AN ANALYSIS OF ERRORS IN LONG DIVISION
ty
Jessie Voigt
A Thesis
submitted to the faculty of the
Department of Education
in partial fulfillment of
the requirements for the degree of
M a s t e r of Arts
in the Graduate College
Univer s i t y of Arizona
19
3 8
2-
Acmmnxmrrg
To the late Florence Oerdlner, critic teacher of the Eastern Illi­
nois State Teachers* College, Charleston, ey appreciation is especially
due because of the splendid example she set as expert teacher of arith­
metic •
I
wish to thank the administrators, Superintendent C, £• Bose,
and Assistant Superintendent George Peak, of the Tucson Public Schools
for permitting me to give the teat in the schools.
I desire also to record ay thanks to Dr. Austin Repp, Associate
Professor of Education, College of Education, University of Arizona,
for helpful and scholarly supervision throughout the conduct of this
research.
1
I t tarsi
TABES ®
x*
XI*
III.
nr.
^
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■ .
/
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xi^THODicncn #****»*»*«****«»*****#»*»#*#***#*»***»»****
1
KTO 1HK DATA uEBS SICCURSD
4
The Bxperl»Mit •••.<
Tbs Test ........ .
us
Chnptar
CCKCTTS
TnSiiTHIJKT CF 5HT DATA .................................
7
Srrorsln Isolated Situations
Srrors in The Total L m g Division Teat ..............
CocporlGon of Combination Errors In
Isolated Situations and in The Total
Long Division Test
.... .
7
IS
CCE1LTOI0M3 ADD HECCUDKBATIC^S ........................
26
B I M I O W A T O ...........................................
34
a w
;
-
, - :
.
11
22
I A
NUMBER Aim P m
GF
KAD3 BY 1## M K O K
IN TRE FUimtiEWAl, PBCGE33E3 IIJVOI.VED IN 130l a u d a i t i M f i a n /an> i n -thh total icrzo a m ................ ......... .
164 PUPILS in 7Zt rUKDAimZTrAL PHCCE33E3 lilm m ) m i s d ^ m simTioiis Aim in see -to ­
tal LCMJ BIVI3XCH TEST ............................
6
9
I C -KUiBBR AJU) PiTH CENT 0? 2RBCS3 ACCCRDUKJ TO'SSC ....
NABS BY 164 Pt5> m IK XEE PQmAL^iTAI. FHOCiiSSSS IRY(3uVED HI ISCLiVTED SITUATIGNS AND
IN TUF. I’CTAL MBS BimiCIl TEST.................... -• 11
II A roaSR CF TYPE ERP.CB3 THOLCH OCCURRED-IN IKS'
TCTAl IOKU DIVISION TEST 0? 164 PDPII3*
1CRK ..............................................
IS
II B 'R«BSR' CP.?¥?22 ERB(1R3 BT W S S S U S S B 06c r am) IN 588 TOTAL LOIKJ DIVISIW USSR
OF 164 PUPILS' %0R% ...............................
ii o m s a m or '
t ype 'msoss AcocRDiso to sse mncn
OOOURRED IN UTS TOTAL
DIVISION TEST
op 164 vurnar nork ..... ....... .........
in a
B
so
m e r a cr ccsaiKA-nm s k r c s s n? i s o a t e d srr-
UATI0H3, III THE TOTAL LCNO. miSlCW TS3T, F
AND IK BOTH ...... .
III
If
ITUiaZR OF COSINfiSZOR ERRORS BT GS/mS3 II? 1 5 0
USED B im r iC K S , IN TEE TOTAL LGB BIVISI(3J TEST, AND IN BOTH ........................ ..
88
24
in c m m m qp '.ccnbzhatiqn errors aocordinc, to sex
TM I3OTATED 3 i m T I C » , IN IBS TOTAL LOKO
DIVISION TEST, AND IN EOTK ....................................................
ill
25
CHATTERI.
iwmmvzTtm
It la said that teachcra of arithmetic find that long division is
a aubjeot most difflcult for pupil mastery.
Such should not be the oit-
uation if the teacher does her part by presenting the subject in such a
nay that errors will be anticipated and thereby avoided, and if the
pupil’s mental equipment is adequate.
Monroe says,
.
"The purpose of instruction in arithmetic is to eneKOder
in pupils the mental equipment needed for responding satis­
factorily <to certain types of quantitative situations vchich
they will encounter,in advanced school work and In life out­
side of oohool. This mental equipment ia frequently called,
"ability ln arith3S8tic't. Sometimes the plural, ’'abilities’* v.'
is used to indicate that the equipment is not a unitary thing,
but consists of a large number of elements, many of which ore
independent in the sense that a pupil may acquire certain ones
but not others." 1
In the long division process the equipment needed is indeed not
a unitary thing.
Hot only must the child possess u m b e r concept and
on arithmetic vocabulary, which in themselves require long and tedious
preparation forming en important and necessary background for later work
but he also must-know the facts of the four fundamental operations, the
steps of long division in orderly sequence, the proper procedure, the
tiao of zero in all situations in the quotient, and finally, he must
knew when the total process is complete emdcorrect.
1.
Monroe, Salter 3. "The Teacher’s Responsibility For Devising Exerolses In Arithmetic.” University of Illinois Bulletin. V o l m e
28, Number 31, (June, 1926).
ssnt numbers of articles or sizes of groups.
ber as 4, aeans 1, 1, 1, 1, or four
also
For example, stick a-num­
ones; and 40 raeans four tens and
toxtf ones. It la evident that a child rust develop the ability
In a m b e r concept to know at a glance just uhat every figure represents
according to Its place value, and to be able to compare umbers as to
magnitude.
That Is, every child ahould be able to knew at a glance or
at the oral sound of tho figures 4 and 8 that eight is larger and four
is crsaller.
Also In number ecmeept falls the ability to .distinguish
between adding, subtracting, multiplying, and dividing.
This trill nerve
ae a fen oznnples to cake plain uhat la Ecant .by number concept.
By m
arithcotic vocabulary is meant tho corroct use of arithmetic
tsortie with tbolr underlying noanli^s clearly understood.
A child uses
such tsords as '♦times*, ’♦plus’*, "ndd**, "quotient•*, and m a y others in
uorklng long diviaton exocplos.
The facts of the four fundamental operetioes tdilch ho must know inW # W e the familiar one hundred addition eomblnationa, tho one" hundred
subtraction facts, the one hundred multiplication facts, and the ninety
division facts.
