vere
d by
each
chi
ld.
(One
child
is w
alki
ng o
n th
e in
ner
circ
les
whi
le o
neis
wal
king
on
the
oute
rci
rcle
.)A
zI a
ttem
ptsh
ould
then
be
mad
e at
mea
sure
mnt
.T
he u
nit o
fm
easu
re c
ould
be
afo
ot,
stri
p of
tag-
boar
d, e
tc.
'tat e
ach
chil4
la a
sm
all g
roup
use
his
ova
stan
dard
so
that
cca
ftsi
onw
ill a
rise
and
.th
e ne
ed f
or a
sin
gle
stan
dard
will
bec
ame
obvi
ous.
58
44,6
'
-
Measurement using non-standard units to lead to
standard units.
Des
k to
pis
thre
e st
icks
wid
e.
Rounding off (approximation)
,59
.1"
e 01
yr
,17,
1 r
'I.r'i
r II
IRO
VIV
,V7L
kilir
lii.q
i
Show several unmarked. sticks of various lengths.
Have the children, working together in pairs,
mea
sure
the
leng
th o
f th
eir
desk
., th
e ch
alk
tray
,th
e fl
oor,
etc
.T
hen
com
pare
thei
r re
sults
in"s
tick
units
."D
ecid
e w
hich
stic
k le
ngth
is th
emost practical for each situation at hand.
9;449-450
It is important that units of measurement be
appropriate.
(We
don'
t use
inch
esto find the
dist
ance
fro
m L
.A4
to S
an F
ranc
isco
or
try
tow
eigh
gol
d. b
ars
on o
ur b
athr
oom
sca
le.)
The children
will
mea
sure
something in stick units
which will not fit an even number of times but will
have some "left over."
The
yshould estimate the
amount of the stick unit left over and round it
to th
e ne
ares
t who
le u
nit.
6:12
2012
6-12
8
Distribute several unmarked. foot rulers (wood,
tagboard, chiPboard).
Compare their lengths
to th
e le
ngth
s of
a y
ards
tick.
Dec
ide
whi
ch, 1
7:70
-71
the
yard
stic
k or
the
foot
rul
er, y
ou w
ould
.us
e to
nie
asu.
re th
e ch
alk
boar
d. a
nd. w
hich
you
wou
ld, u
se to
mea
sure
a p
iece
of
pape
r.
Put m
aski
ng ta
pe s
trip
s of
aff
eren
t len
gths
on
the floor and walls in vaxtous parts of the class-
room.
Use
leng
ths
rang
ing
betw
een
1 ft
. and
8 f
t.Have the children
mea
sure
them
'with
unmarked foot
rule
rs, i
n te
ams,
then
record.
and
=T
are
thei
rresults.
Ask
one
chi
l.d to
mea
sure
a 1
0 in
ch s
trip
usi
ng h
isun
mar
ked.
foo
t rul
er, r
ound
it to
the
near
est f
oot,
and.
rec
ord.
it a
s I
foot
.A
sk a
noth
er c
hild
tom
easu
re a
3 in
ch s
trip
with
his
rul
er.
Rou
nded
toth
e ne
ares
t foo
t, it
wou
lc.be
cal
led
zero
fee
t.Is
-
Ilen
e&
12
it re
ally
zer
o? M
iss
hope
fUlly
, will
lead
. Int
oth
e id
ea o
f us
ing
wai
ler
units
.C
hild
ren
can
man
ipul
ate
cme-
inch
-cub
e bl
ocks
todi
scov
er h
ow M
any
will
fit
an a
n un
mar
ked
foot
-ru
ler.
We
can
now
say
that
one
foa
t 'un
it of
17:9
7,m
easu
re is
evi
vale
nt to
12
one-
inch
uni
ts o
f12
8m
easu
re. N
ow th
e cl
ass
may
be
at a
poi
ntw
here
they
can
use
sta
ndar
d ru
lers
with
inch
or h
alf-
inC
h m
arki
ngs.
Tea
cher
s sh
ould
. rem
embe
r th
at w
hen
we
add
ve d
ono
t add
. Inc
hes,
any
=re
than
we
add
appl
es; a
llw
e ad
d ar
e nu
mbe
rs.
If w
e ha
ve 1
1 in
ches
of
stri
ngan
d. 2
inch
es o
f E
rtri
ng, w
e ha
m 6
inch
es o
f st
ring
alto
geth
er--
caly
bec
ause
:
+ 2
is 6
Cal large togOoari draw tatangliss
suutrilitherals.
