MODERN
ARITHMETIC
AND
METHODS
PROBLEMS
By
THEODORE
^INDQUIST
KANSAS
STATE
NORMAL
EDITED
SCHOOL
BY
.
GEORGE
THE
SCOTT,
CHICAGO
UNIVERSITY
FORESMAN
MYERS
W.
OP
CHICAGO
AND
COMPANY
NEW
YORK
^
COPYHIGHT,
1917
BY
SCOTT,
FORESMAN
AND
COMPANY
PREFACE
AUTHOR'S
below
Students
from
benefit
good
senior
class
a
than
text
in
for
this
review
of
is to
of
that
while
author
teachers.
plans
and
the
found
been
actual
the
makes
touchstone
scholarship
the
the
search
for
basal
the
also
the
only
to
in
such
results
the
has
by
The
have
as
in
teachers
of
the
derives
text
attitudes,
highest
eflficiency the
it demands
efficient
'*big idea/'
"
the
"
"
111
426918
a
plans
aptitudes
and
ization
social-
importance;
ideals
broad
methodology
"big
its
and
motivation
sane
;
all
such
form.
are
the
conditions.
real
method
of
material
syllabus
offered
best
the
upon
as
to
in
by
instruction
this
metic
arith-
of
evolved
been
years
interests,
It looks
and
as
is
as
a
gave
teacher
the
two
contained
good
in
varied
arithmetic
of
has
herein
of
under
economy
sider
con-
presentation.
methods
average
results
suggestions
children.
of
the
past
excellent
from
methods
and
of
engaged
the
treatment
dents
stu-
to
which
inadequate,
volume
productive
practice
The
of
actively
with
used
been
of
discusses
needs
present
During
of
a
subject-matter.
the
the
upon
need
drill-book
is
which
all
supply
arithmetic
form
old
greater
majority
great
methods
as
arithmetic
grade
exclusion
It
the
methods-book
purely
the
well
as
purpose
of
teachers
subject-matter
For
The
much
is based
instruction
lectures.
upon
-classes
which
in
receive
rank
college
thing/'
the
and
and
;
it
sound
it makes
always
the
AUTHOB'S
IV
first
a
of the
concern
review
of
teacher.
seeking always
become
an
effective
how
The
development
considerable
in
which
been
of
sine
the
ough
thor-
a
of
subject-
in
non
qua
be
teacher
units
and
broken
is put
in
to
teaching
example
an
The
serviceable
more
The
live
as
will
of
with
will
two
is
to
a
The
to
adapted
as
the
text.
the
at
suggest
to
both
this
needing
how
view
re-
a
from
standpoint
upon
the
texts
one-semester
a
end
object:
needing
not
second
to
only.
course
time
much
subject-matter.
chapter will
been
well
in the
of
also
exercises
teachers
material
twofold
a
teachers
hours, depending
each
has
as
book
five
review
which
references
found
the
in this
and
the
the
found
exercises
with
study
from
them
be
to
Appended
treated
will
the
supplement
to
experienced
study
of from
in the
Prospective
later.
while
will
text
given
which
been
of
subject-matter
subject-matter
standpoints,
work
of
less
adopted
selected
been
has
use
been
fully
longer. and
structure
contained
review
a
material
teacher
This
accordingly
chapter
serve
the
has
of
sort
a
more
are
are
chapter
for
subject-matter
each
than
short
form
it has
used, and
retained
has
appear
given
time
of
period
it would
be
can
forni
teachers
beginning
that
outcome
The
experience.
before
work
short
better
is the
book
ii"portance than
of
a
unities.
in this
classroom
greater
Small
grasped
to
along
need
prospective teacher
the
is that
varied
completeness
of
all
reorganizing
A
presented
material
found
be.
to
thinker.
study
to
demands
train
to
teachers
young
student.
a
of
most
and
recasting,
matter,
of
As
subject-matter, it
reviewing,
children
PREFACE
for
selected
for
be
their
found
the
short
a
raphy
bibliog-
accessibility of the
bearing
upon
the
topics
chapter.
is suggested
teachers
in
as
a
text
for
experienced
college, normal,
or
or
experien
in-
high-school
AUTHOB'S
training
is
suggestions
Miss
R.
Avice
State
Lena
of
Parker
Chapters
IX
Emporia,
October,
Dr.
V
and
acknowledge
which
Mr.
Flora
and
Kansas.
1917.
who
Keller,
H.
of
problem
the
J.
the
Miss
and
Kansas
the
of
a
the
material
X.
Theodore
leagues,
col-
my
and
Cooke,
contributed
helpful
many
from
Lull,
G.
vidual
indi-
"
come
W.
;
School,
;
the
have
Herbert
Miss
circles
courses.
contributed
who
Chapter
W.
to
Hansen,
B.
Wright;
Normal,
part
here
criticisms
and
reading
correspondence
and
;
privilege
my
V
institutes;
teachers'
courses;
self-help
It
PREFACE
Lindquist.
large
cis
Franfor
EDITOR'S
characteristic
The
indeed
as
social
worth,
ideals
of
they
and
are
implied
in
of
problems
with
these
high
a
the
and
actual,
the
broader
following
tioned
men-
ideals
of
ical
arithmet-
of
treatment
claims
pages
to
notes.
implies
material
it
of
In
results.
cational
edu-
means:
training
the
that
similarity
close
teachability,
usableness
in
children
and
skills
of
ready
a
Instructing
knowledge
into
them
the
arithmetical
of
measure
judged
has
race
of
fund
the
metical
arith-
preserving
worth
them.
for
2.
Giving
children
of
vocations
and
3.
native
Doing
enthusiasm
for
experiment,
the
under
Let
subject-matter.
material
of
the
this
shaping
either
gauged
objectively
or
book
occupations
foster
the
pupil's
learning.
be
may
will
that
methods
the
and
adulthood,
modern
by
for
preparation
some
this
all
Teachability
of
specifically
educational
degree
procedure
1.
oft-cited
vx
^
to
the
in
The
not
characterizing
Practicality
"
The
worth.
tion,
educa-
practicality,
are
soundness.
efficiency
are
social
and
accord
education,
modern
and
arithmetical
modern
psychological
economy
methods
in
all
and
practicality
of
notes
of
because
be
PREFACE
has
by
it
be
grown
of
influence
"
Vll
"
empirically,
intrinsic
noted
into
extended
in
by
room
class-
characteristics
passing
its
present
classroom
that
the
form
trial,
EDITOR'S
VUl
form, herewith
final
the
marked
PBEFACE
offered, having been
with
attended
success.
is directed
Attention
the
to
following
internal
of
marks
teachability :
The
book
in
presentation
later
the
seeks
great
fully
the
It
material
to
the
for
supplies
of
reading
supplementary
teaching
degree
and
practicality
After
go
aids
far
have
the
material
exemplified,
and
exemplify
to
topics
study
practice
lists appended
ample
sources
topics treated.
the
on
dren.
chil-
with
well-graded
exercise
teachableness
realizing
toward
plain
it
the
to
to
high
a
three'meanings
the
of
from
stimulating
is
of
sources
**big'' thing is
the
life situations
copious
and
and
snappy
real
that
"
able
consider-
that
pupil-teacher
is essential
exercise
drill
drawn
the
guarantee
systematic drill
to
of
first
above.
stated
making
in
sort
and
bibliographies furnish
The
chapters.
These
abundance
is the
study.
to
discontinue
to
every
**big" idea,
This
of how
judge accurately
an
right
the
to
purpose.
in
search
to
his pupils.
secret
which
at
and
controlling, the
into
the
to
attempts
motive
gauges
essential, the
stages
guide
usability the guide in
and
teacher-pupil
the
acquiring
in
book
The
pupil
this habit
implant
step
the
as
the
the
struggles continually for motivated, and
habituate
for
to
early grades
It
work
to
situation
and
the
work.
purposeful
It
simplicity, childlikeness,
makes
"
^problem
prescribed,
such
material
and
are
cited.
The
text
commercial
of
men
material
both
treating this material
Sometimes
is
shown,
of
affairs
a
teachers
to
adopt
as
social, industrial,
from
fully
so
sources
actuality.
used
draws
to
better
and
give it
way
are
even
the
better
for
guidance
a
close
the
than
counseled
way,
but
the
approach
way
to
actually
one
to
and
try
the
to
get
habitual
EDITOR'S
of
way
the
treated
Tidbit
the
methods
of study/
into
working
editor
The
marks
feels
chapter
this
In
A
of
phase
well
material
social
of
reason
distinctly
to
secure
of
this
show
poor
book
and
its
other
Its
so
nate
fortu-
empirically
conceded
to
not
to
to
to
prove
moreover,
sess
pos-
it of
explicitly.
to
that
teaches
how
adapts
adult
and
of
subjects.
The
interests
and
that
all
make
of this
use
its
to
is
for
this
erally
gen-
arithmetical
school
as
a
by
concerns
is
It
opportunity
interest.
The
matter,
author
and
to
profitable.
presents
fmiciion,
its
by
selves.
them-
is not
treatment
do
fully
pursuing.
are
consider
psychological
through
speak
employment
high
excellent
an
values
socializing phase
point of view
material
motwation
of this group
well
this
both
as
arithmetic
school
subjects
done
ter
of
common
practice
make
to
over
significance of arithmetic
to, is the
school
has
sStoow^
It takes
social
certain
A
modifies
and
social value
the
how
sociaUzaiion
the
the
be
treatment
exemplified.
has
to
as
carry
are
judged
said
been
socialization
from
who
must
is,
attended
group
book
There
the
and
the
this
of
worth.
out
perforce
will
whether
that
already
method
chapter
and
practicality.
treating
brought
then
has
social
results
teachings
merits,
Enough
beget in teacher-
children
the
ship
scholar-
of
degree
to
as
usableness
of
the
text, must
both
use,
This
of high
^^^"^ and
by the
habits
internal
from
or
of
such
have
to
as
sufficientlywell
always
with
presented,
so
fostered
and
urged
the
IX
exemplified.
material,
pupils
is
world
workaday
and
Such
PREFACE
office.
doing.
actual
knowledge
its departure
to
mat-
It
It
immaturity.
from
what
the
EDITOR'S
X
child
it
that
will
and
can
of
way
three
will
in
not
social
with
few
a
the
The
pupil.
is
conceived,
pupil.
These
found
again
be
will
merely
take
isolated
that
of
logical
psychothe
always
of
ear-marks
and
again
but
places,
that
a
in
clerical
by
out
through-
attitude
attitude
toward
of
this
growing
present
October,
It
of
number
education.
arithmetic
who
take
and
We
a
need
function.
1917,
who
those
education,
past
class
book.
modem
for
book
George
Chicago,
viz.:
soundness.
a
modem
the
the
meets
education,
termed
reducing
toward
volume
modern
be
this
rapidly
that
this
psychological
and
propriety
all
aid
augment
capable
claims
worth,
moreover,
teachers
books
the
characteristics
essential
feels,
will
rightly
to
therefore,
editor,
then
He
to
such
book.
practicality,
may
learning
reader,
this
teaching-technique
makes
teacher,
soundness
careful
The
the
of
way
psychological
the
It
do.
study-technique
becomes
normal
PEEFACE
W.
Myers.
gressive
pro-
more
CONTENTS
1.
Introduction
I.
Chapter
for
the
as
to
A
Seek
following
II.
Chapter
6.
broad
of
the
for
ia
metic
Arith-
4.
of
formal
in
Improvement
pline
disci-
arithmetic
subject-matter
pensable.
indis-
idea."
**big
Exercises.
chapters.
9.
of
Use
Bibliography.
^Notation
History
Early
doctrine
knowledge
always
arithmetic
Why
arithmetic.
of
The
5.
arithmetic.
7.
2.
value
life.
larger
8.
teaching.
arithmetic.
utilitarian
The
3.
taught.
the
of
leaching
Poor
1
tion
Numera-
and
"
10
Why
10.
history.
the
study
number-reckoning.
Tallying
and
17.
system.
Arabic
notation
Boman
III.
Chapter
being
The
Notation
18.
Hindu-
the
world's
ing
count-
distribution
and
origin
system.
in
the
Arabic,
and
Development
of
numeration.
19.
Exercises.
superseded.
13.
Hindu-Arabic
things
big
and
bases.
14.
counting.
Hindu-
The
counting
different
to
two
iTie
16.
system.
Hindu-
The
15.
number-system.
Arabic
of
forms
other
Primitive
11.
Numbers
12.
raphy.
Bibliog-
Arithmetic
op
18
.
20.
Early
history
arithmetic
upon
24.
27.
got
Topical
25.
Exercises.
of
Spiral
in
26.
of
United
the
early
States.
teaching.
Object
Rational
28.
teaching.
object
Hindu
influence
The
system.
How
21.
Europe.
22.
Arithmetic
23.
system.
views
Two
in
schools.
the
arithmetic.
later
Chapter
arithmetic
of
into
method.
Bibliography.
IV.
Subject-Matter
The
28
Arithmetic
op
.
'
29.
Duality
31.
Elimination
duty
teacher's
son
for
matter
subject-matter.
and
by
Many
should
37.
grades.
should
fit
the
in
in
aid
Interest.
season.
in
32.
arithmetic.
curricula.
selecting
38.
subject-matter.
arithmetic.
in
changes
material
Basic
30.
curtailment
regarding
arithmetic
35.
work
of
34.
word
material.
Motivation.
Exercises.
A
of
36.
39.
The
Bea-
33.
tion.
cau-
The
Subject-
Bibliography.
CONTENTS
XU
V.
Chapteb
Socialization
C!orbelation
and
of
Artthmetio
40.
36
Socialization.
class.
fourth
Illustration.
41.
Sources
43.
Correlation
44.
material
of
with
other
illustrative
46.
example.
the
Motivating
teaching
47.
Sources
48.
VI.
Chapter
material.
of
42.
An
for
example
subjects.
of
of
such
45.
An
problems.
in
accounts
Exercises.
the
metic.
socializing arith-
grade
Value
of
arithmetic.
Bibliography.
Problems
45
.
.
of
51^ Solution
good problems.
arithmetic.
in
53. Interpreting
English
Step one;
silent reading highly important.
54. Step two;
in solving problems.
of
exercise
55.
''Casing
judgment
Inverse
idea
off"
57.
of
56.
The
problems.
processes.
arithmetical
58.
jeciprocity.
Step
three; computation.
orderliness.
59.
Short
and
60.
61.
cuts.
Step
Accuracy
preting.
inter62. Visualizing and
four; interpretation of results.
of problems.
63. Formulation
64. Changing
bers
numin textual
Office of difficult problems.
filroblems. 65.
66.
"Fake''
Unmotivated
67.
practical problems.
lems
probExercises.
are
Bibliography.
impractical.
49.
of
Historical.
,
VII.
Chapter
68.
engage
73. Drill
of
drill.
79.
monthly
Number
drill.
A
82.
team
for
game
86.
Foreword.
92.
teacher.
to
for
and
and
class.
team
A
83.
of
scores
''field
cises.
Exer-
tests.
72
of
97.
save
Bibliography.
and
thought
thought
89.
Confusion
of
arithmetic.
Unification.
Later
matter
plan.
How
devices
competition
Value
85.
of
Language
teaching.
Symbols
94.
a
Mode
87.
88.
90.
the
small
a
Skillful
Miscellaneous
arithmetic.
in
Group
drills.
for
77.
Other
78.
individual
Keeping
Approximations.
Bibliography.
VIII.
99.
81.
exercises
subject-matter.
Forms
cards.
drill.
tion.
calcula-
Drill
unify
74.
drill
84.
Chapter
to
brief.
80.
games.
records.
Mental
71.
Drills
72.
for
numbers
69.
drills.
good
activity of afi.
arrangements
in
of
drills.
and
be
periods must
76.
Using
example.
An
meet."
reviews
for
Essentials
70.
75.
61
Drills
Necessity
must
Value
50.
52.
problems.
instances
of
93.
of
and
language
children
91.
Early
with
instances
of
95.
teaching, not textbook
writing.
Waste
98.
assignments.
time.
100. General
suggestions.
Lesson
tions.
frac-
cautions
Some
unification.
of
guished
distin-
be
must
fication.
uni-
tion
Unifica96.
of
son
Lestime.
Exercises.
CONTENTS
Chapter
IX.
Primary
Basis
101.
work.
of
in
requirement
of
Chapter
arithmetic.
minimum
Illustratiye
107.
detail.
Exercises.
Chapter
Addition
XI.
History.
115.
The
teacher's
duty
118.
Advantageous
Short
Austrian
127.
130.
work
in
No
nines."
Other
Order
124.
of
division.
139.
Order
Cheeks.
142.
143.
Number
relations.
145.
Order
of
priority
Checks.
129.
ber
Num-
Division
and
addition.
of
troductory
In136.
Two
138.
140.
forms
division.
Long
in
checking.
labor-saving devices.
signs. Exercises.
Some
144.
plication.
Multi-
132.
observed
be
to
.
multiplication. 134.
division.
of
.113
.
symbolism.
in
work
Cautions
141.
first.
at
Example.
126.
checks.
step. 135. Steps accompanying
Division.
137.
two-digit numbers.
With
Difficulty
tens
subtraction.
Arithmetical
131.
by
binations.
com-
Bibliography.
MuiiTipucATioN
History.
addition
120.
of subtraction.
128.
addition.
in
up the 45
addition.
grouping
of
116.
operations.
Ck"mbinations
taking
of
Exercises.
133.
of
fundamental
117.
Order
125.
the
relations.
...
here.
of
97
Subtraction
four
The
order
method.
XII.
with
113.
3.
Lesson
121.
process.
123. Methods
cuts.
"Casting
Chapter
Descriptive
a
strong
son.
Subsequent les-
109.
1.
Lesson
development
and
Order
119.
"carrying"
122.
88
.
Bibliography.
114.
of
112.
nificance
Sig-
106.
Exercises.
(Continubd)
further
2.
Lesson
111.
class.
108.
minimum
projects.
Arithmetic
lessons.
A
104.
requirement.
Possible
110.
Social
105.
construction
Early
102.
work.
number
early
Primary
X.
work.
number
of
the
84
Arithmetic
early
Forms
103.
XllJ
arithmetical
Bibliography.
Chapter
XIII.
146.
History.
introduction.
150.
151.
Business
Addition
and
numbers
in
The
introduction.
introduction
idea, the
XIV.
of
to
usage
subtraction.
way
followed
be
Weights
with
Division.
157.
159.
objective.
writing
All
What
fractions.
fractions.
Multiplication.
153.
Formal
148.
fractions
to
shorthand
155.
multiplication.
of
numerators.
by comparison
cancellation.
158. Aliquot parts.
know.
Exercises.
Bibliography.
Chapter
129
Factions
Informal
147.
149.
the
After
Common
152.
Mixed
154.
156.
in
Division
one
step-
pupils
should
Measures
and
143
....
160.
162.
History.
Standardization
161.
Choice
of
of
weights
units
and
by
measures.
early
163.
peoples.
Mone-
Xiv
CONTENTS
standards.
tarj
ideas.
166.
should
tal
.
Chapter
170.
Decimal
175.
First
both
on
169.
introduction.
method.
methods.
work.
188.
Advantages.
198.
The
Decimals
Bib-,
Thermometry.
vantages.
Disad185.
problem.
of
the
Development
Exercises.
Bibliography.
Percentage
186
of
cents
per
teacher's
The
of
important.
cent-forms.
per
inverse
XVIII.
Application
plication
Ap-
fundamental
statement
Special
ject.
ob-
193.
percentage.
194.
197.
cents.
The
191.
of
199.
Problems
for
problem.
200.
as
Percentage
unitary analysis.
percentage.
a
common
corruption of unitary analysis.
excessive
analyses. Exercises.
Chapter
to
Exercises.
187.
190.
Introduction.
History.
192.
Teaching the language
phase of the language.
195.
problem.
Variety of form
Per
ductions.
Re-
173
189.
196.
181.
traction.
sub-
179.
Fundamental
184.
XVII.
Chapter
and
System
Metric
186.
Formal
method.
Division.
178.
History.
172.
Addition
176.
Multiplication.
Circulating decimals.
fractions.
182.
common
Approximations.
liography.
183.
fundamen-
Second
174.
180.
XVI.
Four
that
158
177.
Chapter
units
Bibliography.
Informal
173.
Remarks
Fundamental
Isolated
Fractions
171.
History.
introduction.
165.
years.
167.
Reductions.
Exercises.
operations.
XV.
four
measurements.
168.
learned.
be
First
164.
Indirect
socializing
Avoid
201.
Avoid
202.
Percentage
op
199
.
203.
Motivation
brokerage.
205.
208.
through
activities.
Discount.
206.
Profit
and
life
Sick
Exercises.
Chapter
XIX.
benefits
204.
and
Commission
loss.
insurance.
.
and
207.
209.
surance.
In-
nues.
Reve-
Bibliography.
Banking
Interest
and
207
....
211.
ing.
bankbanking.
Learning directly about
214.
213.
212.
Interest.
Interest
Savings banks.
215.
Abbreviated
forms
an
as
application of percentage.
tial
Par216.
of computing
interest.
interest.
217.
Compound
219.
notes.
218.
Discounting
Exchange.
payments.
checks.
220.
Banks
collecting agencies. 221. Travelers'
as
Exercises.
Bibliography.
210.
Chapter
General
XX.
Partnerships
and
Corporations
217
.
222.
Partnerships.
statements.
224.
223.
Treatment
Corporations.
of
the
Stocks
interpretation
and
.
of
bonds.
CONTENTS
225.
Stock
227.
Bonds
stocks
defined.
228.
bonds.
229.
and
XXI.
Chapter
Bills
Bills.
230.
The
232.
literal
literal
ography.
Bibli-
223
Bibliography.
227
Arithmetic
A
of
Problems
in
teaching
teacher.
the
236.
Treatment
for
Reason
233.
duty
equation.
237.
company.
difficulty with
Exercises.
Exercises.
equation.
literal
stock
a
child's
a
Accounts
and
234.
of
Bankruptcy.
Literal
equation.
the
Dramatizing
Cause
Accounts.
231.
XXII.
Chapter
226.
companies.
XV
later
materi"l
as
238.
grades.
the
ducing
Intro-
235.
for
Literal
translating.
solution
of
239.
The
literal
problems.
equations in the
ratio
240.
The
solution
of
with
and
proportion.
equation
with
241.
of
literal
the
equations.
Advantages
problems
translated
fundamental
relations
242. The
equation method.
numbers.
into
without
243.
Problems
sign language.
lustrations.
must
be followed
244. General
by specificquestions. 245. IlExercises.
Bibliography.
237
Graphs
XXIII.
Chapter
Introducing
versus
graphs.
251.
Essentials
of
250.
graphs.
Exercises.
Bibliography.
246.
Why
247.
taught.
continuous-line
Chapter
Ratio
XXIV.
252.
245
....
of
Change
definition
249.
Proportion
and
253.
History.
Common
255.
ture
subject. 248. PicLocating points.
Material
for
problems.
the
notation.
of
source
much
Ratio.
254.
confusion.
256.
Omit
257.
Proportion.
complicated proportion problems.
than
258.
treated
Proportion
equation rather
by the
by
259.
direct
inverse
and
unitary analysis.
Graphing
portions.
pro260.
A
261.
Solving problems by proportion.
262.
Ratios
that
of proportion.
use
plea for the reasonable
263.
seem
equal not necessarily so.
Majorities, pluralities,
Exercises.
Bibliography.
averages.
Mensuration
XXV.
Chapter
History.
264.
essentials.
of
265.
four
Construction
267.
formulas.
First
256
269.
Fundamental
266.
years.