The steps of long division which he must know in orderly sequence
are divide, multiply, compare, subtract, compare, and bring down.
The knowledge of the proper procedure in long division includes
the ability to perform correctly the steps of long division In orderly
sequence, the proper placement of quotient figures, and knowing when
the example is correct and oreplete.
3
Possessing ell of the chore equipment and relotod okllls end con- cepto, a child should he able to carry to successful completion the
long division process.
Still, the fact remains that many children do
not solve long division problems successfully; therefore,, an analysis
of pupils* errors in long division m y prove to be a subject of fruit­
ful study.
11th thin thought in mind, the present study xras undertaken.
The purpose of this study Is to find the loci of m a t errors is
long division so that teachers may better knom rhore to place more teach­
ing emphasis and drill,
furthermore. It Is tho purpose of this study
to discover mhother there is a close relationship betreen errors made
In fundamental processes In Isolation end those made in the total long
division process.
The writer also hopes to learn whether the frequency
of error differs greatly botwoon the sexes, and finally, to note whether
errors have a tendency to decrease la frequency as pupils advance from
lower to higher grades.
CHAPTER II
m ; THE DATA sins SHCORSO
The HxpcriEont
Thie study In lens division t?as cade possible by the coopomtlcm
of the administrators of the public schools of Tucson',- Arizona.
was oonduotod in four bleneatary schools in the city.
It
In School I, a
4A group was tested; in School II, a 5A group was tested; In School H I ,
a 6A group was tested, end in School 17, 4A, 8a , and 6A groups were
tested, naking a total of six groups tested in four schools.
A teat was raide and adninistored to 1 # pupils.
The teat was given
May 10th, n t h , 12th, 15th, and 14th, two rooks hafore the close of the
school year, lt36-193?.
Tba ond of the semester was chosen as the best
tiiao to administer the tests because the fourth grade had by that time
severed all the work in long division.
laeluded in the pupil count rare
a total <^i
55
■ ; ' . •
4A
## __________ sa.
53 — -- —
- 6A
The test reauifed two sittings of thirty minutes each, given on
two consecutive days, twenty-four hours apart.
All pupils took the test
and all papers were analysed, except throe, which wore not completed be­
cause of absence of three pupils at one sitting.
Pupils were not told
to hurry, but wore encouraged to do their best work.
The regular
5
clasaroost teechcr was present at the tlse the test m s adKinlotereS
hut did not give the test.
The writer assunea responsibility of giv­
ing all testa end of scoring nil papers.
The Test
In order to learn what errors pupils rsako and in what elezsnts
pupils lack skill in long division, a sixty-ninute tost m s built.
Tho test was constructed so as to Seelode types of lcng division
;
.
:
.
■■ ,
.
■
■ -
-
'
'
- ;
... _
^
- :
■
-
'
'
exanpleo which would compel the pupils to deal with situations such as
thorn with:
'. • i.
3.
d.
5.
6.
7.
llo reminder
,
Cno ncro In the quotient
Two neroa in the quotient
.
rinnl soro in the quotient
Scattered zeros In the quotient
Trial quotients
'
■
:
■
There wore ten examples in the long division test; the complete test
appears at the end of this study.
Since Interest lay not alone in errors, but in tho cause of errors,
the total long division tost was broken up into its elements.
These
element a wore grouped into several sections which would reveal errors
of a particular type if pupils wore susceptible to making such errors.
To show exactly how the examples ware broken up, follow the illustration:
______1 _
54} 1 3 9 1
The firatquotient figure la 4.
Therefore, the first example seen in
the first quotient figure test Is 139 divided by 24.
Turthermoro, one
of the first multiplication examples seen in tho grouped sections, which
shall be referred to hereafter as the isolated tests, is 4 times 4.
'
,
Another rmltipllceition eauwrole la 4 tines 3.
After nultlplying 4 tines
Is
4 the worker smot carry the 1 anfi.aftd It to the product of 4 tines 3,
Thus the first exanple in the Isolated addition test is 12 plus 1.
The
product of 4 tines 34 Is 136, which lo written under the dividend, 1391.
Consequently, a subtraction example, 139 nlnua 136 results and composes
the first example In the Isolated subtraction test.
In this manner each
of the t en long division examples was broken up; this breaking-up served
as a double chock on addition, subtraction, multiplication, and division
Teachers of nrlthnstle often believe tlmt pupils can solve.eiapaUt
problena in the four fundamental operations such as 18 plus 1, 139 minus
136, and 4 tines 4, with ease and accuracy when they are unable to solve
eoereetly cm example.such as 1391 divided by 34, which involves quite
the sane operations and figures.
Perhaps they are prone to believe this
because the latter example Is longer.
The breaking up of tho ten long
division exampleo was done deliberately to learn whether there was a
close relationship between errors made in fundamental processes in
isolation and those node in tho total long division process.
«
After each long division example waa broken up and grouped in the
isolated sections, there were 17 addition examples, 26 subtraction sxanples, 43 multiplication examples, and 33 division examples.
These
isolated tests m y bo seen in parts 1 and 2 Is the sample tost found at
the end of this study.
(atoms n i
TBKATHERT 07 TEI DATA
Errors In Isolated Situations .
A critical analyois was m d o of errors from a total of 1C4 papers.
The isolated sections of the test yielded thenoolveo readily to objec­
tive analysis.
The long division test, contrarily, required subjective
judywnt in many cases.
Tor instance, in the long division tost, sup­
pose a child Multiplied 4 tines 4, had 2 to carry and wrote 19 as the
result.
It was not evident from analyzing such work whether the error
were a mistake in multiplication, one in addition, or one caused from
'
’
.
'
a mental lapse In which the pupil forgot to write the correct response
after it had been thong#out.
In caaos of this type, the writer’s
best judgment was used, and it was assumed that errors in judgment
would have a tendency to balance themselves.
The rmribor and per cent of errors made by 164 pupils in the fundatomtal procesaea involved in isolated eituetiona and in the total long
division test arc shown in the fcllotting table, page 8.
be read ns follows:
test in addition.
The table should
There were 17 addition examples in the Isolated
There ware 164 pupils who took the test; therefore,
there were 8788 chances for errors in addition.
Since the table shows
that just 36 errors wore made in addition, then 1.29 is the per cent of
error made by 164 pupils in addition.