Mans tase cbildiem
mea
sure
the
leng
thsce tam line
segm
ents
for
min
g th
e si
des
of th
e M
ures
. Dra
w 1
2:11
0T-
thes
e fi
gure
s, in
Izz
ch la
rger
for
m, o
n th
e pl
ay-
438
grou
nd. w
ith c
halk
and
mea
sure
with
a y
ards
tick.
._Se
lect
gro
ups
to w
ork
toge
ther
mea
suri
ng o
bjec
ts6:
129-
the
teac
her
has
alre
ady
mea
sure
d..
Nav
e th
en13
0re
cord
. the
res
ults
and
che
ck w
ith th
e te
ache
rto
see
if th
ey a
re r
easo
nab
corr
ect.
Res
szbe
r 11
-60-
that
mea
ly f
irst
grs
iie.c
k tb
ie61
,98
dext
erity
and
coo
rdin
atio
n ne
eded
to b
e pr
ecis
ein
thei
r m
easu
rem
ents
and
sho
uld
not '
be e
xpec
ted.
to b
e ac
cura
te.
-
The
num
ber
line
as a
mea
suri
ng d
evic
e.
01
2ie
IMO
61
Chi
ldre
n w
ho h
ave
not b
een
intr
oduc
edto
the
num
ber
line
in k
inde
rgar
ten
shou
ld s
ee a
flo
orm
odel
fir
stan
d. h
ave
the
expe
rien
ceof
hop
ping
in e
qual
hops
.L
ater
ther
e sh
ould
.be
a n
umbe
r lin
e on
the
blac
k-bo
ard
(low
eno
ugh
to r
each
) as
wel
l as
indi
vidn
alnu
mbe
r lin
es ta
ped
onea
ch c
hild
's d
esk.
Num
ber
lines
can
be
foun
d. s
ever
al p
lace
sin
the
clas
sroo
m. M
e cl
ock
onth
e w
alk,
whi
ch m
easu
res
time;
the
yard
stic
kan
d. th
e fo
ot r
uler
whi
chm
easu
re d
ista
nce;
the
cale
ndar
(in
num
ber
line
"chu
nks"
) w
hich
mea
sure
sda
ys, w
eeks
, etc
.
Put a
ver
tical
num
ber
line
on a
wal
l so
that
child
ren
can
mea
sure
thei
r he
ight
with
apa
rtne
rto
ass
ist.
-
Col
ored
rod
s as
a f
ora
of n
uMbe
rlin
e.
Example:
Cuisenadre Rods
Com
pari
ng le
ngth
s.
Equ
al le
ngth
s(E
quiv
alen
t tra
ins)
R+
G+
W=
G+
G=
D
Use
of
the
colo
red.
rod
s is
a c
once
ptua
l18
;24,
approadh to math through algebrarather
than the traditional approada through
59,
counting.
The rods are a model of the
81-8
2,ra
tiona
l num
ber
syst
em.
The
y pr
ovid
e a
84-8
5,concrete model of abstraat numbers and.
87-89,
of r
elat
ions
exi
stin
g am
ong
thes
e nu
mbe
rs.92-93,
The
re a
re 1
0 co
lors
end
eac
h co
lore
d. r
od 9
7is
ref
erre
d. to
by
its c
olor
.T
he r
ed.
99-1
00ro
d. is
wri
tten
"r:"
the
whi
te"I
V e
tc.
Aft
er w
orki
ng w
ith th
ese
rods
the
15:22-23
child
ren
cane
to r
ealiz
e in
tuiti
vely
that
there is a standard unit of measurement
invo
lved
.: an
d th
at it
can
be
veri
fied
..If
the
leng
ths
of tw
o "t
rain
s"(a
term
used
.fo
r jo
lukE
the
colo
red.
rod
.s to
geth
er)
are
the sane, we say the trains
are
equivalent.
The sign
is
used
. bet
wee
ntwo expressions
representing trains to
indi
cate
that
the
lengths of the train are equal--that is,
that
eac
h tr
ain
has
the
sam
e le
ngth
.
We
tan
wri
te "
R +
G+
W =
G+
S=D
" to
indi
cate
that
the
trai
n m
ade
with
a r
ed r
od: a
gre
en r
od:
end
a w
hite
rod
is e
quiv
alen
t in
leng
th to
atr
ain
mad
e w
ith tw
o lig
ht g
reen
rod
s or
one
dar
k gr
een
rod.