268.
Inductive
work.
proof
for
Suggestions
Exercises.
problems.
Bibliography.Chapter
270.
274.
forms
XXVI.
Powers
History. 271.
**
Carpenters '
that
deserve
and
Powers.
rule''
study.
272.
for
272
Boots
Roots.
square
Exercises.
273.
root.
Square
275.
Bibliography.
root.
Badical
CONTENTS
XVI
Geographical
XXVII.
Chapter
Measurements
and
Time
276.
278
Province
278.
time.
of
The
280.
Exercises.
277.
Introduction
problems.
time.
Standard
281.
279.
of
the
Longitude
International
date
ject.
suband
line.
Bibliography.
XXVIII.
Chapter
study.
the
fundamental
/
Series
285
.
282.
series.
Arithmetical
Exercises.
and
geometrical
Miscellaneous
series.
Bibliography.
283.
Forms
of
2
MODERN
ARITHMETIC
METHODS
these
young
people
high schools.
Some
of these
fault-finding,
but
employ
This
apply
he
of
lack
to
is
ability
of
the
the
subject
important
second, the
life brings
all business
it does
making
lack
of
for
other
the
no
they
the
subject
indeed
in
obtained
International
Report
on
United
States
as
cited
be
less
standing,
Notwith-
criticisms
these
to
that
edly
unconcern-
go
metic
portion of arithif
Or,
allotted
evil that
by
time
to
in
they
good
died.
reme-
fifty large cities
of time
results.
satisfactory
ject
sub-
investigation "
arithmetic.
amount
be
not
can
late
a
the
to
While
to
arithmetic, it is^at the
where
give
this
such
time,
same
instances
Many
have
results
been
time.
Commission
Mathematics
Bureau
devoted
produce
to
cient
ineffi-
plays in
subjects.
time
found
exorbitant
an
an
school
was
in
of
the
on
the
Teaching
Elementary
Education.
:
consideration, they attribute
it
was
total
the
entirely sufficient
*
to
twofold
be
responsible.
are
momentary
a
States
means
important
may
comparison
this branch
other
improve
to
which
hand, it
of
United
the
an
spicuous
con-
more
is often
grades
that
efficiencyto the limited
15.25%
by
in
therefore
attention
no
and, therefore, consider
is
is the
superintendents
and
attempt
no
stop to give it
of
principleswhich
practical importance
in the
of the
paying
way,
teaching
that
to
person
inefficiency into the lime-light,
case
teachers
their
on
On
warranted.
young
criticism
must
part
this
in the
do
not
many
the
mere
inseparable relation
criticism
this
for
reason
;
do
the
of the
first,the teaching of arithmetic
and
of course,
world.
The
as
are,
school
to
its
itself,and
subject
business
in
subjects because
other
teachers
arithmetical
mastered
more
of
part
the
of
the
entirely
are
many
the
on
have
to
and
the
criticisms
general j)ractice the
supposed
with
great
a
from
and
PROBLEMS
AND
of
Mathematics.
Schools
Bulletin,
1911.
of
the
No.
American
United
13
:
17.
States.
3
INTRODUCTION
it
Therefore,
grades, from
the
himself
concern
the
upon
effort
raise
to
the
helping
been
2.
in
clearing
of
**how
of the
arithmetic
situation,
to
put
personal
a
work
to
is for
It
point
a
the
purpose
especially of
and
has
*'how
and
children
of
civilization
Our
questions
involving
much"
know
what
cake,
or
a
to
savings
the
then,
game,
machinery,
necessary
3. The
and
measurements,
to
The
put
to
into
a
necessary
the simplest
complicated
most
computations
and
as
suflScient
From
building of the
the
whose
quantitative
is not
case.
our
three, heap
two,
interest.
in each
and
savage
ingredients
pays
counting
Utilitarian
written
"
mechanic,
bank
crude
and
some
enter
as
a
element.
arithmetic,
need
what
use,
jects
sub-
eliminated.
wholly
these
It
attention
other
the
the
of
many."
a^^mount
to
full
relations
**how
this
more
of
questions,
element, and
to, **One,
are
number
medicine
to know
modem
lives
and
that
thing is
child's
daily
be
much
that
be
limited
is
many."
be
would
number-knowledge
*'how
would
ments,
measure-
the
essential
an
titative
quan-
measurements
curriculum,
music, would
as
of
Without
the
lack
subjects, such
general
of
side
with
answers
much?"
grades would
games
comparisons
with
is the
It deals
Arithmetic
quantitative
of these
Arithmetic
Taught.
grade curriculum.
quantities.
of the
The
this
Is
size, and
many?"
the
said
this stigma
for
arithmetic, by making
up
Arithmetic
Why
and
to
superintendent,
cause
importance.
the
prepared.
with
The
with
inexperienced teacher, that this little volume
the
side
to
its
the
to
the
of the
status
with
aiding
of
connected
one
teacher
removing
teacher
grade
every
primary
in
commensurate
of
behooves
of
the
a
of
**
Value
late
bread
good
of Arithmetic.
the
regarding
and
command
butter"
of
Much
utilitarian
side
of its
arithmetic
by the up-to-date farmer,
has
by the
by
been
value
of
importance.
the
skilled
progressive
pro-
4
MODERN
fessional
AEITHMETIC
is
housewife,
cient
has
also
obvious
so
for
will, however,
depend
how
far the
more
upon
The
be
found
of
problem-solvings.
for
call
from
drawn
its first
in
the
in
training
earliest
through
have
passed
such
inferences
must
receive
subsequent
taught
music
receive
to
or
the
the
extent
But
aid
the
pick
we
It
pupils
as
pupils
for
Just
is true
they
draw
to
arithmetic,
the
great
with
might
should
English
to
ourselves
that
larger
and
they
but
more
is
which
the
concerning
argued
taught, then
lish
Eng-
A
child
altogether.
believe
of
first of
must
they
must
all
also
complete
the
life.
material
the
that
fimction
sary
neces-
baker,
''get along'' through
to
only
schools
the
in
or
thereby
and
butcher,
be
abolished
be
is the
the
is sometimes
as
musician,
painter
that
is not
child
great
it
why
reason
English
teaching
to
no
A
a
metic
arith-
because
reason.
If
arithmetic
bread-winners,
the
that
a
is
value
become
only
force
render
teachers
Life.
returns.
enough
not
can
schools.
which
grades
up
Larger
conversation
on
to
degree
unable
are
not
become
may
restricted
carry
receives
pupils, when
teacher,
may
candlestick-maker,
in
can
be
to
were
he
he
financial
big
be
must
greater
a
the
solely for
that
that
drawing
the
for
studied
be
in
If the
essentiallypractical
should
that
blame.
4. Arithmetic
an
daily
first little problem
the
eight grades,
properly,
the
of
grades.
the
through
with
of judgment
exercise
to
long series
one
Conclusions
continues
and
is
met
are
solution
the
Life
judgment.
This
grades
all the
has
Situations
value
only utilitarian
the
of arithmetic.
given data.
in his vocation.
progresses
is not
but
followed;
vocation
individual
of
arithmetic
of
amount
the
upon
study
exercise
an
the
Much
discussion.
no
practical affairs of life. Thia
these
ability to compute
in the
to need
as
effi-*
by the
and
mercbanty
regarding
conducting
PROBLEMS
AND
successful
said
been
necessary
the
by
man,
METHODS
American
equip
prepare
life; that
their
these
which
5
INTRODUCTION
of
esthetic
will
Space
of Formal
Doctrine
5. The
permit
not
of the
Doctrine
training
in the
study
the
of
comprehension
of mental
establish
to
established
of
number
long
mentioned
the
may
many
of
keenness
keenness
in
her
whether
to
No
can
one
of
mode
in
and
of
that
found
that
other
subject
judgment
to
it offers
degree
a
deciding
of
afforded
so
on.
valuable
a
the
by
grade
for
opportunity
an
have
in cook-
quantities and
possesses
no
girl
a
will
recipes, in
small
or
that
will, however,
arithmetic
in
also
of
by its
it
seasoning
keenness
size
large
for
argued
of
jection
re-
first be
arithmetic
use
the
development
made
be
insight in her
the
for
not
not
can
Before
It must
made
are
modifying
thought
exercise
arithmetic
arithmetical
dispute
curriculum
claims
large
a
of
matter
for.
looked
be
discriminating
food
which
be
can
studied.
the
velopment
de-
sary
neces-
are
theory
any
been
in
Her
buy
namely,
;
general
as
time, during
have
It
the
in
of
disproof
the
and
in the
developed.
serve
of
validity
the
conclusions
Such
or
Arithmetic.
Discipline
subjects and
modifications
supporters.
developing
ing
as
particular subject will aid in
a
instances
study
ardent
most
her
well
as
to
as
concerning
Formal
long
a
theory
that
through
other
after
decided
of this
of
proof
varying
very
of
brawn.
the
only
Discipline
discussion
a
invalidity
that
ethical
the
and
material.
the
or
the
of
cognizance
takes
the
other
no
subject.
Arithmetic
in
a
it have
an
in
the
pupil
his
or
This
has
can
skim
either
over
that
for
right
or
the
and
makes
which
results
it
training it affords.
mental
is unable
because
is
absoluteness
teacher
he
definiteness
a
he
to
each
lesson
a
has
do
he
wrong.
must
It
a
In
the
lesson
secure
has
study
a
some
passed
unsur-
subjects
yet fool either
and
accomplished
with
obtained
of
a
an
desired
himself
results.
arithmetical
definite
ethical
lems,
probwhich
result
value
as
well,
6
MODERN
which
and
a
valuable
the
teaching
below'
children
the
been
grade
be
relative
inculcate
these
difference
the
in
in
ceiving
re-
of
study
opportunities
have, opposed
the
is far
of
advantage
numbers,
of
study
a
more
his
into
portuniti
op-
oxygen
When
you
arithmetic
phlegmatic
foresight,
of
it these
ties
qualito
necessary
done
have
class.
inertia
and
proportions, of
into
it the
of two
"
discrimination,
Breathe
your
and
over.
without
the
what
in
this,
of
Instead
it will
alive
be
finger tips.
the
in
Improvement
furnish
no
the
nevertheless,
vital
given
cautioned
method
which
and
be
never
A
an
Broad
Arithmetic-Teaching.
for
panacea
take
never
7.
is
living, vital subject.
a
of
study
It
breathe
passed
their
children
take
system,
habits.
conclusions.
at
gives of
keenness
be
from
or,
to
a
four.
acquiescence
6.
merely
and
will
you
it
dull
failed
arriving
and
watch
has
is not
the
stubbornly,
too
to
that
mind
either
that
values, of keenness
of
make
in
it
schools
benefits
borne
and
four
and
two,
A
the
teacher
Arithmetic
is
through
advantages
to
to
lightly
go
there
likely, the
and
not
it must
arithmetic
these
surface, is
above-mentioned
the
have
the
and
truth
trutii is
this
thoroughness,
neatness,
care,
simplicity, perception,
industry,
accuracy,
seeing
of
need
of
short
opportunity
the
Moreover,
one.
of absolute
lesson
accepting nothing
of
PROBLEMS
AND
The
emphasized.
necessity
most
If
little
is too
the
METHODS
ARITHMETIC
place
to
obtain
scholarship
the
being
is to rob
the
make
a
extremist
it
or
Knowledge
comprehensive
At
the
knowledge
requisite of good arithmetic
common
led
away
by
this
be
must
or
teaching of arithmetic
Always
can
sense;
time, the
same
joy forever.
a
of
or
and
results, methods
best
consideration.
about
methods
difficulties of teaching,
the
of
Mere
teacher
that
new
of its perplexities,
be
rational
;
faddist.
of
Subject-Matter
of the
Indispensable.
subject-matter
teaching.
It would
is
a
prime
be a^ absurd
7
INTRODUCTION
for
who
one
piano,
for
as
have
must
only this,but
Not
of
the
of
arithmetic
word
he
least
at
Only
geometry.
comprehend
which
their
have
8. Seek
of
workable
well
is
ideas
may
cases
where
later
be
they
constantly
essentials
latter
of
must
method.
a
period,
idea
even
or
have
is to
The
of
form
at
by
if
cannot
be
teacher
constantly
cause
with
compare
future
harmful
will
to
children.
present
teacher
to
past
get
on
into
The
drills
recitation
the
The
ideas.
This
sane,
quiet
in
a
no
revolutionary
the
of
all
such
teacher
all
spirit of the
pupils,
help
contrary,
Above
big
life.
of
new
possible the
the
to
the
details.
in his work.
grow
results.
that
day.
minds
characterize
must
special
advised
so
full
out
the
and
Whenever
with
the
and
the
on
the
These
whether
produce
in
improve
in
is also
of
carried
will,
but
revolution''
not
that
as
course,
ideas.
during
or
trying
ever
a
distinguishing features
time
bewilderment
or
usable
incidental
drills snappy
is
by
prospective
the
big idea
the
end,
other
some
stages
results
the
the
at
teacher
easy
this
student
the
as
In
adaptable
the
with
experimentation,
manner,
be
for
accepted
all such
successful
for
thorough
followed
teaching,
The
alert
beginning,
principles
be
instance, it is immaterial
For
the
at
come
needed.
be
never
of
to
confused
not
are
quately
ade-
he
can
This
should
lookout
the
on
the
of
year
a
mathematical
teaching.
work
modified
are
on
of
later
to be
urged
sense
knowledge
and
'*Big Idea."
the
methods
the
during
teacher
be
in
ject.
sub-
arithmetic.
subject-matter
the
course
as
for
portion
biggest
thorough
algebra
the
try
field of the
preparation
such
impart
in
root
of
to
of any
in the
this
year
by
and
Always
knowledge
a
arithmetic
teacher
successful
be
to
teach
to
expect
of the whole
grasp
supplement
must
by
successful
a
to
of
knowledge
a
The
subject.
of arithmetic
plane
lacking
piano
the
played
never
one
that
teach
to
has
**
the
lution
Evo-
ments
experishould
it behooves
work
first,
8
MODERN
while
ARITHMETIC
student, and
a
later
the
9. Use
this
remaining
book
be
may
be written
for
what
text
is recommended
in ink.
this
to
pick
out
chapter
This
care
and
in
and
future
in
fixing
mind
the
I.
1. What
States?
2.
is the
Wherein
How
To
student
deavor
en-
in
incidentals
in
emphasis
proper
here
as
aid
under
sented
pre-
and
schoolroom,
brought
each.
as
for
up
cussion.
dis-
in elucidating
consideration.
EXERCISES
status
lies the
cause?
should
matter
the
chapter
of greater
be
useful
contained
the
point
the
present
time
much
each
else will
each
in
use
found
material.
the
the
serve
pre-
always
be
let
from
to
should
similar
placing
in
find
to
for
Nothing
in
later
possible illustrate
as
will
big idea
the
grasp
Try
idea
some
far
to
will aid him
teaching.
essentials
the
Any
gestions
Sug-
desirable
and
inspirational value,
for
problems
consultation.
notebooks
and
metic.
arith-
problems.
be
size envelopes
of
the
acquainting the
good
it may
paper-clippings
of
each
while
which
Legal
work
for
for
of
subject-matter
few
a
constitutes
in loose-leaf
kept
preserving
make
contain
eight chapters
phase
the
to
used, primarily,
discussions
and
made
be
with
arithmetic
ing
dur-
each
nature;
topical
some
which
be
to
are
student-teacher
good
will
exercises
These
review.
with
deals
the
general
a
first
The
Chapters.
of
are
reference
No
enthusiasm
spirit and
in
grow
Following
ones
in
to
PROBLEMS
AND
of teaching.
work
of the
of
except
METHODS
be
of
arithmetic
How
can
devoted
each
teaching
it be
in
United
the
remedied?
day to* arithmetic
in
the
grades?
3.
State
subjects
4.
grade
5.
of
Discuss
essential
the
the
standpcHnt.
between
arithmetic
and
the
other
grades.
arithmetic
curriculum
Discuss
differences
the
and
value
as
of
of
illustrating
the
quantitative
side
of
the
life.
arithmetic
from
the
"bread
and
butter
'*
CHAPTER
10.
Why
real,
the
since
teacher
devoted
to
system
and
history
of
will
taken
be
Of
of
any
attention
of
ability
to
various
the
of
form
counting
count
to
each
clothes;
of
of
only
fingers
beyond
other
today
in
ten.
secretly
mien
three,
Most
vised
ad-
are
mentioned
by
the
the
of
touching
two
to
men
are
is
needed
communicated
of
hands
the
but
many;
counting
this
lowest
the
limited
five, and
board-of-trade
10
only
be
four,
merchants
Early
The
accomplishment
an
may
tribes
many
on
subject
possess
ability
nevertheless.
and
that
today
two,
one,
is
extent,
This
civilization.
the
Number-Reckoning.
and
limited
a
tribes
counting,
the
on
to
of
complete
histories
the
brief.
very
more
a
the
each.
17.
p.
Counting
count,
of
engaging
are
in
of
topics
discussion
a
which
of
some
be
must
into
phase
The
teaching
interested
historical
list,
separate
enter
questions
further
Primitive
to
Those
any
various
the
be
number-
our
arithmetic.
references
historical
mooted
of
of
with
impossible
appended
11.
is
be
the
connection
these
consult
to
it
in
historians.
development
of
of
many
of
the
development
will
chapters
two
to
quite
manner
a
evolution
the
development
up
will
the
of
general
the
the
it
Therefore,
race.
consideration
a
necessity
Hence
the
the
especially
in
develops
more
through
appeals
study
a
becomes
apparent
more
individual
the
of
that
to
Such
history.
its
science
Any
value
its
NUMERATION
AND
History.
and
vital,
of
similar
the
Study
more
study
in
NOTATION
HISTORY"
EARLY
n
done
to
prices
their
under
signal
to
each
EARLY
by
other
of
digits.
has
of
means
Further,
conclusively
Arabic
in
except
found
been
that
base
of
Numbers
12.
that
ten
five
base
a
as
be
being absent,
to
numbers
the
is five
100
express
as
we
now
fives,would
five, twenty-five
we
small
Its
to
use
now
for
necessitate
two
20.
express
Twenty
twenties,
expressed by
been
as
base, but
a
about
13.
a
It has
100.
have
ev^n
also
so
after
Tallying
notches
record
in
One
the
cut
to
hundreds,
upon
each
a
of
and
as
five is too
is too
large.
be
be
it would
that
selected
been
had
to
would
hundred,
recognized
used
to
bring
ages.
of
Counting.
similar
the
so
units,
a
was
interested
into
were
split in order
parties.
ments
instruof
recording
second
Accounts
on.
Pebbles,
crude
and
developed
was
stick which
the
four
sticks,and
counted
man
by
1000, which
by
10, would
early counting
the
20.
twenty-
futility of trying
Forms
Pinger-eounting
sj^tem.
third
in
used
numbers.
cut
be
symbols for 10, 11,
if twelve
the
Oth^
sticks,nuts, notches
were
has
all these
and
or
long been
fortunate
more
change
figures, or
9
written
hundred
base, twenty
19, and
etc., through
be
expressed
Just
additional
ten
be
One
tens.
number
of
10, would
would
ten
expressed
hundred
one
convenient
a
would
use
state
would
then
tens.
be
With
consist
would
for
fives,would
ten
fortunate
symbols 6, 7, 8, and
symbol
the
Twenty-five, which
just
The
the
as
twenty.
or
they represent
14, while
11, 12, 13, and
ten
figures,or
Two
five.
express
founded
was
It is quite
number-system
0.
quite
mystery.
place of five
1, 2, 3, 4, and
required
great
a
base,
or
showing
origin of
the
called
are
counting
Bases.
the
present
our
only the symbols
then
in
selected
was
of
form
Different
to
numerals
radix,
the
twenty,
or
is not
number-system
our
the
even
cases,
Hence
finger-reckoning.
upon
two
earliest
the
11
NUMERATION
number-system
five, ten,
be
to
AND
finger language;
a
Hindu-
universal
the
NOTATION
HISTOBT"
From
regular
a
the
tens,
a
kept by
to
give
this
we
a
12
MODEBN
have
ARITHMETIC
counting
the
upon
tablets.
times
The
recording
before
numerals
used
are
The
' '
Two
of the
symbols
simple; V, of
C
while
which
gives
of
the
to
express
in the
system
Arabic
each
the
well
are
today
for
separate
of
few
They
the
so
other
about
had
the
nine
All
on.
its
sixth
the
for
ten
symbol
such
origin in words
century
and
a.
d.
is
Japanese
They
have
together
and
so
on,
page.
and
that
for
hundred
for four
number-systems
computation
7,
principle
thousand,
for
by
and
numerals
opposite
symbol
with
hundred
place value
the
on
forty with
it, and
first
several
twenty
system.
a
used
was
expressed
was
Chinese
ten, hundred,
nearly useless
course
of
such
coming
of
this additive
from
tion
posi-
one
the
value
represents
The
shown
are
it ; five
five beneath
for
Until
?
1000.
and
in
value
the
5
as
of place value
idea
forty
upon
and
simple
100
that
system.
zero
ones
so
which
away
the
which
write
beneath
idea
Roman
breaking
it took
based
system
known
characters
in
is the
form.
especially
special character
a
or
together
fifty by ?^?,
Another
with
number,
added
and
ten.
system
in
System.
as
easily than
the
Like
not
are
fully
is
of the
the letters
Hindu-Arabic
in another.
have
number-
words.
hundred-fold
a
Arabic
These
greatly changed
more
or
it would
Hindu-
characters
9?
digit ten-
a
as
form
notation,
made
are
what
over
the
M
and
ing
Count-
Hindu-
have
first big thing of this system
The
of
Roman
the
early
extensively
on
The
just
Arabic
Hindu-
coins.
and
to
Big Things"
upon
those
used
number-symbols
letters, the
15.
are
of
alphabet.
numbers
alphabet
common
are
number-system.
present
form
to
forms
strewn
carried
thus
arithmetical
the
sand
tablets
were
our
in
Number-System.
compose
symbols
a
of
Later
records
stone
numbers
Hindu-Arabic
more
writing
and
PROBLEMS
tally."
only permanent
invention
the
14.
The
abacus
inscriptions upon
were
and
AND
expression **keepiiig a
the
stone
METHODS
purposes.
written
The
one
second
and
are
The
after
biflf
EARLY
HISTOBY"
thing was
These
invention
the
ideas
two
numbers
zero,
by
out
The
million
Italians in the
completed
was
Lyons, France,
who
The
on.
use
present
for
of arithmetic
because
16.
The
of
The
by
added
the
terms
makes
in
the
now
it the
the
comma
of
four
the
adoption of
the
being introduced
great world
different
is used
Nicolas
tion
numera-
of
Chuquet,
billion,trillion,and
has
years.
retarded
usable
World's
Hinduinto
the
are
decimal
The
a
science
long time
number-system.
Arabic
both
very
for
in universal
been
Counting
System.
number-system
China
counting-system.
countries
for
by the
present
our
hundred
simple and
a
needed.
introduced
simple number-system
previously been
lack
carried
not
was
the
NUMERALS
Hindu-Arabic,
universal
^it is
"
the
and
1520
only about
had
zero
was
about
of
computations
the
century
CHINESE
so
and
simple
invention
the
later.
two
or
present
our
Before
(great thousand)
fourteenth
century
a
in which
abacus
13
NUMERATION
constitute
expressed
be
of the
term,
zero
integers.
could
means
of the
practically
for
number-system
AND
NOTATION
and
Japan
Variations
slight.
point while
In
in
"
found
Germany
England
14
MODEEN
it is the
ARITHMETIC
dot
2*5.
thus,
placed higher
In
southern
large numbers
while
in
periods
would
be
and
of this
the
Hindu
in
the
The
to
the
were
the
not
was
the
eclipsed
of
hands
the
developed.
writing
or
The
system
learned
with
historians
extending
symbols
years.
the
develop
the
often
further
left for later
of
perts
ex-
is
As
into
passed
they
tem.