The sara method was used In de-
temtalng the per cents in aubti-aetlon. In multiplication, in the first
quotient figure test, end in the total long division test*
TABLKI/v
ran msiBHR Aim per
oeht <f BRitcms made b y 164 pupils hi the
ikvclib> nr iscnvam siyuatiqs
ram total Loirs c m sio r test
Txmimtma, processus
Aim in
tion
. ^tipli:
eatioa
: First. :
scpetistti $
206
t
$
Addition :
limber
OT
Errors
M
Per cent
of
Errors
1.29
/;
:
4.83
' s
89
1.26
• -V
920
: W
OO
MTlalon
:
%
:
;
%
:$
i
:
'
26.40
_v
:
Per coat of errors in the fimflacentol processes involved in isoleted oitnatlcns fresa greatest to leest frequency mere found in this
order:
v
-
■
. '
.. - ....
-
:
Division
Snhtraction
Addition
Hnltlplicetion
■
'......
This agreed exactly trith the study rodo by Busmoll e M John.^
They.oondueted an experiment in 1926 in schools ©f Chicago,
Vinnetke, JMLeacoe, Illincio end Kenosha, Wisconsin to find where the
%
' 'moat errors in the fundamental operations oceurrcd. To l e e m this, .
.
they analyzed the erithcotie moric of hundreds of children*
The coaclu-
slons shotted that the frequency of errore occurred in the anno order as
i n Table I A.
Table I D shows the results by grades of the isolated tests and of
the total long division test.
*»
It should bo read do follors:
There wore
Busuoll, Ouy T., and John, Lenore. "T^Lefcoetle Studies In Arlthme-.
W . g^.ment.ry
Kom.rmA.
Tm.
9
55 pupils in tho
4A grade and there wore 1? addition exnnploo in the
isolated addition test.
935, therefore, represents tho possible number
of errors which might have occurred in addition in the 4A grade.
Since
just 5 errors did occur in the 4A grade, i53 is the per cant of error
as shown in the table.
In the earn manner, the per cents In the other
operations were obtained.
T/JiXZ I B
m m m Mm
cmr c? -kbbcss by ohabss iiadb by is * h p iis
IH W E FtSBAMS»AL PRCCES323 iHVCLYI2) BJ ISOLATED
a i m n m
a®
nr m
t o t a l lciio d i h s i o m t e s t
Addition : Subtree- ;: Multipli­
cation
tit*
.4a ;
maker
Per Cent
5:
Total Long
Division
Process
:
5
:
. . .55 , :
SA
Number Per Cent
First
Quotient
13
'
6a
Number
Per Cent '
.
08
]:
!
16
419
lit
6.15
Jt
.68
83*W
50.73
64
36®
200
i
w
:
5.15
18
;
2.00
:
-■ 2
9
43
3.18
is .m
2.66
I
:
:
.39
35.71
133
64
7.60
12.00
s
Here it seemed that the higher the grade, the poorer was the addi­
tion.
This eight have been the case because there was less drill, on
addition in the sixth grade, or it night have been because the pupils
were careless.
Cooveraely, subtraction appeared to be better as the grade advanced.
Anf explanation of this situation was highly speculative but the most
defensible one is that subtraction was bottor ca tho grade advanced
Quite Interesting It is that suitiplioation tms boat in 6A, next
beat in 4A, and pcoroat in 5A.
Perhaps grade 5A ranked lorrest because
fractions had occupied m a t of the t i m while whole masters had been re­
viewed occasionally in speed teats and In self-test drills as found in
the text.
drills.
Also nuoh noro tins was spent on fractions in the maintenance
'
:
:
In 4A the multiplication facts wore supposed to have been mastered
anti by the tiiaa tho teats were given, the 4A, pupils had the facts well
in tiled and for that reason, (the lam of recency), might be expected to
surpass the 5A pupils in nultlplication ability.
As for 6A, because of saturation pies drill, the pupils night
reasonably be expected to excel in Bulti^lieatlcm.
Conwrolng errors in the flret-qnotient-flgure teat, the pupils of
5A did best, those of 5A did second boot, end those of the 4A did poor­
est.
Because 4A pupils had recent, iateasive drill in long division,
it would seen logical that they would surpass 5A pupils in that opera­
tion; just why they did not, no evident cause could be deducted.
Tho pupils of the 6A did best, and those of the BA did poorest in
the long division test.
One explanation of this la the fact that the
4A pupils had concentrated on long division for e whole semester and it
mas fresh in mind, while the 5A pupils had not received Intensive drill
In long division since they loft 53.
The 6A pupils, of course, had
plenty of time since 4A rozic for the gradual ripening or mastery of the
process to develop more cmnpletely.
Eapecially did the iA pupils shorn
superiority in cothoda of procedure; that la, in placement of ths first
qaeUeat figure. In coeparing before and after subtracting, and in tho
various poaiticna of sero In the quotient.
TABL5 I C
W C W B R /SB PltR CEI5T CP 2520113 ACOORDIKG TO SSX liADS B Y 164 P U P H S
IH THE FUKD/iHETsTAL PR0CBS5E3'I K T C m D IN TE? ISCIATHD
simncHs /dm rn .nn: total lceo dihsicii tzst
Table ! 0 should bo read os follows;
There wore 93 boys who took
the test, and there were 17 addition examples.
Therefore, there were
1571 possible chonoes for error anpng oil tho boys in grades 4A, SA,
and SA.
The 93 boys cade just 17 errors in the isolated addition test,
fherper cent of error for girls was found in the same manner.
T/iBLS 1 0 shows that boys were superior la addition and subtreetiom;
and that girls wore superior in multiplication, in division, and in the
Icmg division process.
Even though there is a olight difforenco in early
ability between the saxes. Bell,8i» an experiment found that as individu­
al3 reach adulthood, tiieir ability in all operations is about equal.
2« Bell, J . Carlton. '•A Class Exporiixsnt ln Arithmetic*.
Eduontlonal Psychology. 7. 467-70, ( M o ^ r , 1914).
Journal Of
"
12
iSrrora In The Total tong Division Test
AO the errors In the 164 papers nare analyzed, each type «ao writ­
ten down and every time the same type or error occurred. It ras counted,
TABLE II A should be read as foilera:
quotient wan one type error.
counted and resulted in 57.
Failure to write zero In the
Each tin® this error occurred, it rao •
'SSicrofore, the error of failure to write
zero in the qwtlent occurred 57 t i m s in all 164 papers.
Each type
error naa tabulated and counted in the sane way to get all the data as
shorm in TABLE II A which follows.