"ft.+
-
4
i,,,,'
,. .7
..W,,"
0.,,,
"77'
'''. ''
''''''''
''", c
..-",
".r
'''' '"
.'"'"
''''-"
'-'7.
"'''''
'''' ''
''-' I
. '''"
'''''''W
"'''''
''''''
Ineq
ualit
ies
with
rod
sW
hen
any
rod
is c
ompa
red
to a
sec
ond
rod,
one
of
the
follo
win
g si
tuat
ions
mis
t occ
ur:
1 3
Tim
e (t
he *
cloc
k)
The
fir
st tr
ain
is lo
nger
ant
we
wri
te
A >
B
The
fir
st tr
ain
is s
hort
er a
m/ w
e w
rite
A 87
-
7117
2.t.-
V.S
.70.
7Tt-
,,,--
7,,:f
.,,T
Ci7
.7,3
4,E
r,,it
i,,,,4
1`t,,
,,,,,,
,. T
,c-o
rar,
04"
,,,,w
,,-; .
,:!...
.77,
--no
s,...
7,r,
,,,,.
- yr
,,,e,
7,, ,
" 1,
,i,-
,,,..,
rw
. -..,
r. '
-. '1
, ,-
,m
-mr,
.4.,
ve
,,,,,,
n .r
.,'-,
1,.
Vir
,. ,
.7,s
,"--
.,,
.-,;,
,,,,v
,T,r
;-,m
9kr,
In`p
y 1,
,74p
.r.,,
;*1-
-Y,V
v`V
cAN
MV
.VL
`r1,
Constructing geometric figures
Give students paper marked with equally spaced dots to
represent points.
(Easily mark by placing onion skin
paper over a sheet of 1" or
graph paper and marking
dots over the intersection of the lines.
Trace dots on
master and ditto off.
(Do not give graph paper to students
as they have difficulty thinkia of line
intersection as
Let students experiment with the dotted paper, drawing what-
ever geometric figures they wish using a penciland straight-
edge.
Insist only that all paths drawn biAIERIRALAELE221
through at least two points.
When the class is comfortable
with this activity have them draw specific figures according
to detailed description:
1.
Can you make a triangle with one side?
2.
Can you make a triangle with two sides?
3.
Mhke a three-sided triangle.
4.
Construct many different triangles.
5.
Construct a three-sided square. (impossible)
6.
Construct a four-sided square.
7.
Construct many different sized squares.
8.
Construct many different rectangles.
9.
Construct a.four-sided figure that is not a
square or a rectangle.
(many possibilities)
88
-
Concept aline segment
Concept of line segment
4
Students choose two points on ageo-board and
userubber
bands to show different paths
of points between the two
selected points..
Some possibilities are seenat left.
Ask:
1.
Which is the shortest pathbetween the two
chosen
potn
tsT
2.
Which is the simplest pathbetween the two
points?
Whidh is the straightest pathbetween the
two points?
4.
Ls it a line?
5.
Does the line bave a beginningpoint?
6.
Does the line have an endingpoint?
Generalizations:
The shortest pathbetween:two points
lies along a (straight)
line.
This line including itsend-
points (beginning point plusending point) is
called a
line segment.
Use the term, line segment,
in talking with
the children.
Acceptable
definitions for seven yearolds
inolude:
"A line segment is the
shortest path ftom one
point to another,"
"A line segment Is stopped onboth ends,"
or "A line segment
is a piece of line."
Let children lodk for examplesof line segments in the
elassroom (edges of books,desks, chaiktray, crayonbox,
intersection of front walland ceiling, etc.).
Have them
name the "beginning"and "ending" points
Ethe segments
fbund.
89
-
line segment AB
line segment BA
line segment CF
line segment FC
AB, BA, BC,
Points, lines, and points on a
line
Give sheets of papermarked with dots to
students.
Have
them choose a pair of points
(dots), label them with capital
letters, and connect the
dots using a straightedge.
Ask
"what name can we give a line
segment so that othersknow
which segment we're talkingabout?" Ask them to name a
line
segment in two ways
ie:
line segment AB andline segment
BA.
Have students drawtriangles on the dotted paper,
labeling
their endpoints.
Ask them to name the
line segments in one
triangle.
Help them to see that
there are three line seg-
ments, each having two names:
line segment AB, line seg-
ment BA, line segment
BC, line segment CB,
line segment CA,
and line sIgment AC.