Sys-
themselves
wholly
soon
were
whom
into
came
spread it
the latter and
very
simplicity
simple number-system
purpose
knowledge
also
this fact
Arabic
wrote
the
idea
with
contact
the
India, tliey naturally adopted
of
tion
adop-
Hindu-Arabic
computations.
hundred
Arabs, through
disseminated,
was
for its
in
fact
also
that
retarded
the
its
much.
very
Arabs
world-wide
a
of
this
development
not
difference
considered
class
several
States.
needs.
whoili
by
class did
the
From
digits,
separated
are
marked
Such
their
records
meager
period of
progress
The
States
three
in United
as
any
began
out
merchant
trace
same
"
numerals
class
the
educated
As
they
speaks volumes
secondary
one.
class,very
a
United
the
periods of
Distribution
and
carrying
merchant
which
States,
23,567,486,356,892,139,504
astronomers
astronomy
Because
United
139504
universal
The
primary
over
to
Origin and
case,
the
PROBLEMS
the
Europe
forms.
number-system
The
aid
into
9, shows
written
adaptability
in
to
in
that
million
"
character, the
one
17.
and
356892,
'billion
"
printed
and
in
northern
digits so
567486,
trillion
the
than
Europe
and
of six
AND
read
23,
Only
up
separated
are
England
into
METHODS
to other
treatises
gained
origin, although
properly acknowledged
the
of the
times
of
merchants
the
simple number-system
parts of the civilized
on
this
of
world.
notation.
the
notation
early Arabic
writers
currency
the
simple
was
invention
that
as
coming
from
HISTOBY"
EABLY
the Hindus.
inventors
used
were
before
18. Notation
general
and
the
middle
this line
is done
however,
confuse
not
expect this development
a
teacher
the
on
is in like
and
attention
must
the
of the
of its
child's
the
growth;
as
work
best
with
excellent
early
a
along
The
teacher,
accidental, and
without
accord
own
A
forms
The
hence
as
show
to
will
money
Numbers
be
must
be
must
also
given
up
in the
be
is
initiative by
cautioned
telling
after the
child
he should
be
has
him
too
been
permitted
against
told
to work
much
the
out
be
ber.
num-
always
zero
destroying
numbers
the others
in
give
thoroughly,
consideration
directly.
used
helpful
20, 200, 304, 460, 7000, 6800, 4060, 3007,
teacher
of
each
written
found
make
ideas
can
slowly and
especial
order
which
chalk,
containing
taken
in
the
in
444,
as
race
Much
toothpicks,
as
the
the
this matter
Sticks,
of
child.
the
to
explaining place value
States
which
among
difficult to
to
number
ment
develop-
this
closely that
differently'colored
in
teaching place value.
difficultyand
that
proved
material
illustrative
in
United
follows
child.
the
to
figure is written
advantage
child
mind
stumblingblock
a
hundreds.
and
in
therefore, be given
must,
it home
bear
of place value
manner
bring
tens
ever
of
part
idea
The
race.
to
his
a
counting,
to
as
episodically.
happen
chapters
Let
in
incidental
to
century.
directing hand.
The
to
Some
of north-
children, only
given
need
a
his games.
of
numbers.
incidentally, or
must
be
during
separate
work
of
as
come
fifteenth
As
here
writing
and
of
part
by those
early number
development
necessary
all
of the
no
numerals
Hindu
merchants
at
in
and
Italian
hardly
suggestions will
reading
number
the
Numeration.
the
to
The
system.
but
the
and
devoted
few
the
century,
Europe
will be
of
distributers
the
were
15
NUMERATION
AND
extensively by
very
the thirteenth
em
Arabs
The
the
sense
NOTATION
such
etc.
child's
the
For
instance,
fifteen-,
through
of the
**
teens"
16
MODEBN
himself.
by
The
reasoning
exercise
to
it.
for
the
best
in
child
the
of
notation
and
at
is
in
numbers
clocks.
some
Number-symbols
in
such
a
symbols
Roman
letters
been
notation, will
notation.
LION,
lOOOW
Thus
and
so
found
be
Why
to
and
special
after
the
the
on
tion.
atten-
other
numberthe
through
letters
in
teaching this
50I0N
becomes
on.
EXERCISES
should
teachers
the
study
of
history
subject they
any
are
teachf
2.
Describe
3.
What
Hindu-
the
the
is the
Suggest
5.
Make
a
f
system
counting
of
radix
or
Why
"f
better
of
the
numbers.
ajj^drecording
number-system
a
is
What
is it for
What
f
likely origin
most
the
are
are
the
using
numerals
universality
two
5
better
as
a
of
this
big
called
of
that
with
tilings
the
in
the
f
base:
multiplication using
Hindu
its
Compare
What
addition
f oUowing
the
is this
Why
base.
following
the
Make
401.
7.
base
Arabic
4.
6.
methods
early
f
system
+
Arabic
the
after
one
COW,
becomes
II.
1.
as
Roman
The
by
stimulus
a
to
on.
chapter headings
into
translated
given
be
so
replaced
words,
out
reasoning
"
arranged
spell
to
as
have
also
not, therefore, be given
and
manner
of the
tunity
oppor-
^including spelling
"
in
A
such
Superseded.
mainly
It need
is there
must
being
the
hundreds.
fifty-six,and
Being
is found
present
words
more
the
cultivation
a
hundred
and
more
to
for
by stimulating him
study
Attention
Notation
Roman
for
developed
other
no
forty, thirty-five,two
19.
be
PROBLEMS
is recommended
also
on;
arithmetic.
in
writing
so
can
helping
as
process
and
And
AND
procedure
same
twenties, thirties,
child's
METHODS
ARITHMETIC
5
as
313
a
2^1
+
base
231
:
''Arithmetical
the
Hindu
literal
+
1^
14.
X
bet
alpha-
alphabet.
number-system?
Illustrate.
8.
Compare
9.
Give
called
Arabic
a
the
brief
f
Homan
and
history
of
the
the
Hindu
Hindu
number-systems.
number-system.
Why
is
it
CHAPTER
DEVELOPMENT
THE
20.
Early
of
adoption
from
and
not
the
learned
the
learned
did
in
class
Greece,
wholly
either
the
;
hence
of
idea
world
The
abacus
the
be
not
merely
in
of
vacant
after
the
invention
zero
dust
was
strewn
a
expressed
antiquated
had
passed
Hindu
complete
first
used
upon
in
away
stone
18
the
with
tablets.
d.
could
by
for
10
's.
persisted
invention
ing
contain-
number-system
computation
small
with
407
abacus
abacus
the
custom
a.
Japan.
and
that
column,
bol
zero-sym-
1100
to
the
on
the
ized
civil-
the
the
like
number
that
in
China
in
putation
Com-
time
400
from
second
the
its usefulness
The
zero
be
could
another
zero.
the
the
with
extensively
of
Schools.
the
numbers
time
some
used
leaving
long
of
philosophical
known
universally
they
as
extent.
from
general
was
adopted,
still
into
of
development
the
great
any
Oot
writing
the
but
many
the
is
to
became
written
Like
of
value
finally
Before
abacus
until
was
art
in
putations
com-
those
the
to
aid
not
Arithmetic
the
place
the
or
Hindu
with
did
in
arithmetic,
in
themselves
they
merchant,
the
Wherever
interested
was
interested
mainly
was
devoted
they
that
matters.
became
science
How
21.
class,
fact
the
to
arithmetical
and
side
due
the
after
development
the
numbe:^'-system
new
satisfactory
the
centuries
several
metic
arith-
required
labor
great
the
to
in
progress
the
to
For
computations.
making
far
owing
Prior
Europe.
number-system,
slow
extremely
was
in
Arithmetic
of
Hindu
the
ARITHMETIC
OF
History
of
invention
in
in
figures
The
ten
writ-
school
of
THE
DEVELOPMENT
using these
men
dust
algorithmic, from
For
tended
the
for
the
and
used
was
a
the
of these
took
salts.
the
A
merchants'
eon-
give
to
of
astrological
and
cities
in
the
to
way
algorists.
tables
aided
in
TOBM
01*
did
records.
THE
their
as
the
The
use
church
ABACUS
little
very
calculating
tween
period of rivalry be-
legislate agaiust the
business
knowledge,
in
this
during
have
to
XAKLT
revival
of
study;
this
potent
more
at
time
this
only
correct
they
in
which
schools
subjects
the
learning,
to
in
interest
time
of
the
mote
proputation
com-
yearly-
was
productive
which
encouraged
arithmetic
taught.
These
of
and
excellent
felt
was
recognized
the
institutitms
higher
influence, however,
o^anizationa,
arithmetic, and
principal
method
arithmetic
Easter.
the
up
had
monasteries
lay
With
aiacists
numerals
arithmetical
of
the
scientific
methods
new
coming
until
thing
common
and
of
algebra.
on
algoristic numerals.
AN
schools
schools
called
was
writer
Arabic
an
two
almanacs
two
numerals
these
more
establishing the
It
of
19
ABITHMETIC
gobar)
or
name
supremacy
simpler
Widely
(gubar
centuries
several
OF
the
establishment
accounting
special schools
re-
in
value
the
of
of counting
were
for
the
teach-
20
MODEBN
ARITHMETIC
ing counting
arithmetic
The
until
introduced
more
in the
included
steps until
The
22.
prevailed
was
work
the
Influence
designed,
material
much
useful
from
one
much
of which
on
table
the
through
written
writers
arithmetic
The
issued
with
the
are
Riese,
TnjJOH,
and
(Nicolo
Simon
period
that
in any
the
to
way
rithmic
algo-
first English
The
in
rare
connected
1522
and
nently
promi-
most
Gerbert
:
Pisa
op
Christopp
John
(FibonRudolfp,
Napier,
States.
P^talozzi,
in
incident
school.
The
as
1788
At
to
^.n
the
find
teacher
two
did
books
was
Pike's
used
were
of
of
the
United
lished
pub-
mostly
were
the
conduct
nial
colo-
Complete
very
copies
not
in the
arithmetic
and
first,texts
the
adoption
taught
was
1729
During
the
after
Although
early
as
publications.
one
gave
Leonardo
United
poorly.
appeared
a
largely treatises
ToNSTALii
little arithmetic
Boston
Arithmetic
was
activities.
early arithmetic
first years
the
and
States, and
it
were
Fontana),
Stbvin,
in the
constitution, very
English
d.
it had
Grxtbe.
Arithmetic
in
commercial
men
of
development
naci), Tabtaglia
23.
of the
some
Sylvester II), Boethius,
Adam
interest, etc.,
in
by Guthbert
volved
in-
measures
long after
numerals.
dust
as
terms,
and
of
books
later
time
in Latin.
following
(Pope
their
and
was
written
was
abacus, which
of the
use
of weights
a,
The
contained
large
rules
The
that
it at
with
various
1000
easy
mind."
schools
studying
changes
about
by
lems
puzzle-prob-
the
quicken
i"ersistedin the later
obsolete
books
**
commercial
another,
to
grades.
Arithmetic.
complicated
to
those
to
the
into
increased
Later
many
the
when
ago
prevail.
exchange, reduction
money
The
been
Early upon
for
century
extensively
declared,
was
PROBLEMS
a
partnerships, fractions
complicated
become
of
written
works
about
grades has
contained
it
AND
conditions
present
early arithmetics
later
METHODS
scarce
and
same
text
recitaticms
THE
DEVELOPMENT
arithmetic,
in
''set
but
individually.
out
manuscript-book
the
pupil
its time
than
was
success
took
child
of the
his
writing
and
life of
schools.
Rules
related
for
been
In
of
fact
that
of
the
cording
Ac-
him
chief
formulated
the
the
as
from
each
other;
puzzles only
Special
texts.
unfortunately
as
ceas
pana-
have
they
considered
be
the
or
methods
offered
were
will
basis
of affairs is still
state
which
teenth
nine-
subjects of
conditions,
this
these
of
part
the
arithmetic
to
not
under
paragraphs.
Topical System.
of the
is treated
as
application
or
reference
Such
been
used
topic
any
from
such
the
as
interest, is considered
comprehensively
As
of arithmetic
teaching
it is studied
pupils will permit.
accidental.
The
topical plan has
time, when
'*
given
of
day
time
to
Some
its extreme
one
His
guided
taught
commercial
ills but
numbers,
at
of
desires
long, complicated
present
our
arithmetical
separate
form
he
latter
were
often
time
panaceas.
24.
the
Unfortunately
from
arisen
ahead
material.
problem
was
one
existing
to
quite prevalent in
have
7n-
since.
the
to
rigidly.separated
were
were
child.
the
in
become
problems
slightly
his
children
the
it had
texts; topics
the
of
yeiars, until
the
grammar
of the
appeared
measure
increasing attention
century
the
farther
impulses and
natural
his
book.
Constantly
throughout
in large
selection
in the
has
vate
pri-
duty of the
published
was
that
been
due
his statement,
to
in
doubt
no
1821, which
has
text
any
account
normal
in
his
examples, but
the
was
Colburn
Warren
Arithmetic
it
work
to
from
the
only
not
pupils
selected
he
well, which
as
reproduce.
to
tellectual
he
solutions
complete
examples
containing
his
for
examples"
These
21
ABITHMETIC
OP
soon
for
as
as
to
it thereafter
a
plan excludes
a
considerable
the
a
by
a
first.
very
nominate
fractions, de-
exhaustively
period
understanding
so-called topic is
will
some
be
slight and
mature
and
of
the
**
pleted
com-
wholly
treatment
of
22
subject taken
any
is deferred
topical plan
later
to the
certainly
early
PROBLEMS
and
in the
to hold
the
direction
extreme
the
to
concentrate
give methodical
not
cludes,
ex-
topic material
any
It is well
conserve
also
and
years^
of
use
years.
sufficiently to
energies of the pupils and
study, but
the
years,
AND
the
during
up
these
during
which
METHODS
ARITHMETIC
MODEBN
their
to
heretofore
measure
mentioned.
25.
Spiral Systooi.
of the
topical system,
times
during
the
mensuration
in
the
and
later
of various
numbers
instance,
thoroughly
be
considered.
10
during
found
a
secure
for any
time.
treated
of
good
In
the
regularly
rotation
results.
to
up
a
tion.
construc-
the
up
the
taken
12, should
as
up
to
second
the
trial in
have
could
13
numbers
during
Germany,
or
more
appeared
from
these, a topic will be
fifty or
sixty
pages.
will
be
The
propriety of calling into
found
For
up.
texts, based
of
in its
tions,
factors, combina-
to
Several
views
known
pupil before
100
spiral
radical
such
thorough
extreme
every
their
be
be
various
spiral system,
most
volumes
very
confined
was
unsatisfactory.
the
and
with
number
the
by
given
was
had
number
one
and
the
general.
more
should
all the
work
upon
each
number
first year
theory
extremely
that
that
The
found
He
man.
next
in
solids, and
in connection
known
one
of
areas
complicated, the problems
conditions
comprehended
the
This
method
the
operations
be
to
finding
in
next
stance,
in-
first grade
very
of surfaces
more
or
insisted
believed
he
and
time
other
subject and
was
the
eral
sev-
For
principle, is brought
or
the
widely
any
entirety before
less
larger,
are
than
year.
work.
grade
rectangular
topic,
a
ifemore
system
of
areas
site
oppo-
topic is studied
each
in
lengths,
of
difficult,and
Grube
but
years
finding the
time
used
more
the
eight
just the
geometrical figures, together with
Each
on
that
volumes
in
grades
in
of
is
spiral system
is considered
measurement
rectangles
are
The
far
too
Such
mechanical
to
use
a
DEVELOPMENT
THE
principle
operation previously learned, whenever
or
clearly recognized.
arises, is, however,
of
combination
trying
are
23
ABITHMETIC
OF
these
will
embody,
to
plans which
two
rational
a
of
texts
most
doubt
no
So
far
by
prove
sion
occa-
today
the
most
satisfactory procedure.
26.
Object
of object
an
extremist
the
more
of
teaching,
and
appealing,
who
those
than
extremist.
ardent
had
it
carried
the
used
It
Speer
using it; and
not
much
far
as
was
found,
method
were
hence, it has
pupil
a
and
further,
might
as
teaching
Tillich,
pupils.
not
was
his
make
to
object teaching
Speeb, of this country,
pupils
his
to
He
it today.
objects only
nator
origi-
really the
was
know
we
used
Pestalozzi, carried
an
as
and
real
of such
Pestalozzi
Teaching.
be
expected
that
however,
less advanced
versally
uni-
almost
been
discarded.
Two
27.
two
of
ways
is to
Views
base
a
measuring,
for
on,
looking
the
and
speak,
idea
second
the
use
of
will connect
an
of the
value
be
and
motivating
a
The
no
is the
in the
In
necessary
that
that
one
addition
to
will be
in
Difficult
solved.
of
solution
of sketches
use
to be
discontinuance
longer
against
so
and
to
so
work
the
of good
productive
illustration
or
drawings
simple by this
made
been
one
blocks,
as
clarifying
of objects throughout
use
over-estimated.
problem
objects, such
warned
graphic representation, w^hich
of
objects,
jxermanently with
second
as
concrete
and
not
is hence
object
idea
later, there
are
of
one
teaching;
of
use
teally
are
paper-folding, paper-cutting,
teacher
arithmetic, such
hardly
make
to
There
of object
upon
noted, the first has
and
results
of numbers
purpose
As
Teaching.
this matter
at
jjra wings,
the
in hand.
Object
of
such
tinued
con-
a
children
the
object. The
work
the
in
problems,
drawing
to
problems
scale
in
considered
picturing
can
tail
de-
tions
condi-
the
often
are
means.
of the
is
use
urged
of
objects
just
as
as
soon
strongly.
as
A
they
crutch
^
24
is noti meant
be
to
most
mensuration
and
fully
Such
a
based
upon
the
of
use
28.
the
text
and
at
the
formula
for
arrives
arrived
and
of
part
it
the
is
one
far
as
and
how
far
the
much
the
It is
so.
older
a
the
working^
there
for
a
has
;
to
themselves.
of the
Each
pupil
different
model,
of the formula
much
very
is grave
to
mend
com-
of
danger
of discrimination
deal
great
a
rules
development
with
and
children
the
truthfulness
the
was
these
triangle.
a
in which
to
teacher
be
doing,
over-
the
on
far
how
to
that
much
the
gain
more
younger
to
rationalize.
The
;
to
extent
of
any
particular class.
all of
the
the
can
pupils
processes
the
have
of
will
all
majority
course
pupils.
depend
be
of
to
largely
grasp
the
of the
done
do,
far
how
will
members
expect
process
art
wn
ahUMy
the
the
as
carried
be
may
inquire into
to
studied
far to carry
ahility to understand
decided
with
all times
at
should
work
science; how
individual
Again,
than
procedure
of
reasoning underiying
most
of
which
rationalization
is too
do
a
habituate, and
the
from
teacher
explain
to
stimulate
area
of the
part
any
far
upon
grade.
work
the
by
to
method
a
for the
how
which
first
teacher.
and
of
the
requires
It is necessary
how
duced
intro-
arithmetic
down
mentioned
be
result
same
it is
be
instruction
arithmetic
essayed
methods
may
This
at.
it,but
to
laid
rules
establishing for himself
thus
is
tions
transac-
to
in the very
be
This
grade.
grades, needs
earliest
The
finding
the
at
is to
work
topic, as business
insure
endeavor
and
illustration
an
passed
principle of proceeding
teacher
we
rules
it has
abstract.
arbitrary
formulate
will
MethocL
present
advancing
counting
as
for
objective
in the last
the
later
;
new
fundamental
to
upon
Each
objects
the
Bational
based
each
PROBLEMS
need
the
with
objectively
as
concrete
As
idea.
erroneous
AND
that
hold
(gradually discontinued
a
after
used
teachers
Many
away.
METHODS
ABITHMETIC
MODERN
with
It
all
time.
class
the
26
MODEBN
11.
DistingiiiBh
12.
Diflcuss
Illustrate
14.
A
drawing
a
John
16.
riding
in
hour.
How
the
far
of
giving
will
the
they
be
object teaching.
to
a
6"
of
the
of
rate
of
3
and
In
f
hours
5
into
drawing.
bicycles,
on
miles
15
the
lead
a
time
same
12
If
water.
with
Illustrate
at
Show
ascertained.
irregular lump
an
in
class.
it is wide.
as
contains
A
at
in
be
can
leadf
of
from
directions
least
at
three
each
per
Solve
f
hours
four
other
which
in
above
suitable
problems
drawings
for
objects
or
work
class
will
aid
in
solution.
Discuss
18.
dimensions
lump
long
as
^'' diameter
start
apart
a
clear
a
times
the
arising
tioned.
men-
drawing.
the
to
of
Henry
opposite
use
similar
its
and
is four
by immersing
7%''
volume
Suggest
17.
of
object teaching
presentation
objects
its
PROBLEMS
of
dangers
in
acres
simply
to
is the
it, what
320
how
is raised
water
use
glass cylinder
A
15.
by
of
of
AND
forms
two
and
advantages
the
farm
the
between
the
13.
by
METHODS
ABITHMETiC
rational
the
method
of
its
teaching;
and
advantages
dangers.
19.
for
20.
21.
of
use
How
selff
should
various
the
Explain
the
by
far
How
a
work
f
grades
how
fully
far
the
habituated
be
far
how
and
ized
rational-
Illustrate.
should
subject
a
introduced
be
inductively
illustration.
concrete
should
pupil
each
required
be
to
do
the
work
for
him-
to
take
up
their
Illustrate.
22.
Illustrate
arithmetic
work
how
the
do
they
as
lead
would
you
pupils
your
difficulties
which
arise
in
their
play.
BIBLIOGRAPHY
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27
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Spelling
XIY:
Lambert
Arithmetic.
14.
8.
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Century
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OF
DEVELOPMENT
THE
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Journal
of
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Teaching
pp.
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of
Arithmetic.
Primary
42-68.
on
the
Teaching
of
Math.
U.
S.
CHAPTER
Duality
arithmetic
composed
of
basic
element,
first, the
The
future
and
until
and
of
textbook
comprises
and
but
later
instruction
object
pupils
in
as
fractions,
these
addition
useful
used
This
applied
to
relations,
;
the
the
it should
information,
four
of
that
dealing
vast
with
28
value
the
tic
arithmeto
language
various
forms
principles
of
interesting
quantitative
the
of
of
derlying
un-
ments;
; measure-
principles.
these
store
tials
essen-
object.
of
operations
all
his
of
between
the
the
of
more
the
its
of
tinct;
dis-
recognizes
matter
various
fundamental
application
include
and
the
and
and
this
include
integers
quite
are
who
secure
because
must
studied.
is
lesson,
basic
The
problems,
games,
between
each
to
primarily
life.
number
and
motive
selected
after
arithmetic
of
element
eflfectiveness
the
metic
arith-
tion
atten-
second
more
distinguish
parts
the
all
teacher
The
for
understood
elements
partake
Subject-Matter.
be
to
the
and
Basic
will
he
of
material
two
increase
transitory
the
and
is
will
of
basic
material.
because
form
the
problems
divisions
two
30.
the
The
these
years
basic
of
nature
real
earlier
the
teachers.
useful
acquire
to
commonly
is
the
which
needs
nearly
in
of
mass
receiving
and
agencies
through
on,
During
these
the
all
so
the
writers
elements.
subject-matter
what
been
has
that
child
the
is
this
to,
recently
includes
the
distinct
less
or
of
subject-matter
The
more
When
use.
referred
is
two
which
knowledge
quantitative
present
Subject-Iffatter.
of
is
ARITHMETIC
OP
SUBJECT-MATTER
THE
29.