.TABLE H
A
.........
i-rmmim of t y p e e b h c e i i i e c h o c c u k p e b h i t e e t o t a l
LCSIO Division TEST £F 164 PUFIIS' L'CHK
f indioatee the place of error
TABLS I I
Failure to
tract all
0%,
■
:
#
Illustrationo
;
-
306)13^
W
f
w
:..\y.
:
. 3
-S
:
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:
of
:
«nroro ... :
s
i
$ Indicated the place of orrox.
t.
f indicates the place of error
=
0
*
i f
30
TABLE II
failure to
yelli
s failure to
:•
;
780}19B900
780)196900
#
mwrta*ttono
:
l.URbor of
crrora
: .■■■■
.
m
46
14
:V
:
:
:
t inaicateo the place of error.
: 34)1391
•!
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Illustre:
tions
;:
■:
■. l
failure to add
correctly
i l
failure to mul­
tiply correctly -
. 30S)E4«92C4
360)3^0
5S&1
2880
- '.
-■
T
IJuabor of
errors
49
—•* . • -
"
• , ,
:
f infiicatee the piece of error
60
48
•
■
. -::
:
15
TABLE H
A shoxra tlwt
rtf
subtraction* Running b close second in ntsaber of errors traa failure to
,
;
nrlte zero In the quotient. The exanples In this teat tmro so con­
structed that one zero, two zeros, and final zero appeared In the quo­
tient, and zoro was found final la tho dividend and not final In the
dividend.
The zero error was no at frequent in 5A, and least so In GA.
To explain just what Is meant by failure to Trite zero in tho quotient,
the following illustrations m y serve to make It clear:
Simple I— Final zero
im quotient and In dividend.
'
1260
329)412020 :
654
1962
1962
000
ISany pupils failed to vrtte a final zero In the quotient.
Semple II— Final zero In quotient, no zoro In dividend.
34)13^""
:
::
^
Many pupils failed to urito final zoro in the quotient.
16
Sample III— Zero Intermediate in quotient, zero in dividend.
360)38680
5 § „
^
:
Hany pupils failed to vnrite the zero in the quotient.
Sample 17— Two intormodlato zeros in the quotient.
-, :
■
7
'
/
:
■ 2448
1224
•'
_
Many pupils failed to write two zeros in the quotient.
All of these omissions were grouped as "failure to write zero in
the quotient".
...
Grossnickle
. ...' .
..
.:
.... ...
..
who made a study of errors in long division having
one-figure divisors found that zero was one of tho greatest causes of
error. Grossnickle*s study showed that ths greatest number of zero
errors occurred when zero was final in the quotient only as in Sample II.
He found that the frequency of error caused by this zero type was greater
than for any other form of error ereept for errors of combinations in
the division facte.
3%
Grossnickle, F. E. "Errors And Questionable Habits of Work In 1c
Dlvlsicm 11th A One-figure Divisor". £<
Research. Vol. XXIX, pp. 965-68. (January,
If
The next exoateot DUEtor of errors occurrea In w i t i n s the first
quotient figure over the wrong nunbor In tho dividend,
56 errors of
this typo rere found.
.
;
■
?
•smt£ II B, the reading of trhleh io self-explanatory, follova:
tabu:
II B
H B S E B OF TYPE 2RRCB3 B Y GRADES U H C H CCCURRED HI THE
TOTAL u m DIfISiai TEST OF 164 PUPILS * “IORE
..■■■■■ "1
4A.
Bustiber of
errors
5A
Dumber of
■ errors
6A
Dumber of
errors
■ ..
'-:*
- .. ■-"
4A
Humber of
errors
: Failure to write : Uniting zero In s'. Writing quotient
figure in wrewg
quotient when ;
3 : zero in quotient :
%
not
needed
:
. 3
Piece.. .
*
%
2
'
■ X
:
31
9
3
3
X*
s
■:
2
.■
:
2
i
■X ’
3
s
25 . ; ..
3 .
■i
26
:
6
3
•
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:
"-S
2
:
%
tl .
' :
15 _ _
.2
0
3
3
o
' ' '■..''
"T
"'V
-V. » ■
-
•• .
Failure to sub- : Failure to bring
tract all
*
Sown correct
the way
:
3
number
:
:
.
;
6
$
3
•
2
5A
#
H m b c r of ■ :
errors
:
6A
H m b e r of
error#
'
%
2
.3
2
: Failure to bring
dom
:
2 '
:
*
S
:
3
.2
*
9
6
:
,
.x
:
9
•
. 3
0
3
•
:
3
3
2
:
2
7
is
TABIS II B— Continued
■
: Uniting the wrong
8
quotient figure
■
-»»
-
-
j
i
..... SA.
-...
Kusber of
errors
0
7
$ ' ;■
■ m
■:J
6R
U m b e r of
errors
• 1
%
a
2
0
•
.... -
s
K -
1
■„
.... #& . .
litadior of
errors
; Attempting to
% divide when it
:
:
2
0
#
:
.:
*
2
*
:
:
V 8■
0
1;
■V
3
: Bringing down
i
anno nunber
twice
t
......
•
!
:
6
-
-
1*
Failure to sub: Failure to m l - : ' Failure to add
-correctly
- '! tract correctly
tiply correctly
4A
Uunbcr of
errors
5il
KuEber of
errors
2
2
2
20
2
:
Ja
em i r s
2
::
2
%
16
s
19
t'
21
2
.2
24
2
t
.
2
2
21
t
$
2’
.
8' "
-'
....
2
2
2
'•
-
; / •
'' :
. 20
-
Locking at TABLE II B it is evident that ns pupils advanced by
grade, they overcM» this error of writing the first quotient figure in
the wrong place.
6A oomitted no stick error, which showed that their
procedure habits were superior to those of 4A and 5A.
The feet that
pupils do Improve in procedure M b i t s as they advance in grade is further evidenced by Oroesnlekle,e study.
Be believed such wee the ease
because saturation in both judgrant and skill had occurred through
drill and advancing age of the pupils.
The fourth most frequent error was in Rultiplication.
Because nul-
tipliectlon caused the lowest per cent of errors in the isolated tests,
it ray be that errors in rultlplioation in the long division test,
occurred as a result of lapse of attention, or because of lapse of neecry when the correct response trao made, or because of. failure to add in
the correct m m b o r which was carried.
In other words, the errors seoned
to be caused by inability to multiply when nore than one digit was
4.