Say "line segmentAB."
Do not write
symbolism AB at this level.
Refer to line segments
verbally.
Have student draw squares,
rectangles, and otherquadri-
laterals (4-sided
figures) on -Ole dotted paper.
Ask them
to label the endpointsof the line segments.
Place figures
on chalkboardPad discuss the eight names
for the four
line segments of each
quadrilateral.
Ask students to think of a place
in space and to puttheir
finger on it.
"Are you touching the same
points?
(no)
How many points are wetouching?
( A number equal to the
number of students in
class)
Now touch two points in
space.
Now ten points.
How many points are we
touching?
(ten times the number of students inclass or just "lots
and lots of points.")
Suppose every student in ourschool
was touching tenpoints.
(Wow!)
90
-
Yarn repTesenting aline
segment and aline
EIN
IMIIM
INO
NO
Mm
Drawing lines throughpoints
Aal possiblUiles on
geo -board.-
Place two students anopposite sides athe room where
they can be easily seenby the class.
Have them holdand
pull taut a longpiece of yarn or rope.
Ask a student
to think of a pointalong the stringand to place his
fin-
ger on it.
Have him toudh twopoints.
Ten points.
Keep
adding students tothe string.
Continue to ask questions:
"How many more points canwe touch?
If we touched the
points with our pencil
points instead offingers, could we
touch more points?
Suppose the pointscontinued beyond the
endpoints of our line
segment.
Where would .the points
end?" Generalization tobe reached
informally:
We can
count the points
in a line segment or a
line forever orhow
many points are
in a line segment or aline?
(more than we
can
count)
This activity can beperformed by askingstudents
to mark points
along the string with
clothes pins, opened
paper clips orsafety pins.
Give students dittoed
sheets of lines shownin various
lengths and positions.
What do the arrowheadstell us?
(The line goes onforever in both
directions.)
Tell them
to label any twopoints A and B.
Mark with dots as many
points as you can online segment AB.
"How many points
are contained
In line AB?"
(more than we cancount)
Sharpen student pencilsto fine points
and repeat exercise
with other lines.
Use a 9 point
geo-board and:rubberbands to show lines pass-
ing through a point..
"How many lines can passthrough a
point on thisboard?"
(four)
Give the student
dittoed
sheets of dotted paper.
Select a given pointand using a
straightedge draw linesthrough the point.
Remind them to
include arroWheads.
"How many lines.can. youdraw?"
(many-
linswers vary)
91
-
Diameter concept
(without use of term)
Place a dot on thechalkboard and draw oneline through
tt using a ruler.
Invite children todraw more lines
through point C.
Ask "How many lines canbe drawn through
a point?"
(more than we cancount)
Ask students to mark twopoints on a paper.
(Hhke the
dots Ilia.) Label thepoints S and T.
Using a straight-
edge,-diiw a line through
S and T.
Ask--"Can you draw
a different
(or another) line throughS and T?"
(no)
).......-
Lead children to seethat there is only oneline through two
two points--Demonstrateby-using two students
for points
and a piece of yarn or roperepresenting the line.
If the
size of the dot ormark used is too large,
confusion may
result.
If necessary, use aneedle and black threadand
two sheets ofwhite paper to illustrate, i.e.
the line
represented by the threadand the pointsrepresented by
he tiny needle holes
in the two sheets
of paper.
Provide three dimensionalobjects which represent
circles
such as bicycle tires,
hula hoops, embroideryhoops, 3
gallon ice creamcontainers
(cylindrical), etc.
Have stu-
dents measure the widthof the circle shape
at its widest
point usingnon-standard units such aspencil lengths,
string lengths,
etc.
"Does the widest path across a
circ
lealways fess
through the center of the
circle?"
(yes)
Prove visually with ropeor easily seen,
brightly
colored yarn.
92
-
Extensions of line
r)egments and
reflections of curvesaad figures
.1
given line segment
0411
1P. 1
1.11
1011
. 01.
.100
.
studiant
extensitin
Give studentsdittoed Sheetscontaining circles
of various
sizes with their centers
marked.
Ask themix) drawseveral
lines across a
circle that passthrough the centerpoint.
Lab
elwith
capi
tal l
ette
rsthe points wherethe lines
intersect the circle.
Have.them measure the
line segmemt
AB by placing a
shee
t of
pape
ralong AB and markingits
length.