IV
In
and
conditions,
THE
which
SUBJECT-MATTER
does
keeping
laws,
loaning
in
changes
has
matter
modern
considered
of
finding
curtailed
modified:
and
numbers
with
together
proportion
to
be
being
of
as
weighing
one
fractions
problems
too
foreign
Many
business
to
fear
The
Modifications
are
much
teacher's
the
duty
various
outside
liable
more
to
has
for
is also
unreasonable
elephants
or
involved
lems,
prob-
pupils
the
the
of
to
keep
before
from
esses
; proc-
pace
with
these
do
because
of
usual
custom.
the
in
decrease.
such
changes
anticipate
appear
in
our
metic.
Arith-
in
arithmetics
our
than
to
even
persisting
Changes
material
increase
and
simply
passed
away
still
are
Regarding
Duty
textbooks
problems
following
large denominators;
or
conditions
break
to
Teacher's
over
practice of today.
usefulness
and
the
long
mature
number
and
their
immobility
too
or
ratio
ning
day, is just begin-
sixty pounds,
uncommon,
technical
topics
long after
with
weight,
proportion
involving
fifty pounds;
hundred
of
tables
partnership.
as
problems
weighing
;
greatly
troy
as
of
its
had
such
nature
less:
books
numbers,
32.
and
about
been
reductions;
many
progressions
;
general
a
less
used
as
has
ess
proc-
problems
root;
used,
superiority
the
been
Euclidean
have
the
have
ago
cube
seldom
are
(although
recognized)
Material
of
requiring
analysis, which
unitary
topics which
topics
extraction
problems
subject-
factor, intricate
Other
that
the
to
Several
decades
common
discount.
Due
since
alligation,duodecimals,
highest
the
true
denominate
and
:
few
a
on.
modifications
began.
important
eliminated
exchange,
on
arithmetic
of
so
of
living, this part
tal
pos-
banking^
in Arithmetic.
continuous
undergone
study
wholly
of
of
and
procedure,
as
plans,
elements
Curtailment
custom
subject^such
architects'
of
accounts,
business
and
our
grade
reading
general
money,
other
any
simple
Elimination
were
in
of thermometers,
reading
31.
arise
not
29
ARITHMETIC
OP
the
It
as
is the
arise
them
texts.
in
with
A
30
has
fitting
there
as
in
used
is
simpler
and
better
adopting
about
adopting
it becomes
always
in the
as
be
may
cited
excavating,
33.
Reason
main
for
should
at
in
Much
that
a
child
is in
In
the
processes
of
the
tables
*
8ee
Sec.
classes
use
standing
118.
of
is
up
aid
material
have
only.
of the
addition
actually
and
a
found
For
at
them
to
quaint
ac-
in
the
placed
and
in
or
arithmetic
there
for
use
only time
multiplication
class period.
during
This
The
quantitative
the
multiplication
hindrance.
the
ulum
curric-
present
instance, the
addition
recite
to
from
been
order
of
cost
necessary
arithmetic
child,
the
examples
Curricula.
in
in the
activities
to
seems
makes
tables
his
of the
present
arithmetic
the
Material
it will
that
consider
to
standpoint.
courses
Arithmetic
well
methods.
day
present
be
to
as
of the
in
used
text
including material
be
future,
the
the
the
own
As
If
papering.
and
convert
teacher
computation
are
measures,
to
method.
and
man's
from
with
for
reason
notation
and
in his
the
they
as
the
improvements
where
backward
teachers
It behooves
business
pupils
method,
man
adopt
are
men
arithmetic
and
to
of weights
system
for
deviate
business
the
method.
plastering,
the
teachers
the
man's
the
should
teacher
of the
serviceable
of
lookout
:
duty
case
duty
business
arithmetic,
metric
better
the
on
of the
most
the
the
the
to
the
business
the
that
than
In
of
practice
the
it is the
case
about
them
that
methods.
business
teach
for others
discarded
be
this
is entirely
should
schools
the
that
fact
criticism
later
must
of
Much
the
to
the
why
reason
no
occurs
is the
such
is due
into
life.
actual
often
It
is
of arithmetic
practice.
teaching
practices which
arithmetic
able
receiving consider-
sufl"ciently. Such
done
been
not
PROBLEMS
present
business
of arithmetic
criticism
AND
bringing the work
with
conformity
closer
of
is that
attention
is at
which
proposition
general
METHODS
ARITHMETIC
MODEBN
a
*
knowledge
and
many
SUBJECT-MATTER
THE
similar
34.
A
various
Word
of
out
themto
look
arithmetic
Consequently
few
a
instance, fractions
with
10, 20, 25, 32, etc.,
are
uncommon
to
confine
world, yet
business
forms
Many
chance,
of authors
and
vital
a
these
ficial
super-
element
are.
Assistance
first 'four
years
operations
with
devoted
forms
of
of
receive
then
Work
up
by
a
a
considered
educators
present
study
this
in
to
as
study
various
can
suits
pur-
limited
thorough
detailed
for
omission, modification,
be
In
the
learning
and
main
the
the
four
fractions, and
"the fifth and
study of
measures
of mensuration
tions
addi-
and
work
and
is
now
the
fundamental
should
sixth
years
of the
fractions, decimals,
which
of
measures
extent,
common
study,
attention.
proper
Grades.
integers
fractions,
Much
making
engaged
men
be
their
should
to
to
from
later
questions
The
taken
will
topic
various
36.
asked
in
province
will meet
as
thorough
a
haphazard
to
of other
and
tion
selec-
of life.
walks
each
will
be
made
The
It is the
material
select such
needs
profitably
As
of arithmetic
have
very
left
be
not
in texts.
they
these
and
must
after
what
are
made
to
and
narrow
the
and
work
Selecting Material.
customs
teachers
and
needs
in
prevailing
get together
day
be
be
may
subject-matter
to
or
Aid
Should
this
of
the
industrial
the
most
a
2, 4, 5, 8,
than
arithmetical
child
give the
to
other
in
For
quantitative considerations.
35.
to
denominators
training
a
mechanically.
computation
of fractions, which
idea
in
be
would
to accustom
into
degenerate
the
is to
teaching
quantitative standpoint.
not
of
special forms
before
keep
activity and
be
not
may
always
arithmetic
the
must
caution
teacher
the
things from
at
tion
considera-
the
under
of
word
in arithmetical
children
the
engage
later
object in
chief
the
A
Let
here.
that
31
ABITHMETIC
topics.
of Caution.
place
himself
to do
will be discussed
instances
of the
OF
and
taken
three
centage.
per-
late in
32
eighth grade
the
in
METHODS
ARITHMETIC
MODERN
well
may
with
connection
The
business
furnished
be
and
girl's education.
and
short
cuts
should
form
well
be
given
cations
appli-
such
elementary
boy's
of every
computations
systematic
a
thus
ples
princi-
the
of
in
usages
years
can
of
for
part
a
two
years
study
a
Commercial
can
two
material
through
which
procedure,
these
application
Excellent
previously learned.
can
last
the
to
over
PROBLEMS
during
studied
be
measures.
largely be given
very
AND
at
treatment
this time.
far
As
This
unit
of
usable
to
leave
The
possible the
as
is furthermore
material
school
to
as
following is
a
to
in
Bulletin
No.
in the
Grade
r"um6
of the work
Elementary
include
compelled
completed.
given in arithmetic
States
Education,
matics
Mathe-
States
work"
incidental
found
as
on
of the United
others
:
include
multiplication.
addition, subtraction,
Addition
3.
division;
some
unit.
multiplication"
and
others
division.
Grade
and
subtraction
introduction
Grade
4.
Multiplication
Grade
5.
Fractions;
Grade
6.
Decimals;
Grade
7.
Percentage
Grade
8.
Business
completed;
of
fractions.
and
division
decimals
some
and
completed;
and
percentage
some
multiplication
and
fractions.
some
measurements.
interest.
and
applications.
applications,
proportions,
and
mensuration;
algebra.
some
37.
It
Some
and
only
or
been
United
of
Schools
nothing
addition, subtraction,
2.
of the
has
a
amount
is
who
pupil
eighth grade
13, 1911, Bureau
1. Some
Grade
the
be
maximum
the
the
by grades in fiftylarge cities
should
year
encompass
aid
an
before
of each
work
Interest.
gives
to
zest
Interest
work
to
succeed.
energies of
interest must
the teacher
is the
and
imbues
Interest
mischievous
the
hence
must
not
be
a
prime
the
pupils with
eliminates
waste
into useful
part of each
stop here.
He
must
education.
in
mover
a
ation
determin-
by directing
Arousing
channels.
teacher's
direct
the
work.
the
But
energies
34
MODERN
ARITHMETIC
the
As
grows
older
the
teacher
must
in
last
years
and
little
the
the
39.
that
also
interest
be
can
gardens.
Hence
it
written
are
quite liable
promotions.
modifying
little
A
the
The
of
of
and
under
of
use
outside
and
grades
discussed
work
fit the
to
planting of
the
to
As
text.
this condition
year,
the
semi-yearly
the
x"art of
the
on
care
in
having
schools
on
advantageous
problems
the
problems
the
on
work.
situation
teacher
will
in
easily
aright.
selection
the
of
during
in
number
this matter
set
arise
to
December
group
a
be used
to
in
May
child
in
number
the
octjasionally be
may
position of
the
during
during
or
plays
seasons
portant
im-
The
Season.
motivating
developed
interested
enterprises.
the
of the
in
recognized
be
presents
texts
changing
men
simple dressmaking,
and
Fit
little
be
easily
community
Should
the
Christmas
change
cooking
the
During
into
grow
then
can
change, the
change.
children
and
Subject-Matter
life must
is
of
sports
activities
his
likewise
girls
side
in
boys
role
No
The
women.
PROBLEMS
AND
and
the
grades
arithmetical
in the
and
child
of
appeal
METHODS
the
problems
from
activities
of the
chapter
children
Socialization
on
other
the
and
subjects
will
be
Correlation
Arithmetic.
.
IV.
1.
Distinguish
EXERCISES
the
between
elements
two
of
subject-matter.
lUus-
trate.
2.
Point
out
the
in
elements
two
lesson
a
for
of
one
the
first four
grades.
3.
this
Consider
with
4.
similarly
that
Explain
of
5.
Make
6.
What
grades?
for
the
Ex.
advantages
list of
should
lesson
for
the
last
grades.
two
Compare
2.
subject-matter.
a
a
what
of
Apply
should
predominate
distinguishing
to
Exs.
constitute
in
the
2
and
the
earlier
between
the
two
ments
ele-
3.
basic
element.
grades?
In
the
later
SUBJECT-MATTER
THE
OP
35
ARITHMETIC
.
State
7.
basic
the
subject-matter
should
What
8.
day
present
with
together
the
be
in
tendency
the
reasons
for
the
in
the
action
teacher's
of
matter
the
changing
same.
Illustrate.
matter
of
f
changes
Illustrate.
9.
Explain
the
suggestions
Discuss
the
business
10.
teacher's
the
and
State
11.
text's
of
means
of
treatment
the
12.
State
and
13.
State
the. purpose
14.
How
arithmetic
material.
teaching and
from
practices differing
the
subject.
of
value
and
interest.
of
law
the
spurious
the
basic
the
same.
business
two
explain
can
the
overcoming
explain
of
criticism
man's
of
selection
the
on
interest
in
distinguished
be
schoolroom.
the
from
the
genuine
est
inter-
class.
trate.
Illus-
f
's part
15.
State
the
teacher
16.
Show
the
importance
17.
Give
of the
six
school
illustrations
in
securing
of
the
of
motivation
interested
an
arithmetical
second
for
element.
various
times
and
ages
year.
BIBLIOGRAPHY
1.
3.
and
Jessnp,
Walter
Myers,
Scott,
of
Foresman
Suzzallo, Henry,
6.
Wightman,
Mathematics
8.
9.
M.
Wilson,
H.
B.
10.
Young,
11.
Minimum
12.
Beport
1911.
What
J.
W.
II:
metic.
Arith-
for Six
W\"rJc
Years.
Wilson,
and
Beport
A.
on
pp.
Inter.
16-33.
the
Comprise
lY
Subject
Matter
162.
:
Course
the
in
Arithmetic.
147.
and
Com.
pp.
Arithmetic
Shortening
Essentials.
Am.
in
Text
a
162.
:
Should
Teacher,
Schools.
Elementary
13,
J.
Mifflin
Houghton,
the
Bui.
G.
of
of
32-52.
pp.
Teacher,
Wilson,
Work.
Summary
Sherman.
Williams,
XV
Teacher,
Mathematics
Arithmetic.
in
Co., Chicago.
and
5.
7.
in
W.
Course
83.
for Introduction
Grade
School
H.
XXVII:
the
and
39-64.
pp.
A.
George
Superintendent
Review,
Coffman.
Jessup
Elonentary
4.
The
Educational
Arithmetic,
2.
V.
Collins, Joseph
Co.,
the
B.
H.
Boston,
N.
of
E.
School
of
158-82.
pp.
Motivation
Proceedings
Motivation
The
the
A.,
Children
1911:
's Worlc
528.
216-223.
Journal
Com.
of
on
Education,
the
Teaching
LXXVII:
of Math*
576.
U.
S.
OHAPTEB
SOCIALIZATION
V
OP
CORRELATION
AND
ARITHMETIC
Socialization.
40.
of
their
for
materials
of
subject-matter
subjects
school
outside
in
all
of
the
instruments
grades
of
solving
of
sake
in
should
is of
There
sake
the
share
of
the
The
preparing
time
with
ordinary
for
the
On
the
organization
analyzing
the
the
number
By
cannot
the
be
the
of
the
all-efficient
arithmetic.
The
the
the
pupil
of
the
time.
this
but
pupil
to
principles,
and
as
a
of
may
the
very
be
to
would
while
ones,
not
means,
socialized,
but
transactions,
minor
textbook
be
should
separate
use
materials
many
for
work.
arithmetic
source
36
of
socialized
case
text
ciples.
prin-
transactions
used
for
rather
or
into
are
of
nature
very
the
problem
combinations
the
objects
hand,
munity
com-
arithmetic
of
mathematical
same
into
large
part
mathematics,
requires
other
life
teaching
of
the
introduce
the
for
mathematical
a
at
and
but
-school
socializing
collection
heterogeneous
measured.
for
more
textbook
involving
problems
a
place
a
used
problems
home
mathematics,
consideration
primary
course
important
out-of
the
problems
introduced
not
are
the
use
problems
applied
developing
and
in
problems
receive
of
terms
illustrating
The
They
specific and
social
the
not
social
it
does
applied
reality
problems.
problems
the
so-called
in
are
illustrative
sake
The
terms
upon
rarely
solve
in
conventionalized
the
but
to
life
draws
illustrate
to
school.
tipper
but
the
the
think
to
instruction
curriculum,
the
as
fail
pupils
Our
work.
school
phenomena
Our
in
or
the
end.
arithmetic
truly
suggestive
plied,
ap-
in
SOCIALIZATION
furnishing
never
AND
socialized
typical
the real
provide
can
Arithmetization
desired
result
main
The
will
namely,
the
the
and
from
acts
illustrate
only
illustration
is
single
a
only
problems,
in
his various
they weigh
(2) that
and
woman
have
effect.
and
to
This
arithmetic.
to
solved
be
problems,
are
significance
real
a
socialized
for
pupils themselves
the
they
;
is intended
problems
the
forming
and
manner;
cause
problem
they
after
in
that
so
pate
particiall
gathering
required.
following problem
The
what
constitutes
entailed
in
presupposes
improving, repairing,
property,
also, certain
the
property
between
in
expenses
leased
although
division
of expenses
rule
there
these
classes
are
division
lessor
there
of expenses.
1.
Furnished
2.
Unfurnished
3.
Furnished
4.
Unfurnished
are
and
carry
The
:
apartments.
apartments.
houses.
houses.
lessee.
for
of
of expense
with
connected
This
of
upkeep
general
of
division
by the class of property
considerable
classes
of property
usually found
is
provided
four
the
living expenses
is determined
general
and
first,of
knowledge,
a
and, second, of the division
rent
the
the
;
the
(3) that
discovering
at
illustration, however,
an
and
arithmetic.
man
of socialized
real social
pupils,
of
quantitative standpoint
source
requires, (1) that
data
thoughtful
the
expresses
rational
most
following
arithmetic
for the
the
The
the
The
look
to
standpoint
in
things proportionately
Illustration.
41.
perhaps
child
the
act
quantitative.
their
consider
teach
it
course,
given community.
socialization
than
quantitative
think
of any
life
37
ARITHMETIC
but, of
problems
child's
the
is to
OP
problems
clearly
more
from
life he
look at
of
object
activities
CORRELATION
in individual
dwelling
different
four
variation
leases.
places
contracts
classes of
in
leased
as
the
As
a
and
regards
dwelling places
38
ARITHMETIC
MODERN
An
42.
Example
determine
University Addition
from
July 1, 1907,
for
the
The
The
have
it cannot
in
and
house
10,
of
the
city, where
the
the
future),
renting it
than
that
loss
a
north
in
increase
from
of the
house
be
house
the
could
for
1, 1907,
from
July
July
1, 1910,
chase
pur-
for
rate
The
house
1, 1910, for
July
to
the
fair
a
Seattle.
in
property
is
July 1, 1917, for $25
to
:
$3500.
was
than
more
cent
per
follow
problem
July 1, 1907,
sold
Seven
such
rented
month,
a
of
To
10, block
mile
district
class.
certain
and
one
solution
for the
1, 1917.
invested
would
$30
for the
paid
amount
price July
money
or
a
1, 1917, rather
July
necessary
probability
all
gain
a
9
County,
growing
fourth
the
lots
following
rented.
data
1.
The
owning
undoubtedly
to
Glass.
length of time, assuming
same
been
have
King
a
will
values
property
In
in
University,
State
in
lost
or
to
PROBLEMS
economically
was
Seattle, Washington,
(located in
AND
of
example
gained
was
Fourth
the
an
it
whether
much
how
of
with
deals
problem
METHODS
a
month.
following
The
2.
during
against thb property
made
assessments
paid
the
at
annual
due
were
of
time
had
assessment
the
improvement
local
to
of way,
had
which
bore
5%
in
cash
interest
due
either
the
for
assessment
to be
paid
at
were
All
period.
ten-year
notification
of
date
paid
plus
the
the
notification
be
installments
exception of
from
assessments
and
unless
Each
interest..
in
or
each
five equal
time
with
of right
condemnation
once,
ASSESSMENTS
Oct.
May
10,
1909
5,
1910
Nov.
12, 1912"
Jan.
1, 1913
May
5, 1913
June
6,
1914
"
"
$125.00
65.00
110.00
"
"
"
75.00
100.00
37.00
for
trunk
sewer
condemnation
grading
for
right
of
way
street
constructing
lateral
for
sidewalk
sewer
constructing
street
bridge
to
street
SOCIALIZATION
COEEELATION
AND
IMPROVEMENTS
5,
1910
5,
1912
1,
1912
5,
1913
20,
1913
Oct.
2,
1914
Dec.
2, 1914"
June
Aug.
Nov.
Oct.
Oct
Sept.
14,
Yearly
MADE
BY
THE
OWNER
exterior
of
house
"
50.00
painting
exterior
of
house
"
35.00
fencing
125.00
"
"
"
"
avera^je"
sides
two
building
of
lot
walk
cement
25.00
grading
50.00
decorating,
32.00
building
playhouse
60.00
painting
exterior
40.00
other
REGX7LAR
39
ARITHMETIC
painting
50.00
$
"
1915
OF
lawn
interior
RUNNING
of
and
repairs
and
floorfl
varnishing
house
improvements
EXPENSES
Telephone
$2.50
a
month
averaged
75
a
month
Light averaged
1.50
a
month
Gas
1.25
a
month
a
year
Water
averaged
Coal
and
43.
wood
of Material
Sources
for graphs,
net
papers,
For
keeping
chickens
and
hours
interesting
the
later
here
nation,state,
Within
this range
postal department,
natural
so
on.
material
excellent
of
cost
equipment,
activities,such
of
there
material
be
and
of velocities.
and
school,
as
so
on,
and
selling
working
hares, gardening,
or
Foot-races
games.
furnish
first
interest
computation
forms,
out
grades
would
football
and
socialized
for
of
law
The
ing
dur-
provide
data.
supply of community
sense
and
Commercial
games.
vacation
much
and
the
plays
in many
averages
of
costs
of
measurements
Baseball, basket-ball,
of material
with
quantitative side
give practice in
Socializing Arithmetic.
sources
conformity
In
the
comes
for
will find many
alert teacher
problems.
80.00
averaged
included
county
as
would
come
practically
for
problems.
as
to the
railroads
parcel
In
immediate
; express
post
manufacturing
unlimited
an
its widest
referring
problems
well
including
resources;
is
and
to
the
locality.
companies
savings
enterprises;
;
posits;
deand
40
MODEBN
particularly useful
A
comiection
in
arises
of the
other
description of
based
of
the
upon
the
these
above
will
various
the
of
and
so
A
on.
excellent
problems
many
excursion.
each
upon
partment,
de-
description in order
a
solution
the
and
furnishes
Such
gathered
under
mills
excursions
require
mentioned
detail
prises
enter-
postof"ce, fire
lumber
as
English work.
data
sources
in
more
of
will
complete
be
activities
each
for
material
real
problems
various
the
to
stores, bank,
as
local
more
grain elevator, flour mill, creamery,
localized
more
the
excursions
community
PROBLEMS
AND
of
source
with
bakery,
to
METHODS
ABITHMETIC
Many
considered
be
much
will
each
topics where
naturally fall.
Correlation
44.
is the
side of the
quantitative
subjects
Such
include
full
a
necessitates
Subjects.
grade
curriculum.
in
complete
a
they
that
which
the
various
other
nature
plays,
it is
sary
neces-
of them.
one
grade
closely correlated
be
and
of any
treatment
of
development
All
gardening,
as
quantitative side
a
Arithmetic
physiology,
activities
outside
present
games,
to
Orade
history, geography,
as
study, together with
and
Other
with
subjects
arithmetic
with
throughout.
Especially is arithmetic
and
history
part
a
nothing
local
has
conditions
of the
been
grade
done
possibilitiescontained
of
arithmetic
of
place
these
various
give their faculties
exercise.
as
well
the
with
Such
as
grade
an
which
a
to
take
in
a
the
rational,
other
school
subjects
correlation
value
curriculum
opportunity
to
So
will add
to
each
of
the
exceptional
tion
correla-
subjects.
children
will
while
interest
every
providing
apply their arithmetical
In
the
then
be
This
discernment
and
coming
be-
far, practically
comparatively.
and
day
more
common-sense
grade
the
present
and
more
advantage
of observation
educational
are
with
curriculum.
learning facts
merely
studying
with
correlated
be
to
will
constant
stimulus
and
subject
the
of
pupils
knowledge.
42
tain
will
that
does
will
grades
affairs
and
47.
the
present
the
very
For
instance, the
they
they
of
teaching
the
geography
or
made
great
given
of knowledge
arithmetic
not
to
be
world
the
more
its importance
of
at
the
more
of the
vital, and
by
the
Material.
nearly
the
than
that
history
be
may
progressive,
arithmetic
more
teacher
measure
the
usable
that
the
quantitative
this
of
sort
of
teaching-
merits.