Op. cit,. page 3b5-68
20
present In the nultiplicand.
Th* fifth nost froqwet error was the failure to compare after
subtracting.
This caused 46 errors.
It is worthy of note that in
School 17 where 4A, 5A, and 6A were all tested, this error occurred
Je®t six times.
The arithmetic teacher in that school had drilled on
comparing before and after subtracting, which shows that drill on such
spots Is effective.
The pupils In School 17 wore half the number of
pupils tosted, showing that the other half of the pupils tested made
40 of the 46 errors.
ing.
■
''
Evidently they failed to compare after subtract­
:
'
'
The remainder of the errors and the frequency of each m y be seen
in TABLE II A.
Tim distrlbutitm of much by grades cay bo soon In
TABLR II B, and distribution according to sex appwra In TABLE II C.
The reading of all of these tables needs no explanation.
TABLE II C
msaJER GF TYPE ERRORS ACCORDING TO SEX M!ICH OCCURRED
IK THE TOTAL I.®S DIVISION TEST OF 164 P i m S * *CEK
Boys
Number of
errors
Girls
mmtoer of
errors
: Failure to write : Kilting zero in r “Writing quotient
quotient when : figure in wrong
: sero in quotient
place
not needed
:•
:
:
:
:‘
:
;
? - \ 36
no
::
:
:
:
:
.
. 5
%
:
: '
v 8
3
28
:
v,.. a
2
:
:
3
21
Fallt
. Failure to at
*
tractall
t'
the mar
• Failure to bring
maaber
A
______ —
Boys
errors
"j.
Oirlo
Ktaaiber of
errors
Boys
Number of
errors
•
.
'
Girls
NtmdJer of
enroro
(Arls
Number of
errors
•
-
_JL
, :
j
. '
.
6
.1
Failure to
ccnplete
S
:
*
: Bringing down
:
a a m number
:
t^ice
i
:
:
0
:
'
!
:. .
‘'r''.x
'
e
Boys
limber of
errors
18
- .-IS
l
:*
:
.$ lu t i n g the srong :i Attecpting to
:
qmtieat figure
i divide when it
:
: J: is impossible
:
5!
:
2
:
IS
:
5
•l
2
:
:
s
•
‘:1
:
;
f
Failure to com­
pare aftor
multiplying
:
0
‘
Failure to com­
pare after
subtracting
:
:
:
:
:
:
t
:
11
$
:
'' %
*
*
2
:
81
--
:
-*
.
11
■ ' $ ■'
/
:
• -S
•
- -
s
:
1
?
-L_________________
; 1
1 '
' %
'-
*
- ••
■ '
-
.$
3
* '
15
/
‘
•■
Failure to add
•: Failure to e o Icorrectly
% tiply corraetly
1
:
•:
SO
23
:
:
2
3
:
:
i-1
:
12
3
26
i•
:
;
B<qrs
Htoabor of
errors
Cllrls
Hunber of
errors
Failnrs to sub­
tract correctly
30
22
Cosp&rison Or CaaMnation Brropa In Isolated Situations
And In The Total Long Divial on Teat
TABLE III A should bo read as follows:
Tho total nunbor of addi­
tion, subtraction, nultlplloation, and division facts as found in the
isolated testa vore observed; then the total nunbor of addition, sub­
traction , multiplication, and division facts as found in tho total long
division test wore observed.
Every tine a child m d o
a conbination
error in the total long division teat, the writer looked In the isolated
tests to boo if the same child had made tho sane error there.
TABLE III A
mmm
gf ccubihatice hirck? ih isolated situatioio,
IK THE TOTAL LONG D m S I « TEST, AIID
Division . $
Rumbor of
errors
:
:
920
$
:
3
§
36
;
3
:
:
i
:
Addition
s
:
liunbor of
errors
Hi BOTH
3
Multiplication
l
806
-
Total long divi­
sion process
163
89
%
5
-
Both
%
:
:
3 :
33
TABIS III A ahows that tho number of cmbinatlon errors made In
the Isolated toots was m c h greater than the nutiber of combination
errors made in the total long division test.
It may be observed in TABLE III A, page 22, that there mere 920
errors Bade In the isolated test in division.
That mas likely because
the directions in the isolated test in division asked tho pupils to
merely grits the first quotient figure.
They mere not supposed to mul­
tiply on paper the prodwt of # e first quotient figure and the divisor.
There more no written directions or oral orders either-that tho pupils
should not multiply cantally; evidently, few Sid so, and hcnco this ray
account for the groat number of errors in the isolated test*
Since
the pupils rado just 163 combination errors in tho total long division
test, as shown in the table on page 22, it is evident that estimating
only the first quotient figure mas a more difficult task than working
the whole example.
It is interesting to note that there was very little relation be­
tween errors found in the fundamental combinations In isolated situations
and those found in the long division situation.
TABLE III A shows that
in only 31 eases did tho same error appear in both places.
By this is
raant that the some error in the isolated tests and in the long divi­
sion test was found on the anno paper in just 31 cases.
This evidence
_
right lead to saying that errors are not persistent.
Brueokner ° found
that in giving the sona tost six times to students that in only about
5.
Brueokner, Leo I. "Persistency Of Error As A Factor In Diagnosis’*.
Education: 56: 140-4, (Kovenbcr, 1935).
53fS of the cases in 1*1 oh there were errors woo the arae exanple solved
laoorreotly all six tinea.
Hlo explanation was speculative in nature.
He oald leek of persistency of error right be caused by such factors as
differences In motive, doRree of effort expended by the pupils, level
of maturity, and nature of etmtrola.
The number of times that combination errors could have occurred
I* both situations coo possible to have boon as Rreat as several hun­
dred; no exeat computation earn be obtained as it Is not poamlble to predict the possible n m b e r of errors each child might nmke.
'
Out of
a possible several hundred. Just 31 eases where the seme error occurred
in both places was indeed a small number.
The readings of TASIJ£3 III A, III B, end III C ore self-explanatory.
_
TABLB HI B
m m m OF O C M n i M T i m SHRQJB BY GRADES IK ISCIATED s i t u a k c s b ,
IN i m t o t a l zorn d i y i s i c k t e s t , a n d i n b o t h
Addition
- M'- .... .... . %
Humber of
3
errors
... 2.: 'JU.