Place
the
pape
rruler along CEL
"Are line seg-
mea
tAB and, line
segm
ent C
Dthe same length?"
Measure line
segment EF.
"Is
line
segmemt EP the samelength as the
others?"
Continue exercisewith circles ofvarious sizes.
Generalization to be
reached:
The line segments
of a circle
that pass throughthe.center of thecircle and whose
end-
points are points
ofthe circle are
equal in length.
Let
students verbalizethis concept intheir own way andin
words which have
mea
ning
for them.
Give studentssheets of dittoed paperwith line segments
of various lengths
drawn on them.
Ask student to
extend
eaCh line segment
i.e. "Make eachline segment twice as
long.
Use your penciland straightedge.
Draw small arrows
alongside each linesegment so that
student will know
WhiCh direction to
extend the segment.
Be sure to arrange
the segments toallow students todraw them without
causing the "new"line segments totouch each other.
Work
sequ
entia
lly:
stud
entshould extend
a lin
esegment of
one unitlength first, thentwo units,
then three units,
etc.
Extensions shouldbe made right toleft as well as
left to right.
If students havetrouble, encouragethem
to
coun
t the
spaces betweenthe dots
of th
e' g
iven
line
segment. A givenline segment withfour spaces between
its.
endpoints can beextended by drawinganother line
segm
ent
with four spacesbetween
its e
ndpo
ints
.E
xerc
ise
may
be
prec
eded
by
dem
onst
ratio
non
cha
lkbo
ard
orov
erhe
ad p
ro-
ject
or.
Also, a large
dem
onst
ratio
nsize geo-boardwith
100 points maybe useful
93
-
Extension of linesegments and
reflections of curves
and figures
Give students dittoedsheets showing pictures
alon
gsid
ea
dotte
d lin
e w
hich
divi
des
the
shee
t of
pape
rin half.
Ex-
plai
n th
at h
alf
ofea
ch p
ictu
re is
mis
sing
."W
hat m
old
the
pict
ures
look
like
if w
e co
uld
see
both
hal
ves
toge
ther
?"Have students tracethe given halves
with crayon, fold the
paper alongthe dotted line andthen rub the
fold
ed p
aper
with a straightedge.
When opened, the givenhalves
and
thei
rmaps willbe seen.
Have students trace overthe copied
halves with a crayon
different in color thanthe first one
used.
Give students anotherditto sheet.
Have them draw
the reflection ofthe figures directlywithout folding the
paper orreferring to the completedpictures of the
firs
texercise.
Give students dittoed
sheets of paper likeexamples at
left.
Allow them to use
small hand mirrors.
First have
them guess what the
reflected image
shou
ld lo
oklike.
Then,
use thehand mirror to testtheir guess.
If necessary,
students should beencouraged to continueusing the mirror
while drawing the
reflection.
Provide opportunity to
reflect in bothleft-to-right andright-to-left directions
about a
vert
ical
line
in a plane.
Keep figures simple
and appropriate
for both perception andhand skills
ofseven-yearolds.
When students aresuccessful with
reflections about a
vertical line, introducereflecting about thehorizontil
line.
Redevelop concept usingthe same steps asthose
outlined in reflectionabout the vertical.
se
OL
EC
TI
94
0?Pa
-
Sirrtlav polygons
square
rectangle
triangle
Use a demonstration size
geo-board and rubber hands
(36 " x
36" plywood board with 100 pointsrepresented by mils)
show how we can construct.similarfigures by dodbling the
lengths of the line segmentsof a given figure.
Develop
sequentially starting with a square
whose side is one unit
in.length.
Ask,--"How can we copy this square sothat:the
sides of our new square willbe twice as long?"
(Double
the length of the linesegment) TO double line segments
encourage students to count
the spaces between the end-
points of a given line segment
and then dodble the number.
Continue using larger and larger squareuntil student sees
the pattern.
Give students dittoed sheetsof paper with points marked
'by dots 11." apart.
On the top left cornershow squares,
rectangles and trianglesof various sizes. Label a vertex
in each figure with acapital letter.
(A vertex is a
point where two sidesintersect.) Adk students to "copy
the figures so the line segments
in your drawings are
twice as long as the linesegments in thetOpleft corner."
To make sure that thei
figures will be properly spaced
have a vertex of each figurethey are to draw already
marked and labeled with the same
letter given tu the
mOdel i.e. point A in the newly
drawn figure shou