The
all exhaustive
but
from
the
of problems
possibilities
a
is definite,
wants
in
time
correlated
and
when
day
arithmetic
hands
told
more
The
qualitative knowledge,
vague,
Sources
In
of
formal
socialized
real, a
will receive
that
attention
a
to
deal
be
motivates
given
nor
topics.
returns
procedure.
it is realized
When
knowledge,
48.
time
more
a
arithmetic.
kind
a
receives
teacher
common-sense
be
a
proportionately.
decreased
can
such
and
be
can
may
This
slighted
be
through
work
computation
arithmetic.
neither
will
of time
its share
the
be used
may
arithmetic
new
to-
beyond
problems
of expenses
gardens.
time
to
are
gardening
their
in
accounts
Arithmetic
school
accounts
planning
first
are
keep
to
are
in
class
of
of
state
time
pupils
Such
study.
sad
solutions
abilities of the
further
subjects of
in Arithmetic.
from
whose
effectivelyas forerunners
used
that
arise
A
arithmetically.
of Accounts
teacher
by the
this
their feet
upon
Teaching
arithmetical
for
eliminate
simplest
lesson.
other
the
often
so
the
arithmetic
the
tion
solu-
outside
and
solve
to
con-
The
fright
stage
with
to
which
problems
stimulus
a
much
reference
judicious
in
of arithmetic
do
studies
upon
arise
not
Motivating
many
called
arithmetics
problems.
the
remove
place the pupils
to
the
to
PROBLEMS
our
other
from
when
correlation
proper
as
much
pupils
by
problem
the
do
of
distorted
such
selected
problems
activities
shown
of
AND
Still many
practice.
number
great
a
of
A
in
arise
never
METHODS
ARITHMETIC
MODERN
following
merely
other
list is not
tended
in-
suggestive of
grade subjects
:
the
SOCIALIZATION
OOREELATION
AND
OF
Geography
of
Per cent
of
Value
Batios
Finding
increase
population.
in
crops.
value
between
Distances
of
with
between
freight
dates
Battles
and
periods.
killed
cent
; per
Financial
country
one
History
time.
and
Longitude
43
ABITHMETIC
and
so
on.
changes.
products;
Civics
another.
towns;
Parcel
and
car
post.
Support
rates.
of
public
enterprises;
taxes.
Worlc
Hand
Measuring
and
Drawing
material
for
tions.
construc-
Domestic
Per
Amount
and
certain
make
material
of
price
of
cent
sizes
Changing
articles.
farms
sections
Feeding
Butter
divisions
and
and
so
rations
of
cost
feed.
raise
to
Discuss
3.
Solve
4.
Give
5.
Explain
staling
mider
6.
its
fully
obtain
Explain
data
is meant
arithmetic.
of
life.
disadvantages.
42.
for
suitable
should
and
so
on
carried
be
for
work.
grade
practice,
source
mentioned
arithmetic
with
one
any
in
out
the
by
correlation
of
the
grade subjects.
Develop
fully
8.
Suggest
at
9.
Discuss
would
10.
what
and
under
scheme
the
the
what
7.
too
under
how
child
of
stated
socialization
43.
other
of
volumes.
measure.
the
by
illustrations
fully
to
is meant
advantages
problem
other
two
Training
EXERCISES
arithmetization
the
the
how
what
fully
with
2.
materials,
out
Costs.
V.
Compare
Board
30-
a
crop.
Explain
ratio.
economy.
Manual
Areas,
Nitrogen required
1.
recipes,
on.
and
com
of
marking
cost, and
of
fat.
bushel
values.
costs.
Sewing;
Agriculture
of
food
to
Various
Areas
Science
What
affect
45?
would
a
Give
be
how
the
be
may
personal
six
least
fully
of
two
the
other
such
topics. Develop
socialization
a
teaching
the
nature
in
improper
it.
of
danger
illustrations
an
subjects suggested
of
what
question.
data
correlation
would
by
a
metic
arith-
Whatf
asking questions
problems
be
of
danger?
any
offence
for
48.
one.
and
there
giving
securing
of
Is
under
proper
as
suggested
as
well
as
44
MOI"EBN
ARITHMETIC
METHODS
AND
PBOBLEMS
BIBLIOGRAPHY
1.
1915:
2.
C.
Doolejy
in
a
B.
Student
Proceedings
Finances,
Klapper,
5.
Park,
in
W.
A.,
V
Teacher,
for
XI:
Teacher,
Should
Seventh
the
D.
6.
Smith,
7.
The
World
8.
The
Daily
9.
Local
Reports
E.
The
Data
Applied
Mathematics
Arithmetic,
Teaching
The
Almanac.
from
of
Nearest
9.
and
48-51.
Almanac,
News
:
Be
285.
Teacher,
18.
10.
E.
N.
Application
Arithmetic
School
pp.
L.
Practical
Mathematics
Community
Paul.
Louis
of
Arithmetic,
Elementary
Grades,
4.
Minimum
What
Course
W.
Hart,
Eighth
V:
W.
908.
Included
3.
E.
Barnhart,
of
New
Station
News,
Daily
Auditor's
of
pp.
World,
York
The
County
Arithmetic,
N.
24-31.
Y.
Chicago.
Office.
U.
S.
Weather
Bureau.
City.
CHAPTER
VI
PROBLEMS
49,
Historical.
factor
in
the
During
the
of
largely
early
the
of
conform
to
the
to
socialization
by
The
commonly
problem
will
Value
50.
taught
of
are
be
adhered
of
Oood
vital
and
quantitative
and
business
given
for
of
the
with
a
to
For
of
is
which
the
given
furnish
driven
be
home
Any
so
as
presented
as
and
well
principles
of
to
in
well
First
work.
industrial
our
teacher
to
children.
follows.
as
selected
problem
and
problems
involved
struct.
in-
to
example
what
pertaining
the
problems
were
arithmetic
reason
too,
was,
the
application
principles
correctly.
must
in
certain
a
motivation
of
Good
the
partook
these
between
and
here
of
this
the
show
must
each
that
to
see
problem
the
lem
proboutside
from
the
forcibly
it illustrates.
Problem-solving
namely,
to
information
solution
principles
distinction
recently,
for
interest
to
today.
they
elements
seek
them
There
children
Problems.
are
use
text
the
element
life.
is solved
the
today
we
accepted
a
the
lacked
which
all, problems
it that
of
ability
called
arranging
or
always
quite
reasoning.
grading
further
They
in
not
textbooks
the
They
puzzles.
skill
at
attempt
no
in
found
than
from
until
indeed,
important
an
have
they
asking
are
and
of
nature
been
always
but
we
years,
rather
or
that
problems
the
shrewdness
little
result
have
arithmetic
of
study
the
produced
many
Problems
has
exercise
situation.
function
another
of
judgment
In
no
45
it
other
peculiarly
demands
subject
its
own
in
grappling
in
the
entire
;
46
MODERN
curriculum
grade
in
ABITHMETIC
judgment
he
Prom
the
of
confronted
is
in
afifords is
the
Solution
Problems.
of
the
Reading
2.
Deciding what
principles
3.
Computations
and
4.
Interpretation of
Many
take
step
teacher
to
fair
the
to
the
all this
two
lower
grade
of
teacher.
former
strenuous
so
work,
To
will
serve
efforts
will
to
a
It may
not
with
and
cast
the
stimulus
overcome
pupils
selves,
them-
work
answer."
be
The
easier
teaching,
in
work
for
nor
the
is it
problem-
grades, application of the
discussed
be
now
has
fairness
blame
this
in
this
''give
the
upon
more
from
received
with
unacquainted
may
*^
the
unfortunately
first-class teacher
as
at
because
computation
the
effective
earlier
who
children
the
apply them.
work
their
it is not
but
These
speak,
to
problem
do
altogether.
the
procedure
to
metical
arith-
an
:
how
for
to
Really
teacher
in
looking
by
steps mentioned.
The
and
use
steps
pupils
children.
solving requires, from
detail.
to
miserably
this
they omit
do
steps
of
results.
first
check
to
fourth
ship,'*
in
checks.
fail
the
and
mode
vital
This
statement.
teachers
requiring
a
in fact, life
problem-solving
solution
distinct
1.
four
;
daily
are
problems, especially
The
four
includes
problem
only
value
grades.
later
they
"We
which
of
values
price,
The
judgment
situations
great
certain
a
problem-solvings.
of
ing
of find-
that
solution.
the
requiring
with
at
of
that
as
to
inestimable.
is
series
of
boxes,
drill
arithmetical
such
bought
making
situations
coping
one
bonds
judgment
with
practice
51.
from
in
continuous
one
the
in
drill
such
in two
of marbles
is necessary
judgment
like the
of
simplest problem,
the
of income
rate
PROBLEMS
solution
given in the
is
finding the number
AND
pupil given anything
is the
that
problems.
METHODS
rational
up
the
former
inefficiency of the
will
and
he
the
previous
make
even
teacher's
more
sins
47
PBOBLEMS
of omission.
for such
62.
real
The
English in Arithmetic.
irrevocably
parents have
Often
that
thought
because
this
fails
he
in
fact
the
used
made
pupils
clear
with
encounter
their failure
to
that they
not
do
obviate
To
him.
to
understand
taught
to
as
the
the
grasp
meant
and
direct,
even
of
high and
dry
to
that
53.
Silent
will be
the
the
will
ever
is in
importance.
One
and
the
to
relate
reading
among
problem.
in
this
is
form
members
some
The
form
of
notes.
teacher
be
ready
to
fact
state,
pupils
are
to
By
suggest
and
reading
proper
the
pupils
to
carry
of
language.
the
on
English
or
arithmetic
the
The
for difficulties
alert
either
so
page.
printed
of the
to turn
whenever
metical
arithlesson
necessary.
Reading
Highly
general
more
attention
to
given
class
pupil receiving
a
oral
to
read
distributing
(see 66), each
card
an
reads
it
its
article
practice of
Excellent
by
and
with
commensurate
others.
be
Important.
than
use
frequently requested
of the
first
this the
difficulties
may
which
the
to
is not
used,"
gist of it
never
traced
be
is not
teacher
ever
should, therefore, receive
has
and
in the
teacher
much
understand
used
the
of
lack
intelligently for themselves
Silent
Interpreting
reading
a
language
a
to
can
From
contained
language
of language
one
class.
these
over
the
and
"
into
from
as
give illustrations
sympathetic teacher
arising
the
problems
that
but
problems,
to
meaning
this is not
terms
matics"
mathe-
difficulties
the
his
by his teacher.
^which
"
bonds
and
let
difficulties,
these
read
to
of
bonds
recognize
read, the problems
or
technical
the
the
inability
Most
stocks
for
through
not
through
frequently
"
head
no
has
and
work
recognized
not
was
child
a
understand
not
arithmetic,
his
ability, but
language
'*had
he
did
he
arithmetical
been
repaid
well
Many
arithmetical
his
in
lost
simply because
used and
be
always
effort.
extra
Step One;
been
will
teacher
cards
bearing
silently;
then
a
48
ARITHMETIC
MODERN
he
the
states
contents
orally
the
is
unconsciously
of
that
English,
64.
Exercise
Step Two;
far
By
the
the
last
primary
the
grade
drill in
in
how
directly
means
first of all the
range
of
some
simple
numerical
This
has
is
a
order
In
the
to
of
group
shown
of
a
upon
a
treated
and
of
the
make
than
be
told
the
as
In
if they
in
were
subject-matter
it appear
much
solve
to
the
proceed.
be
for
necessary
of the
text
ones.
the
came
different
new
Such
destroy
more
the
pupils
the
one
the
upon
section
and
were
that
under
depending
wholly
each
texts
characteristic
and
which
a
older
common
any
problems
placed
mental
make
to
problems
by themselves
of
the
problem-solving.
it may
simple
problems
how
of
the
pupil
each
to
to
way
modify
possessing
group
were
the
in
of
problems, after the
as
special principle for solution.
divisions
lem.
prob-
suggestions
actually
difficult
Problems.
**case"
to solve
principles
and
to
Ofif^'
Another
text
of his
within
him
himself,
more
additional
in
for
this scheme
out
problems
how
**case."
same
by
sometimes
placed
the
proper
if ever,
part
well
explicit directions
carry
''Casing
the
on
better
counterfeit
introduce
55.
were
problem
poor
to
of
and
seldom,
problems
Far
pupil.
given
teacher
or
of
use
computations
teacher
selection
of judgment
exercise
the
in
pupil should
in
pupil receives
each
questions
of
problem
ones
it that
the
lems.
Prob-
proceed.
to
active
of
page
process
first little
satisfactory solution
way
the
the
complicated
to
see
a
the
but
helpful
be
for
necessary
Stimulus
An
must
of
use
Salving
in
the
more
exercising his judgment
principles
may
the
to
real
written
a
in
step
From
one.
in the
Thus
in arithmetic.
as
Judgment
important
grades
teacher
well
as
solution.
from
meaning
of
most
is this
problem-solving
in
the
others,
to
for
class
getting practice
grasping
ideas
conveying
and
child
the
to
PROBLEMS
AND
METHODS
of
the
depended
highly artificial
unity
difficult than
of
metic
arithit is.
50
MODERN
When
first
the
For
pupils will
thus
of which
solution
inverse
relation
inverse
relation
of
numbers
the
as
gest
sug-
could
' '
price ?
and
its
helpful
The
matter
a
reciprocal
^the
"
either
will
which
question mark
area
=
fl^.
few
written
down
question
marks
or
numbers,
the
pupils
The
to
schematic
letters
the
take
to
permitted
are
with
to
given
the
the
take
will readily suggest
letter
this
Another
such
operation
number
give the
or
=
a
bringing
of
is that
6
for themselves.
in
children.
the
to
breadth
X
has
teacher
this work
and
easily comprehended
place of the unknown
do
that
at
naturally,
home
problems
Z X
solutions, with
chairs
many
dollars
is very
scheme
or,
the
*'How
the
to
they understand.
Length
After
similar
incidentally
may
problem
a
PROBLEMS
problems
teacher
connect
of
accompanying
The
to
solution
the
of course,
the
example,
be led
AND
several
for seventy-two
bought
be
solved
above
inverse.
its
METHODS
has
class
mentioned
one
then
ARITHMETIC
place of
itself to the
pupils.
the
in
work
devices
are,
as
textbook
solutions
schematic
These
soon
of
with
excellent
merely
crutches
arithmetic
of
course,
as
they
today
inverse
relations.
depend
largely
upon
carried
be
may
are
gives
Hence
the
no
very
All
success.
and
much
teacher.
The
Some
will
texts
be
carded
dis-
average
recognition
of what
such
be
to
are
longer needed.
little
throughout
on
to
done
even
these
will
retain
51
PEOBLEMS
the division
where
of
distinct
three
base, and
a
of
sort
have
cases
the
numbers
the other
two
of
the
by
The
been
Idea
inverse
of the
made
These
the
inverse
they
to their
we
three
of
product
simple what
to
for themselves.
are
heretofore
has
of
simple and
so
artificial
such
they
some,
will
when
become
the
at
time
same
it
brought
to find
eager
others
reciprocity^
solution
rational
aiding in the
of
that
by directing
that
find
simple idea of arithmetical
advantage
and
will
groups.
or
natural
so
children
the
teacher
the
This
has the double
Recognition
dividing of problems into classes
notice, and
problems
is the
one
the- introduction
by
relations
their attention
of
given above,
Reciprocity.
readily recognized by
are
here*
p.
=
will make
process
as
""
of Arithmetical
complex
machinery
Yet
;
6 X
57.
that
centage,
per-
enjoying
cases
own.
that
as
fact
its
percentage,
finding of
the
three
these
condition
in
as
for
principality of
same
connected
classes;
given
are
rate, each
independent
exactly
into
problems
unify
to
serves
the
subject-matter.
Furthermore,
the number
is aided
of
not
mto
a
but
that
that
these
generalized type
more
Step
to
of problems
Their
the
also
fact
there
thinking
of mental
Three;
of
This
elements
tion
generaliza-
leads
but
pupils
the
activitywhich
mental
Computation.
all
attention
is thus
and
can
should
Pupils
operations involved
symbolically before
Cancellation
memory
is
a
activity.
express
process
The
fewer
are
related.
are
with
compared
as
really insignificant.
only simplifiesproblem-solving
higher type
58.
is
cases''
only by the
not
to consider
**
principles
of
the number
carrying
centered
later
then
be
wholly
at
in the
out
first
upon
resorted
any
to
be
solution
tions.
computa-
wholly
the
couraged
en-
upon
tions.
computa-
quite often
to
52
ABITHMETIC
MODERN
METHODS
AND
PROBLEMS
simplify the computations, especially where
This
acceptable.
are
of
of whieh
problems
59.
calculation
nothing
will
than
will
be
the
a
for
required
the
inaccuracies
of
the
will
have
computation.
As
repetition
estimates
instance,
a
been
$253.40,
made
later
in
the
we
Orderliness
Problems
and
and
without
answers
be
bered,
remem-
original
the
discovered
again.
by
that
If, for
topics
insisted
be
found
has
error
taken
be
the
an
brings
Discussion
point.
will
a
inary
Prelim-
computation
be
will
numerical
necessity of
should
certain
various
should
accuracy
the
from
first checks.
arranging
be
consequence
work
the
and
problems
of
not
decimal
the
in
in
with
the
operations
with
fewer
with
are
almost
the
system
this
Illustrations
rough
is $25
are
various
connection
to
good
as
placing
in
of
checks
serve
teacher
accomplishing
common
of
over
first estimate
a
of
result
going
or
in
the
lems
prob-
checks, it
rule, mistakes
a
Let
can
children
Good
in
accuracy
consequences
the
taken.
of
them.
of
solution
Solution
of
means
nothing
or
be
accuracy.
disastrous
impress
little
always
upon"
checks
all
in all things.
accuracy
the
examples, since
usual
practically
time
to
mere
cent
per
in
Absolute
insisted
eflfective
more
of
use
100
be
given for solution, and
be
time
but
should
Orderliness.
always
should
big aids
of the
one
advantage
and
Accuracy
accept
is
approximations
work
of
herein
up
arithmetic.
of
also
are
all times.
at
upon
for
excellent
ducive
con-
ing
teach-
self-reliance.
60.
may
Cuts.
Short
give
the
before
abbreviated
this
which
reasoning
opportunity
more
contracting
the
As
This
work.
the
work
in
can
the
depends
power,
and
teacher
upon
is
soon
of
way
not
be
of
process,
form, is thoroughly
care
the
should
fundamental
understood.
prevent
the
the
progresses
memory,
forgotten.
that
teacher
short-cuts
permitted,
which
Only
it
by
for
ever,
how-
is
an
cising
exer-
artificial knowledge
rather
than
the
53
PBOBLEMS
Step Four; Interpretation
61.
teachers, consider
often
"
answer'*
in
little value
The
each
of
a
equally well by 11, 45,
or
by
4
speed
In
16.
or
45
second,
feet per
the
minute.
train
660,
the
or
of
book
designated
660
book
feet per
represents
dollars
4
was
ingless.
mean-
certain
a
and
hour,
of the
cost
be
train, 11 represents
per
is
There
nation
special desig-
may
cost
and
their
when
absolutely
are
certain
miles
The
for
stands
of the
case
represents
without
Numbers
one
Pupils,
answer-book.
in the
one
work.
what
to
as
such
the
Besnlts.
fully solved
problem
a
tallies with
of
16
or
marks.
necessity of
The
addition
division
these
in
or
subtraction,
it abounds.
difficulties
always be used
surest
method
work
the
demand
for
not
think
using
to
and
teacher
and
first,the
If the
vertical
figure are
drawn
each
and
row
a
m
the
In
the
unreasonable
in
as
the
to
of
case
ploying
em-
company
a
but
one
the
would
teacher
used
horizontal
unit
rows,
most
ever
form
that
of
finding
lines
there
in
that
with
he
the
The
the
be made
volumes.
and
areas
accompanying
the
be
for
solids.
solution
if there
will
ask
and
important
in
of
means
carefully specifying
of
surfaces
apart, and
then
Another
problems
It is quite
process
and
the
vary
child.
to see, at
in
rectangular
again
of the
No
school
habit
the
found
be
of
volumes
progress
rule,
day.
avoid
number
results.
quite
Interpreting.
child
is to
must
in
multiplicand.
as
inculcating in the
areas
and
early number
in the
daily payroll of
per
in
is,undoubtedly,
final
this
thoughtless,
Visualizing
number
$3
at
say
This
it becomes
to
arise
to
abstract
an
designating
of the
3
that
children
the
adherence
men
radical,
each
of
by which
process
multiplier.
teaching
amount
284
usual
not
multiplication
in
but
require
to
the
as
strict
finding the
62.
is
time, however,
a
of
The
necessity
of
course
interpretations does
such
are
n
n
X
squares
w*
units
in
of
54
MODERN
ARITHMETIC
in
surface
the
that
equals
of
units
obtained
length
the
by
this
of units
number
the
AND
From
rectangle.
number
the
METHODS
PROBLEMS
arrive
we
of surface
in
by multiplying
the
of
number
units
of
the
at
clusion
con-
rectangle
a
number
width.
of
This
m
n
general
**
is then
process
length
times
treated
in
the
is that
notice
we
width
in
It is always
number
a
result
this number
number
which
chief
the
problem
in
must
be
the
best
with
never
operations
the
in
care
the
the
to
value
*'
in
but
rather
this
of
63.
more
of
with
the
of
the
final
it is still
computation.
part
lems
prob-
numbers.
In
but
to
solution
a
Hence
of
a
tions
opera-
and
manner
distorted
such
exemplified
in
illustration.
proportion
phase
a
the
perform
solution
labeled
the
in
thing
found.
be
properly
found
one
rational"
up
be
of
operation often
called
is to
the
operations
calculatory
use
of
which
possible
accompanying
The
in
be
the
as
wholly
must
must
in
be
to
are
fundamental
all computations
that
formula
the
Volumes
The
manner.
concerned
are
area.''
equals
same
into
contracted
broader
as
suffers
definition
of
ical
numer-
a
by
too
of
ratio.
problem-solving
and
a
much
of
rigid enforcement
more
It
so-
is not
loosening
is
advocated,
that
ment
treat-
common-sense
it.
Fonnnlation
problem
work
of Problems.
than
is found
Successful
in the average
teachers
text
give
Such
55
PBOBLEK.,
will
problems
will
be
difficult
so
or
their
while
helpful
very
nearly
be
not
found
irksome
so
tion
formulabe
might
as
expected.
64.
with
teacher
new
number
seat
work.
Third,
increased
For
how
the
interest
upon
so
of
teacher
for
8%
at
the
ask
may
$500
has
the
in
greatly
involved.
from
$560
the
pupils.
a
principles
upon
class
6
of
explained
of
substituting
by
range
of the
interest
the
repetition
a
drill, but
mental
pupil
one
be
mental
applications
find
to
in
his
at
give practice in
to
afforded
is
the
only
not
made
not
drill
problems
difficult,at will.
or
be
yet
within
instance, after
months,
and
come
number
solution
thus
excellent
result,
resourceful
the
of
number
simple,
can
relations
which
will
This
made
work
much
numbers
be
can
board
Second,
furnishes
text
unlimited
first, an
disposal which
the
Simply
possibilities.
great
means
Problems.
Textual
in
in
numbers
the
changing
It
Numbers
Changing
board-
a
for
7%
at
the
find, mentally,
to
for
months,
3
months,
5
and
on.