■■, ' Y 3
Number of
:
errors
:
6A
3
:
Humber of
1*
Subtrestlon
3
■ 8
2
*
88
3
13
3
2
75
Multiplication
*
2
__ 3_
' -2
:
3
2
:
16
64
'
TAMS I I I B—Contiimea
: Division
V
:
i
41#
;
%
360
•:
:
:
133
:
":
4A
Basher of
errors
BA
Kosher of
errors
6A
Kutiber of
errors
:
:
:
:
:
2
2
:
:
2
%
Total Long divi­
sion process
65
:
;
:
Both
9
-
60
-SB
2
■2
2
*
:
2
2
13
9
TABUS III B shows that tho 5A had 13 oases of egreercent, while 4A
and 6A had 9 cases each.
:
TAMS III C
i m m m <& czmm&Tim e e r c r s a g c c s d i m g t o swl
laausm ) smfATicus, in m tctal low
xb
DIVISION TEST, AED IK BOIE
:
Addition
■*
Boys
. ;,:;1?
Rafter of
errors.
Girls
Rafter of
errors
♦
2
2
2^
;
:
*
%
:
19 : =■; J
-
3#traotlon .- t... irultlnlicBtion
%
-.
St
114
2
■*
2
2?
... 92
2. '
'’ '‘ '■
T
.
"
: Division
% - --
Bdgm
limber of
•errors Girls
Kucher of
errors
-
$
536
Total long dlvision process
97
t
s
.. si
:!
66
■ !!
Be#
16
■
13
. 1t
TABLE III C shows that hoys had 18 cases and girls bad 13 cases of
agreement of errors in both isolated situations and in the total long
division test.
.-..CKAFEK IV
CCKSLinCCSa AHD K5D£33SKDA” ICIS
Conclusions:
Answering the questions set up in this study sad
within tha licitation of the szperinant, the following conclusions m y
be drown:
1.
The loci of s<wt frequent errors in long division ere situa­
tions involving:
a. .St&trastlt*' ' .. .
- .
•
• ,
■
;
b. Failure to #rlto zero in the quotient
e. Writing the first quotient figure In the wrong pleee
d .
Hultipllcation
e. Failure to ecsspere after etibtracti^
Each of these type errors oeourrcd close to an equal number of time#,
ranging from GO downward to 46.
2.
The agreement between errors found in isolation and the same
errors found in the long division situation la slight.
3.
Errors differ in frequency between the sexesi but the differ­
ence is so small that it seems neglibible.
4.
Finally from the study, it may be said that, generally, errors
have a tendency to decrease as the pupil advances from lower to higher
grades.
Especially Is this true of the procedure errors; that is, those
errors having to do with proper placement of quotient figures, zeros in
the quotients, and comparing before and after subtracting.
Rocomaeadatlona:
A good authority cm arithmetic says,
"In our prevailing method of number teaching there is an
onomous preventable waste, which consists very largely of
27
lecdiag ehUdren to iaako ciatukoa, thereby perturbing them,
e M then in trying desperately to correct those mistakes*
Most drill tsork in number is not only ralmless, but positively
a hindrenceto learning; that children’s errors ere rarely ever
ratters off tiere cnreleescesa; and.-toet the business ran com' plains «dth reason that We do not tench children to do even
sisipleauKscorreotly.
Any test rarely registers the test after the darage had
been doi», ^erefore, the liope <rf improving Instruction in
number work la to develop in the teacher habits off keeping
careful, eoatinucuo records off her pupils’ difficulties, off
anticipating them, of helping then to avoid errors, and off
effectual correction of each error on its first appearance."4'
On the anno subject Gist says.
T o eliminate errors it is necessary to definitely locate then, that teachers and pupils ray know whore the great­
est possibility off error is, and that problems be especially
prepared to give the kind of drill necessary.
•
■ ■'
'
■
;
■ :
■
The importance of improved first teaching Is not easily overes.
.
.
.
tinated. It is nest essential that first teaching be such that pupils
will not make errors in such skills as placement of quotient figures,
in having remainders too largo, and in having subtrahends too large.
Most errors will likely never occur if teachers anticipate them and
teach with their potentialities in mind.
In long division, correct placement of the quotient figure ohould
be taught with the first example on through to mastery of that pro­
cedure.
-
’ : '
^- v
"
■
Teachers should certainly teach long division using six steps in­
stead of the traditional four steps. Children ohould be taught to say
1.
8.
liyurs, 0. C. "Psrslstonco of Errors In Arithmetic.» Journal of
Educational Heaoaroh. Vol. V, pp. 248-49, (April, 1922).
Clot, Arthur S. "Errors In The Fundamentals Of Arithmetic."
School And Society. 71. pp. ITS— 77, (August 11, 1917).
th ese steps in o rd er before they ever begin w ritin g long d lv la l
.'
_
-
1.
2.
8»
4.
5/
6.
Divide
Koltiply
Ooepere
Subtract
Cmsimre
Bring d m m
Using this example for Illustration,
295)755581
have a pupil reciting follow this order of monologne:
1.
Divide
T say, 29S into ySS".
nI say, 29 into 75".
"I think, 2 Into 7 Is 5". Ho writes 5 lightly
over the second S In the dividend.
2.
)
Multiply
"3 * 5 equals 15
3 x 9 oquals 27 plus 1 equals 28
3 x 2 equals 6 plus 2 equals 8, giving 885".
He writes it lightly under the 755.
3.
Conpare
"885 is greater than 755, therefore I write 2
lightly as the first quotient figure".
2.
Multiply
" 8 x 5 equals 10
■
2 x 9 equals 16 plus 1 equals 19
2 x 2 equals 4 plus 1 equals 5, giving 590".
3.
Conpare
" S W la less than 755, therefore, 2 la the oorreet
quotient figure. Tie writes the 2 heavily now
in the quotient.
4.
Subtract
*0 from 5 is 5
9 from 15 is 6
5 from 6 la 1, giving 165*.
5.
Ccnpuro
”185 ie loss than the divisor, 295, therefore,
I b H n g flovm tho next nisAor”*
6.
Bring flotm
For a tlEo, It la b good Idea to have children use arroxs ’
.then they
bring down an it helps to keep oxmsplon straight end to prevent the
oenmn errors censed by btlnging So*,m the nrong nuisbor, and of bring­
ing down the oame nneiber twice.
Also It m y prevent the error of fell­
ing to bring down the naetber et ell.
Continuing with the exnnple. It now has this form:
295)755581:
# 1
doing ahead with tho recitation the pupil cays:
1.
Divide
"X say, 295 Into 1655.”
"I say, 29 Into 165.»
"I thlnk, 2 into 16 la 8.« He writes 8 lightly
over the third 5 in the dividend.
S. Multiply
'
3.
”8 x 5 equals 40
8 x 9 equals 72 plus 4 equals F6
8 x 2 equals 16 plus 7 equals 25, giving 2260.”
Conparo
•
• ”8360 Is greater than 1655.
for the quotient figure.
quotient figure.^
2.
Therefore, 8 is too large
Write 7 lightly for the
Multiply
”7 x 5 equals 35
7 x 9 equals 63 plus 3 equals 66
7 x 2 equals 14 plus 6 equals 20, giving 2065. m
a s m s
*30*5 is greater than 1655, therefore 7 ia too large for
the quotient figure. Write € lightly for the quo­
tient figure.*
2.
Uultiply
"6 x 5 equals 30
S x 9 equals 54 plea 3 equals 57
6 x 2 equals 12 plus 5 equals 17, giving 1770.*
3.
Compare
"Again, 1770 is greater than 1655, so write 5 for
the quotient figure."
2.
Multiply
" 5 x 5 equals 25
5 x 9 equals 45 plus 2 equals 47
5 x 2 equals 10 plus 4 equals 14, giving 1475."
5. Cmimi.
_
.... .
.
*1475 is less than 1655, therefore, I stibtreat."
4.
subtract
*5 fren 5 equals zero
7 fren 15 equals 8
4 fren 5 equals 1, giving 180."
5.
Compare
.
r
,
"180 Is less than the divisor, 295, therefore, bring
Sozn the next nunbor, 6."
6.
Bring down
Hew the example has this form:
295)755581
El
4
m
Let It be said that Is this particular example there uas a great
deal of trying; but it was chosen for the purpose of showing exactly
how the comparison stops proven! error.
By the S i m pupils reach
problem of this degree of difficulty, they will be able to multiply n
Rontally for testing the size of the subtrahends.
Also, they will
learn to observe that the divisor, 295, is. so nearly 300, and that in­
stead of thinking, 8 into 7 or 2 Into 16, they trill think 3 Into 7 and
5 Into 16.
If the steps are continued until tho example Is conploted, it is
likely the child will fall into no bad habits and will m e t fewer
stumbling blocks.
This method has boon found very successful In help­
ing to eliminate errors of procedure such as improper placement of
quotient figures, failure to write zeros In the quotient, failure to
compere before end after subtracting, failure to bring down the cor­
rect number from the dividend, failure to bring down at all, and fail­
ure to eoapletethe example.
As for subtraction errors, it cay be helpful if the digits in the
subtrahend are always spoken first.
sistent.
If they are not, at least be con­
In this example,
755 ...
it is better to say,
-
■' : .^
frem S is
9 from 15 is 6.
8 froa 6 is l.*»
"8 less 0 is 5.
15 less 9 is 6.
6 less 5 is l.w
'
Or
'
Do not nix tap the rays of saying It until pupils have mastered eachI*
This ono suggestion nay prevent a great number of subtraction errors.
HuXtlplioation errors require drill.
tipllcation facts nust to mastered.
First, of course, the *ul-
Often, In long division it Is help-
fid if the teacher trains the pupils to d r ays re-nultlply before ccnparlng and aubtroctlng.
Since from this study there s o o m to be little
relation betraen errors in isolation and those in the total long divi­
sion situation, it t&ll be best to give the drill for nultlplioatlon
and also for subtreetlon in aetual long division oxtmples cade for drill
purpose.
For instance, in the example,
3 4 ) W “
>
suppose a child multiplied the divisor, 34 by the first quotient figure,
4, and _obtained 132.
Before he goes any forther, m k e up demo other ex­
amples oueh as,
34)1386
or
43)1?S8
so as to fix the eorreot fasts In M s nlnd before the wrong response
■ '
•
■ •
can become fixed. The teacher should make the pupil eouaotous of the .
.
eorreot answer, ignoring the wrong response as it Is hoped he M i l for­
get It and mentioning it to him may Impress it on M s nonory.
If these recommendations are followed, it la believed that long
•
'
s
division will b e oow autoratle in a shorter time.
3.
W®t only will long
Upton, Clifford B. taking Long Division Automatic.* Rational
Council Of Toachora Of Mathematics. Troth Tear Book, pp. 251289.,(1935).
' '
division become nutomtlo in a shorter tine, but the pupils trill likely
enjoy long division because they will be r$ore likely to Raster it.
In­
dividuals like to do the things they can.do nail; teachers will find
that teaching for mastery pays big dividends both for the pupilsf satis­
faction and also as a tiro-saving and labor-saving ochoEo for themselves
34
MBIJOCtAFBT
1.
Bell,
Carleton
"A OlesB Kxporlisent In Arlthnatte.*
Journal Of Educational Psychology. Tol* V, pp. 467-70, (Oetober
1914)
'Bruetimsiri 1. J.;:;
.
"Poralotency Of Error As A Factor In Dinsnoaie.”
Bdueatlon; 56, pp. 140-4, (Kovenber 1935)
3.
Buweell, Guy T.
"Dlngonlotio Studlee In A^ t W w t l o . ^
Supplementary Educational MonoRmuha. Ro.
(Chicago: TBaiveroity Prass, 1926)
.
180-87,
4.
Gist, Arthur S.
"Srrore In The Fuadamntala Of Art thee tie."
School And Society. Vol. 71. pp. 175-77. (Auroat 11. 1917)
5.
Grocsnlckle, F. E.
Errors And Questionable Bablte Of Horic ln Long Blvlaton tilth A
{hie-FlgureDiTisor."
Journal Of Eduoatlonal Roaearoh. Vol. JXIX. pp. 355-368, (Jmmw r y , 1936)
.
6.
Moert», Walter
S.
:
"Tho Teacher*s Responsibility For Devising Learning Exercises In
Arlthawitle."
:...
'" V
'
■'
;:
■
miverslty Of Illinois Bulletin. Tol. Hill. Mo. 31, (Juno, 1926)
7.
Myers, G. 0.
"Peraistence Of I r m r In A H tittle."
Journal Of Educational Research, Tol. V, pp. 348-49, (April.
ig22j
:
:
~~ *
8.
Upton, Clifford E.
"Koklng Long Division A u t c m t l e . "
Tenth Ye a r Bock Of The Council Of Teachers Cf Mat h e m t l o a , pp.