Fourth,
the
Fifth, by changing
the
assignment
handicap
greatest
the
namely}
"
arithmetic
basis of
is this
will
a
the
treatment
remove
that
while.
reason
the
the
alone
of
continual
great
will
who
work
home
our
do
found
of
to
the
making
teacher's
is removed
teaching
to
text
It furnishes
home
be
the
teaching,
furnish
as
its
suggested
'*help,"
very
of
teaching
good
to
;
home
incompetent
arithmetic
most
and
text
*'help."
in the
hindrance
be
answers
upon
recitation
the
hindrance
upon
of the
the
injudicious,
greatest
ones
of
arithmetic
good
criticism
much
that, too, by
Such
to
numbers
day
prevalent
By far
'^help.
the
on
children
the
of work.
by this kind
removed
be
can
of
dependence
the
much
and
cause.
above
and
for
worth
56
MODEBN
Office of Difficult
65.
puzzles; if
if too
teacher
the
problems
difficult
those
that
member
of
of
merely
problems
for
driving home
solve
to the
ambitious
average
a
class
balanced
that
assignment
the work
to
the slower
66.
the
to
dictate
largely hy
from
it
year
growing
computed
have
only five
than
of
of the
bright,
as
optional, thus
and
more
is seldom
It
alert
lus
stimu-
a
brighter
the
of the
few
A
pupils
is
is
.an
problems
of
to
best
the
are
serve
six
or
every
Solutions
data
class.
needs
Practical
Problema.
needs
problems
The
in
ability
teacher
pupil
as
that
well
so
equal
adapt
to
well
to
as
little practice.
a
in the
those
local
the
problems
pupils
elsewhere
upon
results
which
Additions
up-to-date.
may
be
can
and
the
be
secured
the
to
in
the
will
be
of
the
into
this card
the
text
sub-
margin.
collected,
teacher.
letters written
the
form
and
preserves
master-card
included
and
should
through
burden
numbers
in the
written
teacher
a
numbers
the
written
made
A
in
direction
is then
data
cards
and
be
desired
may
under
can
This
to year.
text
the
those
to
If
become
to
change
to
of
formulation
The
way
no
similar
problems
pupils.
problems
from
of the
the
for
by the
are
duty
oral
for
Data
pupils who
number
same
selecting
ones.
made
too
ones.
with
stituted
be
It is the
teacher.
come
can
a
are
of
to
and
secure
that
principles.
all.
outside
to
difficult
for
**Pake"
these
Data
the
few
These
if it contains
even
to
better
wisely interspersed
those
attention,
that
as
like
are
simple problems
arithmetical
giving practice for
the
great
so
It is far
a
brighter pupils.
than
the
simple everyday
difficult problems
more
problems
ambition
all
several
upon
the
that
of
formulation
the
hold
not
nearly
attempt
based
PBOBLEMS
selecting problems
difficult.
class
a
them
of
is not
too
are
do
discourage
danger
simple, however,
In
recognize
they
they
The
AND
Problems.
must
simple
too
solution.
have
METHODS
ARITHMETIC
system
containing
file.
The
of
uses
keep
the
copying
58
MODERN
ARITHMETIC
and
Explain
3.
arithmetical
illustrate
and
the
and
principles
Explain
4.
AND
METHODS
of
use
PROBLEMS
in
problems
the
teaching
of
solution
of
operations.
illustrate
need
the
of
in
judgment
the
problems.
5.
Explain
6.
Solve
7.
Why
do
so
the
teaching problem-solving!
in, say,
teacher
a
in
preparation
adequate
no
judgment.
steps in problem-solving.
four
of
procedure
had
have
developing
the
fail in
teachers
be
in
problems
illustrate
to
many
should
pupils
whose
of
problem
a
8: What
value
the
the
trate.
Illus-
sixth
grade
problem-solving?
Illustrate.
What
9.
and
fully
into
enters
problem-solvingf
Discuss
this
pupils
illustrate.
would
How
10.
English
of
phase
to
cultivate
conditions
given
proceed
you
in
power
your
f
Illustrate.
Read
in
and
the
explain
the
following
nine
il.
A
remits
he
the
notef
A
12.
left.
lawyer
has
note
collected
How
What
14.
Corn
How
bushels
of
How
16.
In
bond
issue
carry
by
How
17.
make
A
and
the
much
return
will
what
19.
A
his
actual
per
hour.
20.
65
cents
face
value
was
and
money
of
face
the
had
a
of
$3.75
will
tax
which
gain
total
man
a
give {f
will
bushel
per
if the
sold
was
$208.81!
was
who
pay
gain of 10%.
a
$24,500
owns
wortb
is % mill!
tax
of
and
40
be
added
all the
voted
votes
cast
*'no."
Did
to
the
carry
a
election
votes!
water
must
to
100
of
cc.
alcohol
90%
to
pure!
house
heated
three
with
is
pipe
be
f of 60
by
"yes"
60
by
his
i of
than
requiring 60%
voted
or
it 60%
18.
10"
120
is the
What
first!
at
school
which
$608
collection.
less
cents
at
election
24
5%
for
school
if the
an
less
were
much
property
client
divided
sold
was
his
he
number
many
15.
had
much
desired
results
the
state
:
to
15
spent
man
13.
problems
and
rate
Suggest
a
to
be
air
climbing
of
motion
other
the
the
return
a
mast
at
ship
problems
heating pipes
two
of
area
total
the
of
if the
has
furnace
If
of
f that
is
two
hot
S'' diameter.
diameter
the
sailor
by
the
of
areas
cross-section
the
hot
air
with
of
pipes,
pipe!
rate
of
sailing
at
the
is
for
reading
6' per
the
and
second.
rate
of
Find
12
miles
interpretation.
59
PROBLEMS
Discuss
21.
into
of
degree
difficultyto
permitted
be
to
enter
problems.
should
22.
Why
23.
Where
this
least
at
Explain
26.
State
27.
Discuss
the
into
"cases"!
in
placed
be
Illustrate.
teaching
problem-
accomplished?
three
quite
dissimilar
depending
problems
upon
solution.
of
nature
inverse
the
emphasis
the
fully
classified
be
not
be
for
principle
same
25.
"inverse
for
problems
of
advantages
above.
given
11-19
the
Illustrate.
processes."
inverse
problem-
in
processes
Illustrate.
solving.
Explain
28.
the
can
Suggest
2^.
the
problems
should
How
solving?
a
the
fully
fully
how
would
you
introduce
inverse
the
to
processes
class.
29.
should
Why
out?
carried
are
all
30.
Apply
31.
In
problems
grading
33.
Explain
out
of
any
them
the
value
above.
19
importance
should
disastrous
consequences
of
inaccurate
estimates
preliminary
to
curacy?
ac-
tations.
compu-
before
ing
carry-
Illustrate.
of
estimates
preliminary
given
be
Illustrate.
making
of
computations.
any
before
what
checks?
good
are
Make
34.
the
and
13, 14, 17, 18,
problems
Illustrate
What
indicated
be
Illustrate.
to
32.
computations
results
for
11-19
problems
above.
Explain
35.
from
Discuss
37.
Interpret
38.
Explain
39.
Explain
and
how
to
of
Give
three
handicap
of
is to
of
problems
fundamental
of
final
interpreting
solution
this
treat
changing
the
the
increased
be
the
phase
in
numbers
results.
11-19
of
the
above.
problem-solving.
problems
of
the
illustrations.
resulting
from
essentials
of
"home
help."
How
can
overcome?
What
42.
Criticize
43.
Discuss
for the
various
Discuss
Formulate
primary
the
are
supplying
45.
results
the
use.
41.
44.
work
computation
necessity
the
fully
Discuss
40.
it be
its
how
year.
36.
text
of
to
year
illustrate
and
loose
the
and
use
of
illustrate
the
the
good
term
real
problems?
"practical
meaning
of
Illustrate.
problems."
practical problems
grades.
the
method
such
as
a
is not
problem
of
gathering
data
for
outside
obtainable.
based
upon
such
data.
problems
and
60
MODERN
ARITHMETIC
METHODS
AND
PROBLEMS
BIBLIOGRAPHY
"
1.
Brown
2.
Field,
Jessie.
3.
Hatch,
K.
Practical
with
Howell,
5.
"[lapper,
6.
Richards,
7.
"
8.
Henry
Alva
Book
Co.,
J.
Educational
9.
C.
10.
The
M.
American
Practical
W.
A
New
Textbook
York.
Elementary
Arithmetic,
,m
Some
VII:
L.
Book.
of
Arithmetic.
96-97.
pp.
Sources
Teaching
the
on
York.
of
Failure
Solve
to
a
Problem.
26.
Measurements
Pedagogical
Year
New
Co.,
and
Peterson
Agriculture
Elementary
521.
Edward
Problems,
A.
Chicago.
91-97..
pp.
Bi-Monthly,
Thomdike,
J.
Co.,
Flanagan
106-11.
pp.
XVII:
Stamper,
Stone,
B.
Mabel
A.
Lady,
Row,
71-91.
and
Haselwood,
and
Paul.
Teacher,
American
Com
The
L.
43-65
pp.
Arithmetic,
4.
School
Coffman.
and
D.
of
Seminar,
XXI:
Appleton
and
Ability
to
Solve
495.
Co.,
New
York.
metical
Arith-
CHAPTER
Vn
DRILLS
68.
each
begin
to
of
but
close
of
the
the
studied
in
matter
the
quick
review
those
the
plan
the
of
arise
comes
fix
to
as
unify
to
serve
well
the
at
come
should
vious
pre-
will
generally
one
as
year
good
a
may
principal
This
is
which
parts
reviews
year.
during
It
it
well
mind.
arithmetic
Good
the
various
can
Number
that
the
he
facts
them
alphabet
of
use
be
is
in
of
spontaneously
is the
drills
Much
another
he
of
parts
big
the
thing
;
child's
the
work
an
in
tion
habituadrills
these
into
essential
of
through
such
done
mind
letters
the
way
be
large
a
tions
opera-
repetition
only
also
can
is
simple
does
Ceaseless
words.
daily
in
as
this
in
way
number
same
today
of
other
no
the
application
through
teacher
In
drills.
thoroughly
so
various
the
which
their
and
attained.
connect
of
schedule
full
form
to
intelligent
whole,
as
successful
accomplished
habituated
be
uses
can
the
reviews
daily
The
use
be
With
must
drill.
constant
work
of- minutes.
item.
of
makes
much
so
requires
work
forms
arithmetic
to
of
the
Drills.
a
Shopt
year.
topic,
just before
with
work
coining
each
and
especially
work,
the
during
Reviews
year's
year's
end
for
Necessity
a
in
unified
ful
success-
teaching.
69.
Calculation.
Mental
arithmetical
is called
be
carried
computations
upon
out
to
perform
mentally.
A
that
are
As
the
so
a
61
large
average
simple
rule,
of
percentage
that
however,
man
or
they
can
children
the
woman
easily
show
"2
ABITHMETIC
MODEBN
if called
bewilderment
utter
mentally,
complicated
of
men
the
remove
today.
the
through
cause
acrobats
such
B.
will
B
and
What
2.
each
At
what
be
at
clock
is
right angles
far
investment
and
much
from
the
as
schools.
The
thing
critic to
adverse
an
teacher
drills and
with
70.
drills
which
first of
keep
the
all be
will
Good
of
to
the
"
much
as
How
will the
long
hands
of
a
Drills.
the
and
snappy,
on
business
of
their
public
teaching
questions
lating
re-
room
school-
drills but
They
life,such
The
be
of
possibilities.
results.
full of
do
man
our
drills in the
It is not
will
dren,
chil-
of the
the
ttie alert.
by the children
excellent
an
in the
study
desired
arid
efficiencywill
succeed
with
time
The
supporter
to
child
of the
modem
apply these
to
it,if the drill
to
found
be
otherwise
may
needs.
the
turn
would
pupils constantly
exhibited
o'clock
this mental
ardent
an
produce
quick,
in
dayf
one
development
commensurate
of
of
do
can
development
mental
who
later
energy
Essentials
in
will
arithmetic, therefore, is urged
to
A
be devoted
drill work
general
one
solution
prevalent
and
mow
seven
may
development
any
produce
other!
each
time
in
cluding
in-
work,
to
the
to
:
of his present-day
in the
desire
teaching of arithmetic
much
short
oral
were
days
each
and
six
to
the
good
will
10
teachers
field f
the
between
expended
energy
field
the
in
severely by
the
no
most
before
text
most
much
ago
neglected the arithmetical
will fall
the
drills in
oral
years
field
a
mow
however
be, and
as
to
time
However
work
of
few
a
mow
part
it take
of
can
of
is
following, which
the
as
of
use
There
means
arithmetics
A
1.
as
by
problems
mental
with
putation
com-
the
out
carry
behooves
It
simplest
the
can
paper
upon
approximations.
mental
do
to
they
PROBLEMS
inefficiencyis justly criticized
business
the
AND
upon
though
even
operations
This
them.
METHODS
must
as
life and
directly
good
will
thusiasm
en-
propor-
63
DRILLS
"
tional
for
hunt
must
of
that
to
the
the
teacher.
his
and
drills
Sluggish
the
or
class
require quick, spontaneous
must
dictation
the
pupils
should
addition
examples,
be
to
before
should
or
after
given
all
slow
be
each
of
work,
to
the
in
as
permit the
to
in
as
drills
none
enough
example,
drills
Oral
that
ready
are
swers
an-
written
In
rapid
so
for
annoyance.
drills.
no
of
inattentive
and
worry
who
class
hunt
to
answers.
either
give results
can
result
be
a
become
may
than
barely better
are
have
time
teacher
the
cause
so
much
require
questions,
restless, and
will
drill question
next
slow, listless pupils who
to
slow, listless teacher
A
tion
multiplica-
the
of integers.
Drill
*71.
only
child
time.
is
studied
and
upon
those
which
to
be
the
teacher
children
the
must
number
mixed
child
judiciously
while
habituafted
given
is thus
throughout
the
and
the
to
that
the
serves
of
minute
the
attention
lately been
the
most
out
daily look-
the
on
be
review
create
such
find
with
what
hand, the easier
other
not
variety
be
still have
with
a
Not
have
as
be
must
All.
every
children
the
facts
and
be
attention
wh
neglected,
mly
troublesome
more
of all the
to
hold
animation
processes
his
active
the
during
period.
class
72.
made
Drills
as
thinking teacher
work.
Subject-Matter.
Unify
useful
very
The
the
this
operations
of
must
to focus
teacher
On
The
ones.
do
they
activity
previously studied.
operations
must
the
facts
To
they have
but
for
too
Activity
but
number
difficulties which
for
Engage
number
and
diflScult.
in
engaged
It is well
such
upon
Must
drills be snappy,
the
must
every
the
Exercises
work
New
a
of
means
may
can
do
be
Drills
unifying
much
to
introduced
also
can
be
the
subject-matter.
connect
all parts
by
drills
and
of
oral
"
ft
problems
forms
should
and
on
the
old
with
principles which
be joined
which
it is allied.
often
together in drills.
arise
Those
together
in
number
practice
64
ARITHMETIC
MODERN
Periods
Drill
73.
drills will
be
will
continued
begin
for
fall
of forms
of
the
class.
In
work
from
Pointed
or
teacher
advised
all
Let
of
spur
to
The
in
of
the
In
unnecessary.
teacher
been
able
the
give
read
to
Neither
does
definite
idea
careful
studied
unless
a
to
for
but
from
introduce
it be
it
as
to
to
give his
of
respect
does
subjectnot
mean
miss
and
on
rious
long, labo-
mean
is to
what
to
work.
strongly
the
hit
save
be
drill.
at first
card
drills
able
this
is chosen
may
the
are
paper
of
that
allotted
drills
will gain the
mind
teacher
giving
or
command
work
a
time
be
must
the
drills upon
some
roopi.
must
for
is thus
and
this
in
but
book
class
moment.
inexperienced
outline
has
the
borne
preparations,
accomplished
the
efficieney of
teacher
having
it be
the
it is
before
be
to
forms
the
to
The
drills in which
aimless
the
times.
attention
matter.
of
the
labor, as long additions,
number
reference
pupils for
the
by
board
stand
in order
greatly increase
without
undivided
be
later.
mimeographed
for oiher
the
time, the teacher
paper,
or
fications
modi-
easily
can
at
few
a
and
ones
may
or
only
middle
the
near
some
short*
be
dictation
working
involving much
text
a
permit
comes
seated
requiring
and
Dictations
First
be
may
results
correct
the
he
is sufficient
conditions
local
stand
to
examples
in seat
numbers
the
teacher
children
must
Additional
are
be
grades.
will
drill.
pupils
time
the
dren
chil-
must
the
minutes
Ten
lower
teacher.
drills
after
never
snappy
of the
accuracy
Hence,
and
Space
suit
to
the
times
other
the
for
by the
When
for
well
and
these
out
teacher.
At
for
PROBLEMS
Vigorous,
the
grades and
Drills.
for
illustrations
worked
and
results.
upper
proportionately
Forms
Brief.
time
short
a
give unreliable
to
74.
Be
oflf perceptibly.
for only
drills in the
ened
Must
quite fatiguing
soon
AND
METHODS
jot down
this will
outside
soon
the
nothing
introduce
a
a
skeleton
become
beyond
new
the
text
what
fact.
66
ARITHMETIC
MODERN
of
are
on
the
right
side
the
line.
the
sum
of
The
last
of the
in
the
and
10
or
78.
at
write
teacher
the
as
be
ber
num-
one
the
times
teacher
One
glance.
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board
write
to
children
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down
on
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If
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to
sent
is
will, of course,
sum
5
results
child
each
and
site
oppo-
number
first column.
be
may
number
this
placed
line
third
give the final result
child
result
at
two
sum
the
this sum,
to
is repeated
can
the
or
written
The
the
The
second.
added
for, of
are
line.
and
added
top
then
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the
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numbers
form,
left of
the
product,
sum,
center.
another
In
on
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at
one
PROBLEMS
AND
called
is
operation
whatever
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give the
children
circle, the
the
METHODS
the
down
or
column
dictates.
Other
Devices
Drill.
for
Problems
and
examples
,
placed
be used
can
done
A
as
by each
most
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useful
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alert
means
for
should
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at
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used
problems
teacher, but
practice.
at
In
order
first prepare
gradually
learn
a
to
set
which
The
similar
Such
lems
prob-
practice in the
one
work,
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ciples
prin-
hold
and
teacher's
the
and
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best
which
one
to
the
as
the
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before
lessons
a
teacher
class
of
enced
inexperi-
quite simple after
problems
them
same.
be handled
in 64.
gain self-reliance,the
dictate
work
composition, off-hand,
found
of such
the
for the
can
difficult matter
a
will be
to
kept of
arithmeticians
good
seem
is
work
giving problems
of
much
This
to
pupils
credit
extra
numerical
continually.
may
is
mentioned
in
as
all times.
producing
record
given
time, give
well
to the
using numbers
previously
was
require
class
but
A
is that
exercise
work
is
he
and
out
given
are
of drill.
sort
a
pupil
written
mentally
which
cards
upon
time
proceed.
little
may
and
67
DRILLS
Games.
Number
79.
forms
will
be
found
variety. Children
likely
are
and
pupils,
create
Group
is very
in
useless
out
as
soon
misses
one
to those
of
its chief
received
time,
81.
such
other
Keeping
flash cards
the
work
on
result
the
In
team
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to
with
once,
providing
child
has
other
school
games,
be
may
for
teams
child
for
coaching thus
outside
effective stimulus
more
Team
Scores
be used
with
In
records.
is
given
either
to
his
in Drill.
excellent
the
case
The
results
children
one
point for
eveiy
own
individual
score
score.
room
children
be added
spell-down,"
of each
records
a
drops
provide.
and
child
child
permanent
The
oan
team
or
each
paper
into
can
a
**
clude
in-
which
rivalry
Wholesome
pupils is often
teacher
a
is, of affording drill
that
month.
just mentioned
for individual
or
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in which
in the old
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by keeping
as
number
Often
Games
versa.
or
room
Individual
either
correct
vice
most.
the
the
large class will be almost
a
children
it the
by the poor
any
in
purposes,
competition and
certain
than
he
the
Records.
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fails to answer,
as
dividing
by
group
a
of the
needing
created
and
class, and
part
a
game
a;mong
in
pupils
forms
(2) give life
rivalry
the
great
game.
successful
small
a
only
entire
Oompetition
which
game
of
a
many
operate,
to
good
number
good
wholesome
all
the
Every
few
than
repetition and
of
teacher
a
include
(4)
activity throughout
80.
the
for
easy
class, (3)
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to
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early grades.
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to
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of
games
Number
only
learned.
learn
to
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can
thus
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than
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will be
grade
facts
number
such
This
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room.
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supervised by
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play*
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purpose.
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any
in
at
each
as
incentive
known
younger
twofold
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added
combinations
all may
a
rural
pupils in such
68
MODEEN
A
82.
ARITHMETIC
Game
Team
excellent
an
writes
of
contains
order.
writes
and
after
answer
gives his piece of chalk
his
place
the
at
end
pupil
in
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column
has
of
the
pupils in
is unable
to
pupil
The
answered.
made
by
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takes
and
question
all
interest
to
permitted
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board
is repeated
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mates.
writes
a
in front
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order
pupil
each
his question he
answer
until
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He
in line
of his team
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to
process
In
operation
playing.
line
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goes
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through
on
binations
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first question.
line.
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every
error
any
and
in
captain
the
the
to
of the
by this second
correct
The
number
questions
forms
is
teacher
teams
are
number
then
team
questions.
the
there
same
The
of
columns
as
following
The
class:
small
many
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Each
of
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a
as
PBOBLEMS
Class.
a
questions
other
column
for
board
AND
Small
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the
or
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team
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Each
METHODS
to
pupil
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name
board.
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conclusion
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follows:
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to
for each
unable
The
to
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the
record
daily
board,
of
the
score
social
such
drill
in
of
keen
spirit of
cooperation
stage of
our
to
in
these
number
the
have
children
manner.
work,
threefold
they
motivation, and
the
which
classroom,
every
civilization
this
that
child
given
not
The
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to
serve
spirit
responds.
spirit needs
compose
time
advantage;
possess
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school
grades
to
on,
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place
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are
can
1
pupil
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at
For
paper.
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month
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kept for
10, the
of each
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scored
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from
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upon
The
of
of
all
to
and
they
stimulating
promote
fellowship
In
the
nurturing.
the
in
present
69
DRILLS
83.
referred
may
be
into
teams
printed
total
Each
is
school
when
year
supply
often
to
this
boy
of
useful
23
at
meat
Such
of
estimated.
radius
X
80.
13
is the
85.
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tional
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quite
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25
$2
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3.1416.
in
find
to
pounds
tell about
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This
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check
on
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rough
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per
in
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3.2
is given
of 17
cost
35
-at
fully
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can
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practice
mentioned
of Tests.
beneficial
who
or
also
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problems
was
It is
instance, the circumference
is obtained
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use
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also
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Value
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give practice
product
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For
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results
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;
Approximations
3.1416
The
of
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work.
written
25
will
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computations,
value
form
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drill is needed.
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problems
The
days,
This
day.
give the approximate
can
cents
pounds
many
several
beginning
of arithmetic.
part
girl who
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point towards
Approximations
results
exact
as
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companies.
Approximations.
as
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fractions
Papers
for
four
the
upon
and
integers
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teacher.
reads
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each
general review
a
based
games
84.
is given
at
Work
the
is continued
useful
available.
teacher
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as
result
**meet''
especially
Card
with
corrected
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game
copies
Exercises
done.
copied
that
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room
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given signal from
correct
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game
The
already
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teacher
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stops
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Meet/'
''Field
A
work
in 59,
Short
teacher
tests
and
coming
frequently
pupils.