2 5 1 - % 9 , (1935)
•
™
BZBIJCOILM’HY .C^ BCKK5 A ® AHTIC1E3 BETIEIfEB F m TEI3 3TDDY
1.
Buawell, Oery Tliosas a M J a M , 0. H.
■.'"Smmry Of EducationQl Investlgaticmn EelatlRg To Arithmetic."
tfalveralty OfChloago Bulletin. TMverelty CfChlca^o Press,
Chicago, Illinoio.
8.
Brwsokner, Leo J.
PlaiCHoatlo Anti Hftrwdlnl Teaching In Arlthaatle.
John C. Winston cdnpnny, PAlIadelptoi'a, (1930) u
5.
Fox, «.• A. and Thorndiks, B. L.
"The Relatlcmahlps Between The Different Abilities Involved In
The Stngy Of Arithmetic.".
ColxMfbje Contributions To Phllosonhy^ Peyehology, And Education.
XI, pp. 32-40/ (February, 1903)
4;
Klzfcpatrlek, B. A.
nAn Kxperiiasnt In Meiaorizlng Veraua lncidental Learnlng.”
Journal Of Sdacetlonol Psychology, Vol. V, pp". 405-12. (Soptonberj 1814}
:
. '
5.
Knight, F. B.
WA Kote Cn ArlthEotie."
Journal Of Educational Besenreh. Vol; VII. n. 62. (Jemmry, 1938)
6.
Morton, B. 1.
^An A w l y e l s
Pnpile* Xrrors In Fraotlwe.'*
Journal Of Educational Rosenrch, 7ol. IX, pp. 117-85, (February,
“ 1924)
—
:
~~
7.
Merton, Elda, Banting, G. 0., Brueokner, Leo J., and Souba, Ar%y
*1le®edial.Worlc In ArithiBetlc.**
Second Year Book Of T h e PepartCMaint Of Slegentary School Prln.
d p a l s . pp. 595-429, Waahlpgton:.Kational Sdpcatlonal Assoolatlon, (1983)
8.
Osbum, W. J.
"Errors In FundaESRtals Of Arlthcstie. •»
Jouroal Of Kducatlcnal Research. Vol. V, pp. 348-49, (April, 1922)
9.
Thontflke, Kd. L.
" D » Psychology Of Drill la Arlthnetie: The A m o m t Of Practice."
Journal Of Educational Payohology. Vol. H I , pp. 185-94, (April,
1921)
'
.
-
-
■
.
.
•
"
10.
Thor&aiko, Ed. I.
The Psychology Of Arithmetic
Macmllllan Ocropany, pp. XVI > 299, Mere York, 1922.
Division Of Whole Numbers
Name
Grade________
Age
Boy or Girl
When is your next birthday?________________________
School_____
■__________
Date
Summary of
Pupils' Scores
.Scores on
Test.
Part 3
Part 2
Part 1
•
Part I — Multiplication, Addition, And Subtraction Used in Parts 2 and 3.
(Do not divide):
a
II
Multiply
5x5 =
8x7
2^/S =
5x9 =
9x7
2x7 =
5x6 =
6x9 =
1x7
5x0 =
5x5 =
1x9 =
4x3 =
1x6 =
6x3 =
4x4 =
2x0 =
2x8 =
4x0 =
2x3 =
3x0 =
1x0 =
;
6x4 =
7x0 =
5x8 =
2x5 =
1x5 =
6x6 =
7x5 =
3x 7 -
2x9 =
1x2 =
s
11
'
0x0 =
8x6 =
2x2 =
4x7 =
1x4 =
1x3 =
8x0 =
6x7 =
4x5 =
,..............
--
Add (Do not multiply):
12 + 1 =
0 + 2 a
36 + 2 =
24 + 4 *
14 + 1 =
0 + 1
35 + 4 =
0 + 3 *
0 4 4
S3
18 + i
..
45 + 2 =
4 + l »
54 + 3 =
12+4 ■
20 + 2 =
::
46 + 5 =
45 + 6
Part I— Continued
Subtract (Do not add):
1 3 9 3 8 4
—1 3 6
4 2 9 0
-3 9 0 0
1300
-1224
-3 8 4
79
6 5
~6 4
-6 0
3 9 0 0
-3 9 0 0
5 1
3 0
2 10
-210
388
12 2 4
+12 2 4
2 4 4 9
*2 4 4 8
765
1530
850
-612
-1530
-654
19 8 9
-1 5 6 0
2 8 8 0
- 2 8 8 0
*360
755
-5 9
1 6 5 5
0
-1 4 7 5
2
18 0 0
3 0 1
2 7 8
507
513
4 l
-1 7 7 0
-2 9 5
-2 2 8
-456
-3 1 3
-327
1962
-19 6 2
Score on Part 1 ■ Number right * ______
(Total possible score * 86 .points )
Part 2 —
Estimating The First Quotient Figure
Directions: Write only the first quotient figure in its proper place. Do not
take time to finish the examples. look at the samples before you
begin to work.
6
0
Samples:
.35)20
877581
3*759
30671300
30751
577507
3607^80
307T
647385
2957755
780)4290
3277850
30671530
30671225
3 0 W
30672559
78071989
577278
3277512
36o7^
306712"
64779“
307210
295)1800
2957301
34731
3067755
78073900
577513
3607388
647r
29571655
3067122
32771962
32770
Score on Part 2 * Number right X 2 “___ __
(Total possible score ■ 66 points)
P a rt 3 - - Fundamentals of Long D ivision; Checking.
7
Directions: Work the ten examples below, Check numbers’B, 9, 10, and
^ where indicated.
1.
34)1 3 9 1
3.
30)6 5 1 0 ?
5.
306)2 k 4 9 2 2 4
2.
*
64)3 8 4 7 9
k.
780)1 9 8 9 0 0
6.
360)3 8 8 8 0
P a rt 3 — Continued
7-
.
7- Check.
306)1 3 0 0 5 0
8.
8. Check,
295)7 5 5 5 8 1
9.
9- Check.
57)2 7 8 7 3
10.
10. Check.
■
'
■
327’nr i 2 d iTo
right X 10 »
Score on Part >
---(Total possible score = 100 points)
...
>
<
7 54 2
E
9791
V O I G I
J t i A N A L Y S I
938
S
O F
E k K U k S
682
I N
L O N G
O I V I 8
IN S E R T BOOK
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