Those
are
which
70
MODERN
unannounced
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benefits
The
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to
Tests
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2.
Tests
tell which
3.
Tests
show
Tests
This
the
the
day
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work
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Tests
bring
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Tests
show
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pupils
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mentioned
for
work
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:
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greatest
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problems
two
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weaknesses
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upon
resources.
4.
Tests
self-reliance.
develop
EXERCISES
VII.
should
1. What
2.
end
of
the
Is
today.
4.
What
5.
Compare
demands
be
done
should
be
this
overcome
arithmetic
of
teaching
of
of
criticism
few
a
f
Illustrate.
years
with
ago
the
discuss
9.
When
and
for
10.
Discuss
11.
Suggest
12.
Suggest
Give
an
Make
Explain
numbers
the
two
the
the
use
solved
good
drills
be
drills.
Illustrate.
given?
outlined
in 75-78.
in
school
which
card
as
of
short
upon
the
mentioned
those
or
given
illustration
a
of
drills
the
latter
some
out
should
drills f
daily
forms.
other
how
for
essentials
long
how
satisfy
to
material
criticize
and
drills.
daily
the
and
longer
criticism
arithmetic
of
State
14.
to
value
the
8.
drills.
at
'Illustrate.
mental
the
6. Discuss
13.
and
beginning
the
at
f
today.
of
modified
given
be
to
man's
business
justifiablef
can
7. What
review
the
in arithmetic
reviews
of
object
year.
the
it
the
be
fully
Discuss
3. Discuss
on
tion.
prepara-
's lesson.
2.
own
the
:
self-reliance.
in
by including
enhance
can
follows
as
pupils' daily
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prove
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case,
sununarized
be
lacking
needs
of results.
productive
regarding
are
PROBLEMS
AND
most
may
evidence
wliat
These
1.
teacher
written
should
generally
are
1.
As
METHODS
ARITHMETIC
in
75-78
can
be
condition.
paper
or
book
need
not
be
used
suggested.
drill
board.
problems
similar
Illustrate.
to
those
with
71
DRILLS
discuss
15.
State
16.
Discuss
17.
Suggest
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18.
Suggest
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19.
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What
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iv.
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22.
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miles
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to
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yards
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Illustrate.
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approximations
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of
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at
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235
interest
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for
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75
cents
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teachers
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bushel
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in
approximations
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weakness
greatest
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suitable
problems
the
the
results
travels
iii.
21.
of
cost
sound
If
games.
81-83.
condition.
yalue
the
number
good
games.
for
approximate
Find
in
additional
the
of
essentials
mentioned
games
given
some
the
the
grades.
drill
conducting
Illustrate.
workf
23.
What
is
24.
State
the
25.
Prepare
of
object
the
essentials
test
a
of
on
any
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tests
good
a
topic
test.
of
arithmetic.
grade
BIBLIOGRAPHY
Theodore.
Lindquist,
1.
of
2.
XrV:
319.
Mills,
Educational
4.
College
Games
J.
in
F.
Smith,
D.
Record,
E.
E.
Nov.
Beport
Arithmetic
Appeal
Bimonthly,
Mental
on
Arithmetic,
nal
Jour-
183.
Pauline
Mathews,
and
3.
LXXY:
Education,
Plays
Suggestions
VI
Number
1912.
Drills.
the
to
:
of
Play
Experiments
Elementary
Instinct
in
the
on
School
Teachmg
Value
Teacher,
Arithmetic.
386.
Games
and
Number
Bhymes.
Teachers
of
OHAPTEB
"
Vm
MISCELLANEOUS
questions,
present
with
together
various
for
The
Foreword.
86.
have
reasons
few
a
will
general
suggestions,
included
been
not
contain
chapter
some
which
under
the
vious
pre-
discu^ions.
87.
of
Mode
Arithmetic
which
it
as
**how
many?"
French,
ancient
or
mental
(1)
the
of
terms
language
as
Greek.
of
means
operations
invented
Hindus
88.
between
arithmetic.
fractions,
120%
are
are
new
Thought
notations
encountered
of
mode
instance,
percentage
identical
of
are
Be
or
thought
of
and
ing
showmetical
arith-
simple
before
the
all
old
expressing
children
in
72
requires
the
and
language
ideas.
their
1^,
and
decimal
1.2,
and
percentage
of
Many
study
that
this
unconsciously,
fractions;
Decimals
in
Distinguished
fractions,
common
statements.
by
symbolical
centuries
arithmetic
consciously
the
For
and
Must
teaching
recognize,
for
use
and
thought
number-system.
present
and
ideas
Counting
in
this
of
further
the
the
three
by
of
use
in
thought
statement
deriving
been
Successful
teacher
difference
of
the
Language
Teaching.
the
had
represented
the
ideas.
between
is
the
(3)
and
of
race
(2)
thought,
relations
the
the
process
own
English,
is
as
such
its
also
has
language
itself
to
relation
of
It
a
The
of
symbols,
a
questions
distinct
as
development
stages:
in
is
Arithmetic.
peculiar
thought
much?''
**how
or
which
language,
of
quantitative
answers
of
Language
mode
a
possesses
by
and
Thought
of
the
culties
diffi-
arithmetic
74
MODERN
how
of the
sum
of
the
required
three
method
90.
to
of
will
space
work
of
to
two
in
had
they
were
learn
to
which
be
that
further
choice
taken
of
here
the
these
will
permit.
this
have
of
the
bolically
sym-
sive'
progres-
147-151.
see
the
for
necessity
in
both
results
as
the
From
expressed
are
discussion
English,
to
of
meaning
three-sixths
and
As
treated
been
also
places emphasis
has
stating problems
discussed,
been
number
The
recapitulate.
fully in
II
Chapter
requisites for
The
criminati
dis-
as
a
good
a
consistency,clearness, simplicity, and
are
use.
91" Some
times,
as
Arithmetic.
of
symbols
universal
well
For
of this nature
system
the
they
when
first
one-half
interpreting final
not
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denomi-
explanation
.
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in
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teaching fractions,
Symbols
and
of
equal thoroughness.
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see
as
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=
with
up
meanings
by i
of
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is led
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met
confused
teacher
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and
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S +
process.
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idea
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symbolism
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add
progressive
The
upon
1 they
PROBLEMS
add
over
these
" +
to
entirely
an
fraction
other
AND
would
of each
When
of
text
pupils
product
denominators.
the
METHODS
Some
proceed.
to
the
ARITHMETIC
Cautions
avoid
such
forms
usage
Teacher.
the
to
as
as
are
that
It
is best,
inconsistent
found
with
in the
Denominators
placed above
the
should
done.
a
Too
because
operation, together
the
teacher
on
its
forms
many
good form
One
use,
of fractions.
in any
numerator
detriment
common
never
expression of fractions, as
been
they
for each
be
be
mon
com-
has
here
are
erally
gen-
are
idea
lyith insistence
will
all
accompanying
illustrating the addition
diagram
at
found
fusing.
con-
and
by
best.
75
MISCELLANEOUS
carefully
as
never
be
though
they
* *
3
5
+
in
it is
for
well
the
set
is the
neatness
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habit
it inculcates
series of
him
their
unified
arithmetic
is
unification
in the
as
recently, this
few
has
been
upon
the
pupils at heart,
Systematic
and
accuracy,
A
habit
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principles
for
unification
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and
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generally
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teacher
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will
which
is
value
tematic
sys-
that
in
be
valuable
in-
largely
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and
operations, together
this
reason
the
much
as
end,
the
who
his
or
increased
mean
in the
is
accuracy
is taught
of
a
be
taught
be
to
English
has
own
method
not
as
Literature
clearness, greater
of time.
attention; but
a
in
desired
teaching of elementary
such
as
subject-matter
great saving
given insufficient
but
good
upon
topics. It is,however, founded
possibilitiesresulting from
urged
connection,
insist
child, of being
should
and
phase
this
tion
atten-
prerequisite, the teacher,
a
teaching of History
thbroughness, and,
In
to
arithmetic
necessary
It
Geometry.
conscious
his life.
wholly unrelated
A
no
in this matter.
record,
Basic
whole.
of operation
doing anything, particularly
throughout
symbols,
forms
and
computation.
permanent
comparatively
upon
As
in the
in
Unification.
92.
with
of
careful
a
to
always
first requisite of
foundation
making
teacher
good example
a
very
and
operation.
14.
"
give him
they
neatness.
the
or
even
illustration,
may
an
symbols
any
out
general
himself, must
the
that
carrying
figures and
a
As
56-4-4
"
master
thoroughly
so
8X7
=
pupil should
The
a
' '
be understood.
can
statements
thereafter,
teacher
as
mentioned:
be
of
the
by
accepted
just
learned
be
must
use
principles. Incorrect, slipshod
new
should
their
and
symbols
New
make
the
it
one
only the welfare
selfish interests.
Until
metic
arith-
great
to
be
of his
It is safe
76
to
MODERN
ARITHMETIC
that
say
of results
93.
real
so
successful
make
and
these
number
with
the
discussion
and
fractions
to their
place,
it
its related
and
56.
The
basic
will
fact,7
and
number
25, 9f
,
lOJ, 6J, 484, and
for
much
a
the
| of
only
one
facts,or relations,
with
this primary
9 to
adding
by
tens
based
any
then
one,
with
together
multiplication by
practical value,
practical point
in
arithmetic
that
spasmodic
teacher
must
recur.
And
for
can
be
upon
Let
in
drill
will
constantly
on
textbook
practice,
"
easily supply
it be
this
11, 22,
unification
a
the
it
the
borne
lookout
not
of
additional
for
ever,
how-
mind,
grouping
them
they
as
suflScient
will
problems,
not,
or
or
The
result.
no
or
supply
doubtless
teaching of
in
method
necessity,
the
rather,
bring little
does
as
or
unifying
of
view,
relations
if the
such
of
phases.
all its
number
unifying
teacher
learn
to
Again,
are
on,
with
relations.
number
from
so
fact
8, i of 56, and
-f-
the
as
processes
the first
Thus
together.
aliquot parts
Hence,
gers
inte-
number
connection
increasing
In
every
number
it.
around
unit.
in connection
tion
is to call atten-
here
required
their
from
to
cuts, such
56
7,
-f-
other
The
grouped
one
all short
So
56.
=
amounts
deducting
56
As
operations with
studied
be
to
the
community.
better
that
in this way,
easily grasped
be
mind
putations;
com-
himself
simple illustrations.
8X7,
are,
8
X
treated
is necessary
in
are
for
will
in
accurate
his
fundamental
few
a
computations.
and
of
which
relations
as
them
men
be
can
borne
come
pupils
one,
of
by
facts
should
7X8
business
all that
be
of arithmetic
well
as
for
of the
value
must
alert
secure
relations
relations
Number
teacher, of number
and
of the
productive
be
business
in
especially
thus
commendation
will
teaching
in the unification
keen
his pupils
PROBLEMS
Unification.
of
by the
v^e
AND
lasting.
so
serviceable,
most
The
and
instance
one
of
method
Instances
Enxly
represent
is
other
no
METHODS
"
terial
ma-
the
simply
77
MISCELLANEOUS
the
event,
drill
in
be
should
these
for
the
does
in
of arithmetic
phases
with
division
It
child
the
division
when
the
old
good
which
94.
division
of
any
operation
will
understood.
involving
equation,
percentage
the
of
95.
not
for
necessary
but
teachers
must
and
mutual
out
was
be
and
are
some
of
Similarly,
in
inverse
the
tions
opera-
much
of
do
very
in
plifying
sim-
(See 198.)
treated
the
as
wonders
The
in
amply
be
discussion
55
to
with
graphically
special
recognize
relations
enough
the
suggestions
to
the
are
These
recurring
existing
bring
children.
to
that
texts
needed.
are
the
to
Textbook
it is evident
no
teachers
resourceful
brief
Not
Teaching,
alert
silflSeiently
home
operations
relations
wiU
also
already
one
of inverse
idea
ing
mean-
readily
more
with
plan successfully
also
the
here.
foregoing
this
for
inverse
Indeed
problems.
such
specially prepared
relations
following
P,
=
Matter
the
volution.
in-
and
point
are
grasped
principles together
them
between
decimal
fundamental
6
X
of
a
tion,
addi-
with
familiar.
the
of the
repeated
From
carrying
ideas
be
example,
with
the
as
decimals
be
principles
Unification
Writing.
of
such
r
solution
recurring
will
and
him
to
Unification.
Application
problems
for
;
various
the
subtraction
as
clearly, if it is connected
more
cuts
taught.
so
Instances
much
use
be
short
evolution
and
already
placing
Later
of
and
should
is
he
for
rules
multiplication and
and
presented
are
day.
suggested
subtraction
as
operations
such
they
further
groups
natural, and, hence, easier
more
learn
to
inverse,
multiplication,
is much
operations with
to
its
operation, with
every
carried
any
after
arithmetic
practice of unifying
be
may
day
countless
business
the
In
text.
family
zeal
the
to
the
as
unrelaxed
This
grades.
upper
relations
refer
not
of
problems
with
considered
are
the
number
continued
This, however,
which
in
numbers
change
the
The
teachers
78
MODERN
for
ARITHMETIC
This
first requisite is
includes
language
the
old
after
until
pupils'
Stress
for
lookout
they
of
operations, be
be
a
in
the
short
time
mind
Constant
keen
seeing
these
simplify
numbers
in
attention
3.
New
relations
in this
**the
from
episodes
4.
in
the
class work
the
main
reviews
and
so
inverses
it will
only
its inverse
supplied
his
in
thus
a
will
end
Problems
without
afford
lent
excel-
the
pupil's
in these
than
rather
process
by
ability for
the
only is stated
the
dovetail,
the
is made
are
foresight by
They
of
current
a
upon
the
incidental
the
those
ready
al-
glide into
to
they
must
along
come
of
shock
teacher, and
should
be
as
thought.
necessity.
by taking stock
ascertaining
with
experiencing
introductions
are
it were,
as
pupil
without
accidental.
most,
as
answer."
careful
merely
Constant
count
such
with
planned
the
surest
time.
save
upon
topics incidentally
far
and
Thus
While
constant
relations
developing
and
operation
topics should
newness.
classes
the
on
stimulus
connection, because
understood.
new
in
wonders
the
and
than
work,
will
the
and
is concentrated
numbers
a
pupil.
work
which
become
to
as
reciprocity,i. e.,
operation
each
general work
practice
the
early in the
number
the
grasped;
are
number
and
introduced
the
will
be
to
If
watchfulness
teacher
ones
problems.
until
of
with
connected
principles, rather
on
pupils taught
the
new
new
command
good
a
frequently
so
recurring principles
in
occur
laid
be
and
the
used
languaga.
voc?ibulary.
own
should
problems,
or
be
may
as
tactfully
be
of
meanings
the
these
part of the
should
terms
of the
symbolical arithmetical
the
use
ability known
the
as
New
which
2.
teaching of arithmetic
comprehension
a
abjjity to
well
as
of English.
the
the
PROBLEMS
unifying the subject-matter:
1. The
to
AND
increasing their efficiencyin
through
be
METHODS
what
from
has
These
can
be
time
made
time
to
of the
been
neglected and
79
MISCELLANEOUS
particular attention.
needs
what
concerning
connection,
taking
of
up
from
arises
a
limit.
time
the
work,
normal
The
to
of inverse
use
All
With
teaching should
this
in
end
given special attention.
To
future
requires, of
work
of the
with
it.
school
At
child
the
be
for
96.
Lesson
have
future
well
as
Each
lesson
It
dbject.
own
quite strictly throughout;
it
make
during
impossible
recitation
the
which
but
should
when
the
work
ought
up
will
made
give
with
depend
of
of
the
to
page
to
or,
Whether
how
margins
as
rule
for.
each
this be
much
as
ested,
interRules
said
as
as
To
time,
same
tailed
before, dove-
far
this
lesson
as
possible.
end, let the
before
burdensome
use
the
bility
flexi-
some
the
to
arise
may
of
At
lessons
to
in itself
really
are
the
this
to
that
admit
a
grade.
anticipated.
it.
strive
upon
largely
not
previous
thing
class.
not
amalgamated
the
of
of
unit
hold
pupils
demand
consideration
some
the
be
to
is the
had
as
the
whole, each
a
matter
a
the
teacher
iron-clad
so
that
Unification
it
which
circumstances
with
teacher
the
be
never
in
of
strictly,however,
so
consider
to
be
to
the
teaching
the present
is well
on
sympathy
complete
should
not
eye
development
for
as
be
should
one
unified
one
arithmetical
the
Plan.
its
into
which
knowledge
as
to
application.
years
arithmetic
time, the
coordinated
for all the
well
required
process
thus, with
as
well-known
a
and
thorough
a
of
lengthening it.
of
use
the
unification
time
succeeding
teach
years
same
working
future
of
course,
succeeding
can
teacher
and
in
applied
within
principle and
view, every
work
be
work
of
instead
for
be
the
will
that
and
the
work
prescribed
certain
down
such
teacher
required
processes
cut
less
doubt-
tendency
the
the
topic with
one
training school
in the
school, actually
of
the
cover
of
deplorable
idea
perverted
tried out
as
complete
5.
the
is necessary
course
This
one.
new
a
abandoning
the
timely in this
is
word
A
taking
or
teacher
previously suggested
in
not
has
66.
80
MODERN
ABITHMETIC
Preparation
the
lesson
with
need
not
in
work
the
PBOBLEMS
solving of
it must
twentieth
all
for
acquainted
It
action.
**
slogan of
century
in
problems
becoming
mean
laying plans
general and
itself into
AND
the
mean
but
assigned
the
METHODS
solves
re-
ness."
prepared-
,
Lesson
97.
Assignments.
consideration
demands
This
portance.
need
where
each
ones.
Assignments
lesson
work
be
connected
be
may
given
of
shorn
into
said
nothing
in
98.
that, if put
from
very
there
Three
During
which
drill
examples
in
only
inches
often
in
ing
count-
part of the
the
assignments
make
casually.
teachers
time
waste
their
transform
would
of the
requiring
the
pupils
the
when
board,
every
First,
recitation
There
children
fifteen," instead
some
pupil
is
there
than
on
may
possible
as
new"
largely be avoided.
can
at
topic
discretely and
beginning
five equals
work
new
brought
proficient arithmeticians.
the
repetitions, as
times
thus
time,
same
is
really
to
work
work
little
as
find
to
New
is,at the
wholly
is not
order
lesson.
all the
in order
use,
to
at
in
preparation
Too
proper
mediocre
papers
needless
further
to
Time.
is the waste
out
*'
Careful
material
of
A
Much
necessary
new
Waste
and
tation
reci-
Review
possible.
as
it
;
It is
of the
course
next
counting by quarter
is,however,
introduce
and
the
whole.
new.
inches.
by whole
far
incidentally with
being
newer
as
direction
or
unified
the
preceding
hurriedly.
necessary,
connect
purpose,
a
its
about
teacher
to
introduced
be
if
up,
of its strangeness,
some
together
thus
a
it
im-
arithmetic
in
the
made
during
with
brought
than
upon
be
never
its
son
subject do the les-
no
much
so
made
be
by which
something
is thus
depends
they
and
In
ment
assign-
with
attention
should
lesson
commensurate
more
that
of
matter
rarely receives.
it
assignments
suggested
The
of
in
ing
giv-
are
the
to
merely
teachers
has
say,
**
teen."
fif-
assign
completed
82
the
pupils
will
Teach
1.
do
just
3.
what
such
his
from
the
and
work.
unsettled
it is desired
it should
case
from
that
be
day
end
the
at
teacher
careful
be
must
to
pupils.
assigmnent
lesson
work
which
a
teacher
actual
to
The
example.
and
requires
the
no
to devote
the
but
process
Suggestions.
he
Leave
question
In
time
more
by precept
Complete
2.
attention, because
more
have
PROBLEMS
AND
subject is slighted by this
no
General
100.
a
that
parents
really given
is
METHODS
AMTHMETIC
MODERN
of
a
cleared
up
do
not
unless
consider
it be
further.
of
beginning
the
at
day.
should
pupils
the
to
recitation
the
next
lesson.
Require
4.
explanation
of
what
done
a
working
It
gives
him
gives
and
6.
Do
be
revolution"
any
8.
the
oral
the
be
and
Always
and
extremist
the
look
the
the
essential.
big
illustrate
in
the
text, gives the
in
the
work
of
every
the
to
teacher.
pupil
Let
"Evolution
and
not
to
improve by adopting
and
ing
adapt-
Idea.
Be
careful
to
never
confuse
tho
EXEECJISES
the
difference
between
the
language
and
arithmetic.
necessity
the
to
examples
be
ideas.
with
thought
Explain
dictate
can
resourcefulness.
lookout
the
on
for
and
This
confidence
see
has
of
not
confusing
the
language
and
mode
thought.
3.
for
of
mode
2.
of
Explain
to
plete
com-
English.
to
reference
to
work
possible.
ability
pupils
arithmetic
in
as
faddist.
or
VIII.
1.
far
as
him
his
slogan.
the
practical
non-essential
the
the
in
boy
Permit
mistake
The
opportunity
teacher
continually
new
text
without
boards
interested
an
when
the
work.
himself
an
then
to
general strength
not
7. Be
the
in
the
first and
attention
stop
his grammar.
correct
without
at
confidence
but
to
say
his
of
most
teacher
also
to
teach
to
with
class
has
direct
Learn
5.
problem
a
he
settled
been
English
good
Oive
the
4.
mode
of
Suggest
5. What
i^ymbolsf
illustrations
two
showing
how
the
language
may
be
taken
thought.
two
degree
Illustrate.
examples
of
freedom
of
incorrect
should
use
pupils
of
be
symbols.
given
in
the
use
of
83
MISCELLANEOUS
6. DiBcnss
need
the
of
orderliness
and
system
arithmetical
in
work.
XUnstrate.
is
7. What
8. What
is meant
Explain
teaching
and
system
Illustrate.
unification
the
by
unification
how
requisite for
first
teacher's
successfully f
neatness
9.
the
will
arithmetic?
of
simplify
Illustrate.
work
the
and
time.
save
Illustrate.
10.
Give
11.
Discuss
illustrations
three
and
the unification
of
illustrate
the
Explain
how
13.
Make
a
very
14.
When
and
15.
Discuss
16.
Suggest
in
mentioned
of
are
attaining
for
suggestions
the
various
of
plan^which
the
of
forms
of
Illustrate.
in
used
be
can
time
wasting
other
unification.
lesson
day's
next
forms
discuss
form
a
lesson
should
how
and
each
processes.
be
trate.
Illus-
assigned?
mentioned
wasting
teaching.
under
time
than
98.
those
98.
What
17.
brief
the
inverse
subject.
short-cuts
12.
of
the
are
of
advantages
grouping
work
the
for
suggested
as
schools?
rural
Suggest
18.
eombined
for
19.
Discuss
20.
Discuss
workable
a
rural
the
other
each
of
in
plan
schools
as
suggestions
the
which
the
work
under
mentioned
general suggestions
grades
ia
99.
time
saving
for
the
of
on
in
rural
a
methods
under
school.
100.
BIBLIOGRAPHY
1. Brown
2.
and
Educational
School
How
Bimonthly,
VI:
A.
Elapper,
6.
Olds,
7.
D.
Smith,
D.
to
Get
Greater
Effloiency
in
in AritMnetic.
of Time
pp.
School
Arithmetic.
Elementary
111-34.
of
Mathematics,
E.
Arithmetic,
103.
Opportunity
and
Science
in
461.
Paul.
G.
Work
294.
Economy
XIV:
Teacher,
5.
School
U.
V.
Jessup, Walter
4.
110-30.
XIV:
Mathematics,
and
Collins, Joseph
3.
pp.
Development
Collier, Myrtle.
Science
and
CofPman.
The
the
XII:
Teachmg
Teacher
in
the
Classroom.
355.
of Arithmetic,
pp.
32-37, 60-71,
88-101.
8.
Stamper,
9.
Eeport
S.
Bui.
Alva
Am.
pp.
196-224.
Inter.
Com.
W.
Com.
13, 1911.
pp.
36-42.
on
the
Teaching
of Mathematia.
OHAPTEB
IX
ARITHMETIC
PRIMARY
Basis
101.
of
delight
free,
when
convenient
any
water,
wood,
word.
The
the
and
medium
painting
the
is in
educative
of
seeking
teacher
a
which
tasks,
pating
antici-
in
pleasure
the
satisfy
definite,
own
the
and
cutting
must
her
printed
The
the
sand,
or
action
result.
when
almost
paper,
written
spontaneous
cease
as
with
as
the
or
children,
through
ideals
desired
a
value
imposed
are
their
dolls,
realizing
Normal
expression,
of
ready-made
in
Work.
realizing
in
pleasure
or
Number
Early
quirements
re-
ceived
precon-
results.
tests
which
children
the
That
it
be
that
powers
Thai
3.
it
lowing
fol-
the
stands
adapted
be
can
purpose
a
their
to
saiisf action
or
tends
connected
cultivate
to
physical
a/nd
menial
overstrain
no
unconsciously
it
it
efforts,
best
so
there
in
feel
their
arouses
2.
unless
valuable
not
:
That
1.
good
with
it.
taste
in
children,
the
4.
it is work
That
as
which
the
of
work,
things
designed
it.
based
the
the
children
the
school
failure
For
are
have
in
;
84
in
the
cize
criti-
purpose
for
included
been
constructive
and
instance,
brought
grounds
fulfill the
to
creative
can
themselves
might
grade
the
upon
children.
the
on
in
children
the
its
work
outline
that
discover
they
they
All
an
is
suggested
Work
in
the
contact
garden
in
ties
activi-
nature-study
daily
;
in
with
their
real
trips
p
L
L
f#
"
"
r
typical landscapes,
to
and
so
to
their
continual
unlimited
literature
In
them
The
for
the
stories
dramatized
are
chart
community,
The
skill
102
is selected
the
In
needed
making
plan is often
number
The
102.
suggests
various
different
months
spring
much
grades
such
of
first
of early
as
are
plans the
work
convenient,
most
by the class.
thus
verifying, and
the
daily
quantitative
use.
tions
ques-
much?''
The
the
naturally
here
chart
work
construction
In
ing,
mak-
shown
for
early fall and
be
replaced by
the
late
door
out-
activities.
Early
need
what
to
determine
necessity into
Work.
will
children
guide
continually planning,
of ''How
and
of this work
of
and
power
times
many
making
of real
year.
and
these
under
the
a
best, or the
are
the
Forms
articles
them.
never
child
actual
comes
forms
excursions
103.
the
children
Construction
Early
pupils making
directly applied in the
it may
each
for
many?*'
things which
home,
cases
is
continually meeting
of ''How
many
school
judging, estimating, and
are
and
is made.
which
element
pupils
article
articles
the
comparing,
the
in many
though
selected
this work
that
article
after
demands.
care
classified
however,
This, however, is
the
the
and
the
the
of the
use
one
making,
independently,
In
noticed
making
for
with
for
month
not,
of another.
making
how
in
imaging
only the
presents
each
are
It will be
give them
image-growth
selected
are
the immediate
or
gained
clear, definite
them
by the children.
of section
exercises
heads.
various
help
the effort is to
expression which
actually be made
can
the
to
which
those
are
questions, and
own
opportunity
genuine
and
farm
the
performed
experiments
answer
of
visit to
swamps,
plants of the city.
industrial
The
shore, sand-dunes,
the lake
as
their
in
on:
85
ARITHMETIC
PBIMABY
to
Number
measure
suggested
Work.
constantly
in the
chart
In
in
and
the
primary
constructing
in
making
86
MODEBN
simple
ARITHMETIC
keep
of the
account
tell
to
time, and
order
they
and
value.
foot
should
but
his
in
judging
104.
have
A
Mmimum
should
be
a
minimum
grade,
which
should
of the
in
grade
cooperation with
of
grades
first year,
the
school.
functioned
be
relied
knowledge
105.
in 'each
ready
upon
small,
for
measurements
for
facts
present
and
their
The
106.
minimum
planned
There
for
each
teacher
of
perience
ex-
each
higher
next
the
required
oughly
thor-
be
children
the
may
arithmetical
of
too
increasing
great,
for
will
in
to
constantly
demands
be
will
be
that
responsible
problems
they
struggle
metical
arith-
of
knowledge
these
that
but
children
not
suflSciently
are
the
to
do
full limit
them.
overcome
work
work
projects, there
teacher
Significance of the
arising in the
ment.
measure-
should
fund
children
the
involved
processes
ability to
If
for
and
difficult to require the
of
that
so
factor
a
and
be
processes
growing
social
concrete
obstacles
should
fact,
needs
of the
a
use.
Projects.
operations.
the
and
a
the
teachers
grade,
have
to
upon
Social
facts
for
arithmetic
by
little
Very
certain
but
the
as
in
decided
as
foot
the
in
isolated
an
tool
by
ard
stand-
inches
Arithmetic.
determined
gently,
intelli-
the
in
requirement
children,
culations
cal-
time, weight,
12
of the
necessary
Requirement
be
and
with
as
memory
a
as
learn
to
work
are
knowledge
and
power
numbers
volume,
there
to
a
to
games,
they buy,
acquainted
that
committed
school
work.
lines, areas,
instance,
should
of
this mathematical
do
may
in
be
not
their
in
other
scores
that
recognize
early become
For
keep
materials
and
soon
measure
child
the
they
must
of
units
books
they
that
and
to count
essential
are
In
need
They
PROBLEMS
AND
cooking^ gardening, and
in
apparatus
activities.
METHODS
Minimum
may
the
take
grade,
Requirement.
the
and
children
many
lems
Prob-
beyond
individuals
the
in
OHAFTEB
PRIMARY
107.
three
Concrete
the
activities
aid
to
the
number
early
on
are
teacher
work
The
equipment.
earliest
very
third
the
might
third
have
teaching,
the
lessons
since
for
instead
of
success
might
apply
this
and
their
the
first
or
the
grade;
beginning
or
children
class
were
it
in
the
a
and
in
many
activities.
were
being
kind
and
play.
88
of
the
they
in
given
the
full
real,
a
its
described,
ask.
tj^e
since
situations
initiative
If
the
to
worthy
judgment
little
are
of
type
dominates
stimulated
mathematics,
knowledge
and
seems
too
children
their
lesson
by
gauged
valuCi,
the
of
for
teacher
The
but
highest
the
represent
results
the
be
independently
work
little
among
the
second
the
opportunity
give
hypothetical,
a
the
beginning
of
primary
any
be
would
of
ments
measure-
with
and
lessons
last
not
questions,
responsibility
without
class
the
in
since
primary
these
given
one
but
the
upon
lessons
children.
the
and
questions
do
they
of
part
directions
of
two
the
in
any
grade.
first
The
size
at
come
IX,
be
may
(measuring)
number
Chapter
based
which
first
from
important
lessons
the
whatever
chosen
be
in
so
selected
been
classroom,
the
mentioned
illustrative
three
have
on
order
might
problems
measurements
grades,
of
In
lessons
type
typical
school
linear
no
Lessons.
prepared.
been
of
(Continued)
ARITHMETIC
lUustratiye
following
the
X
gain
arising
of
number
But
lesson
even
for
children
a
will
immediately
naturally
in
PEIMAEY
such
Throughout
with
associated
write
would
they drew.
allowed
In
question
Lesson
would
process
from
attention
not
be
central
the
lesson
indicate
quotation marks
the
direction.
or
purpose:
(a)
To
teach
(b)
To
help
foot
the
which
of
height
as
chUd
each
height)
An
long which
foot
a
1.
Teacher's
Child's
line
dren
chil-
The
lesson.
the teacher's
108.
children's
the
illustrative
each
each
be
would
it represents.
(length)
under
**foot''
word
the
word-association
the
distract
point of the
idea
the word
But
to
lesson
a
the
89
ABITHMETIC
linear
a
find
to
he
unit
measure.
approximate
an
in
use
can
of
(his
measure
distances
judging
and
own
the
objects.
purpose.
interest
merely
in
a
activity
which
this
at
age
is
end
an
in
itself-^
interest.
game
Lesson.
Time,
25-minute
of
Exposition
Teacher
Number
period.
in
class, 20
as
a
maximum.
lesson.
asks:
'^'Who
is the
(One
child
result
"How
tallest
that
requested
John
can
you
is
prove
classf
pick
to
out
the
tallest
child
with
th"
chosen.)
with
agree
many
"How
is
child in the
that
the
choice
John
madef
is the
tallest T"
"HowtaUisheT"
"Who
knows
how
to
find
outT"
'
109.
Descriptive
Defail.
kindergarten experience,
the
children
are
each
As
the
given
foot
foot
the
children
ruler
rulers
have
is suggested,
to
examine.
had
and
90
MODERN
They
ARITHMETIC
then
fpre
of paper,
ones
and
measure
One
asked
on
horizontal
the
line
lines
board
foot
and
Children
class
one
hand
foot
the
between
their
of
number
devices
what
considers
the
a
110.
With
very
Possible
a
return
to
lesson
and
the
Further
"How
strip
long
as
lesson
will
of
depend
ten
"How
other
a
distance
its time
might
ask
that
the
energy.
this
at
point.
Class.
Strong
a
the
the
teacher
would
first part
children
they had
and
end
with
in
the
upon
particular class, and
of children
to
found
of
hold
and
up
the
the
measure
strips
a
foot
long
would
it take
to
make
a
is tallT"
indicating that
answers
in
foot long.
these
John
than
apart,
the
the
spite of the day's experience, the teacher
have
John's
a
of
many
as
In
and
just
were
much
jei foot
measures
of the
presented
next
paper
a
children
above
hand
one
use
group
perhaps
line
lines
all the
exactly
Development
problem
strips of colored
which
best
class, a
mature
more
Then
to
go
how
horizontal
hands
used
of the need
slow
to
ruler.
a
judgment
With
note
quickly
teacher's
he
After
it.
lines, and
the
then
teacher
with
hands
the
The
the
all asked
under
lines.
perpendicular
while
are
long
drawn
to hold
draw
; to
teacher's).
foot
correct
have
teacher's)
rulers, a perpendicular
a
pendicular
per-
a
the
children
opposite the other;
apart,
choosing they
blackboard
the
(like
line
and
asked
are
longer
among
After
room.
strips
yellow
or
hun^
(like the
"
long
horizontal
a
those
upon
draw, without
accurately they
the
long
foot
a
measure
they have
draw
erased, the
are
long and
more
red, blue,
judgments.
to
foot
a
of the
their
is asked
child
these
wall
a
verify
line
a
select
to
PROBLEMS
AND
exaciVyi a foot long, from
shorter
and
METHODS
John
is
anywhere
will
possibly
between
two
feet tall.
may
height
we
T"
measure
more
easily
than
with
the
strips to find
out
PEIMARY
this
At
a
point it is expected
reach
by the
asked
to
teacher
The
lesson
four
bring
and
Lesson
111.
To
review
(b)
To
teach
(c)
To
continue
the
long,
and
work, the
seat
wide
as
bring
Also
tomorrow.
foot
the
fact
new
the
the
as
ruler,
your
ruler.''
your
that
there
of
problem
unit.
finding
inches
12
are
the
in
of
height
(during
pleasurable activity
"
take
in their
form
25-minute
minds)
home,
quickly
up
his
and
at
foot.
a
children.
the
this
lesson
a
definite
.
the
all of
work
Children
wall, until
Teacher
class
is
select
accurate
after
in
class
(20).
trial.
one
is not
child
be
to
Effort
made
paper
Each
ones.
accuracy
is
holds
expected
approved.
discarded.
and
four
has
strips of
examine
teacher
disproved
child
each
children
:
choose, from
then
of
number
strip. Great
correct
Careless
lesson
help the
Children
Same
period.
of
Exposition
colored
strips hung
strips, each
1 foot
on
the
long.
directs:
"Hold
up
"Tell
how
correct
more.
:
Time:
from
tall, and
purpose:
continue
Lesson
is
firmly against this
children, for
foot
one
impress
and
Children's
will
John
purpose:
(a)
problem
this need.
2.
Teacher's
To
class
to
is 4 feet
placed within
spaces,
heels
and
discover
will
:
strips, each
them
foot
by giving the
ends
following directions
''Make
he
that
child
some
foreseen
his back
tt is found
board,
has
who
with
stand
that
off into
marked
plank
planed
91
ABITHMETIC
ones."
the
strips that
many
strips
are
you
right
had
to
which
choose
yourselves
you
in
order
to
made."
have
four
92
MODERN
''Who
like
would
''If
how
ARITHMETIC
each
much
each
"How
can
"Yesterday
much
"How
can
"Please
that
June
find
could
we
tall you
John
is tallT"
he
as
out
jutt
out
inch
' '
year.
how
long
as
in
PROBLEMS
feet
4
was
nowf
are
mare."
and
moret"
find
we
take
"Tell
this
just
out
found
we
strip
grown
find
AND
strip just
a
another
has
one
you
"How
have
to
made
one
METHODS
outT"
rulers
your
what
these
small
examine
and
them
are."*
divisions
carefully.''
points
(Teacher
spaces.)
in
then
teacher
The
much
the
"How
same
The
class
counts
claps
With
way
inches
many
Glass
there
one
12
inches
for
we
shall
are
' '
If
child
children
' '
to
Paste
have
before.
ruler.
the
on
counting
cise
exer-
the knowledge
gained
The
lesson
ends
with
that
lowing
fol-
the
work:
make
to
of
each
the
a
long strips
our
her
better
a
to
see
how
tall
be
the
way,
done.
asks
the
* '
teacher
follows:
as
way,
four
your
it could
which
in
way
suggests
follow
to
'''Divide
tell
each
may
no
.
taught
class repeats the
seat
today
was
idea
' '
now.
You
begin
inches
the
foot.
a
inch-unit
foott"
a
to
the
in
foot-unit
the
points.
children
directions
"We
in
devices
all of the
are
there
girl points
one
as
the
child
teach
to
that
are
as
various
until
proceeds
strips into
correct
1 inch
strips together, allowing
one-inch
the
on
end
spaces.''
of
each
strip
your
long
overlap."
"When
strip
with
you
a
many
"Any
one
the
"All
' '
Those
pasted
the
four
strips together,
measure
ruler."
"How
make
have
inches
who
short
of 4
may
make
can,
long strip just
bring
their
who
make
4
short
a
in
feet
long strips
the
feet
to
is itt"
strip just long enough
short
length. '
class
'
tomorrow."
strip bring
it to
class
also.
' '
to
PEIMARY
laesson 3.
112.
to
a
when
arose
loam,
that
the
another,
that
the
makes
great
deal
a
find
which
out
the
water
need
the
use
make
their
for
and
bring
each
it to
school
home
from
which
seemed
getting
a
convenient
should
morning,
before
in
the
the
paper
or
^the
science
this
was
was
a
tion
objec-
no
things
children
to
of
make
of
cardboard,
cubic
4
to
boxes
at
shape
being that
The
inches.
teacher
lesson, the
value
any
only requirement
exactly
which
lesson
Points
hold
the
certain
asked
"
Although
home
at
teacher, seeing opportunities
the
teaching
wrapping
gestions
sug-
child, who
box
there
order
the
one
sized
same
measure,
in
tests
Among
of
own
they would
that
the
that
was
tomorrow."
it,and
box
for
to
held
his
perform
to
value.
the
make
for
suggestion
number
[
soil
the
by the class, and
them, accepted
each
of
soil
which
and
;
holds
desire
the
to
gested
sug-
seeds
soil
the
teacher
of any
comparisons
growth
of
water,
the
of
child
led
wanted
by
amount
same
This
child
each
soil (sand,
One
which
most
suggested
was
way
made
in the
As
the
water
the
held
could
round-about
of
of difference.
measuring
said, **We
with
do
.plant their
to
best.
grow
to
amount
longest.
it
to
much
soil
experiment,
to
'
had
light
would
problem
in which
to find
given
was
The
getting ready
were
seeds
children.
grade
experimenting
were
clay)
or
third
lesson
number
following
children
the
gardens, and
The
fifteen
of
group
93
ARITHMETIC
next
gave
the
follows:
which
lesson
seemed
the
to
teacher
portant
im-
:
initiative
and
(a)
To
encourage
(b)
To
make
children
responsible
in
independent
for
the
judgment.
preparation
of
their
own
be
exact
materials.
and
(c)
To
teach
economy
(d)
To
make
dear
should
(e)
To
be
the
the
fact
use
that
of
a
materials.
tool
of
measuring
must
convenient.
review
work
in
lines, areas,
and
cubical
contents, hy
pre-
94
flenting
similar
problem
new
a
METHODS
ARITHMETIC
MODERN
others
to
PROBLEMS
AND
children
the
have
previouslj
solved.
Possible
children
it
had
sure
of
part
was
important
interesting experiment
an
had
themselves
they
question
to
the
and
asked,
by
answering
method
a
of
a
measuring
suggested.
in
response
113.
itself
activity
The
(b)
seemed
lesson
:
(a) Because
they
the
why
reasons
interest
the
such
finds
experiment
an
a
children.
of
Lesson:
Subsequent
Time:
in
involved
25-minute
Number
period.
in
children
of
class,
15.
Exposition
The
first
Lesson.
Number
"
minutes
5
in
spent
were
examining
the
boxes
made
at
home.
Three
shapes
The
held
4
by
boxes,
of
each
which
the
of
the
best
teacher
of
different
which
had
they
and
made
added
shape.
No.
1,
4
X
4
X
1, with
open
area
at
top
of
4
X
1.
Box
No.
2,
2
X
2
X
1, with
open
area
at
top
of
2
X
2.
Box
No.
3,
2
X
1
X
2, with
open
area
at
top
of
2
X
1.
Box
No.
4,
1
X
1
X
4, with
open
area
at
top
of
1
X
1.
pick
out
children
for
No.
the
box
paper
the blackboard
after
asked
were
chose
select
name,
four
Box
piece of
to
of
set
two
convenient
most
her
found
They
to
selected
class
made
The
presented
children.
in.
cu.
The
two
then
teacher
the
by
brought
were
in
use
which
hold
and
measuring
this purpose.
2 for
and
to
be
could
4
cubic
recorded
looking carefully
box
the soil.
They
of the
They
from
the
inches.
Each
child
judgment
the
boxes.
under
Three
shape
mously
unani-
then
were
made
his
at
the
asked
smallest
passed
his
own
children
96
METHODS
ARITHMETIC
MODEBN
AND
PBOBLEMS
BIBLIOGRAPHY
Buxton,
1.
Flora
Cooke,
the
in
matics
XII:
Francis
English
in
Beguirements
and
School
Elementary
School,
Parker
W,
Board
Press.
Minimum
J.
Card
and
liathe^
Teacher,
245.
Journeys
in
Larkin,
4.
the
Child*
72,
109,
5.
s
Morey,
Charles
Sage,
Co.,
and
and
Kindergarten
Chicago.
Numbers
Teaching
Fint
Based
First
Grade,
upon
II:
12,
TrohUms
16L
Little
The
Folk's
Number
Book.
Charles
York.
Dorothy
H.
An
Arithmetic
Tournament,
Primary
cation,
Edu-
Primary
cation,
Edu-
162.
Sanders,
Smith,
W.
New
Sons,
and
9.
The
of
McLean.
151.
F.
XXV:
8.
Foresman
Method
A
Experience,
XXV:
7.
J.
Mary
Scribner'e
6.
Scott,
Numherland.
Lillian
Waldo,
and
Stone
Van
Ada
Harris,
3.
XXV:
Wis.,
Faper
L.
Fred
Corran,
Menominee,
Construction.
2.
and
F.
George
G.
A
Lesson
with
Milk
Bottles,
162.
E.
Lucy
First
Ayres.
Grade,
for
the
11
Model
Number
:
and
Word
Sheets,
garten
Kinder-
237.
Grocery
Store.
Primary
Education,
CHAPTER
History.
The
subtraction
and
of
the
the
(1)
MCCLIX
(2)
MMCCLXXVII
addition,
difference
left
A.
to
This
D.
who
are
the
at
right,
process
of
method
with
acquainted
in
seems
the
addition
the
of
sum
is first written
middle
''1*'
5
+
+
^
Beckoning
with
to
of
+
1
2
6
+
7
tbe
over
"
to
below
from
which
etc., thus
5,
actual
the
of
+
written
the
erased,
aid
+
**5,"
In
column
4
12,
"
below
862.
but
4
the
to
written
retained
is
first
the
carried
is
which
over
never
awkward
scheme
down,
column,
'V8''
3
1600
about
illustration
an
from
us
beginning
of
the
old
to
the
left
then
the
:
Thus,
the
is
vital
on
until
very
simpler
following
The
right.
carried
were
be^
A
today.
are
continued
course
Under
multiplication
operations
which
method
a
pletely
com-
processes.
and
operations
the
reckoning*
arithmetical
these
that
was
abical
and
subtraction,
what
practically
consideration
a
MDCCXXXIV.
certain
revolutionized
came
addition
in
CCCLXVII:
+
"
influence
by
cients
an-
operations:
CCLXIV
+
computations
appreciated
number-system
Hindu
this
be
the
by
encoufitered
simplest
best
can
following
two
The
difficulties
performing
in
-SUBTRACTION
AND
ADDITION
114.
XI
which
abaent.
~^
97
and
a
7
"
;
from
15,
'*7''
the
the
the
7
simple
making
then
''1"
goes
leaving
as
which
'*7";
the
giving
practice
was
2
6
a
final
and
matter
be
to
5
sult
re-
were
by
98
MODERN
the
of
use
also
ARITHMETIC
the
as
123
Before
the
minus
the
century;
115.
task
The
of
number
the
four
results
to
work
is
fundamental
teachers, and
least
the
not
grades.
Intelligent teachers
children
under
their
fundamental
burdensome
with
manner
In
about
set
the
first
is to learn
place,
the
underlying
it.
will
seen
as
be
;
but
computation,
this
to
time
the
teacher's
be
of
process
This
the
when
operation
be learned
subtraction
and
appeal
to
the
rationalization, side
questions
during
Let
the
child's
this
find
a
a
ously
seri-
systematic
them.
the
and
give the
to
problem
not
child
the
now
reasons
blindly, however,
division
the
child
teacher's
habituation, and
of* the
the
of
poor
higher
the
not
before
of
use
readily the
object in
business
its science.
years
The
of
perform
to
always
not
main
discouraging
really desire
big thing is teaching
not
an
the
must
The
most
teachers
to
their
main
the
1489.
first four
they study the
purpose
occurs
"
integers.
are
14th
or
in
integers, will
achieving
definite
a
time
ability
providing
matter
and
with
so
with
operations
^^^
words
facility in the
a
who
care
13th
+
was
The
the
children's
this
during
accomplished
after
one
Operations.
operations
Hence,
Subtraction
Widmann
give them
to
bers
num-
symbols
today.
subtrahend.
John
the
during
the
written
symbols
Fundamental
teacher
of
during
the
published by
Four
the
of
of
first record
arithmetic
an
sometime
arose
value,
of
signs.
the
over
lined
out-
years.
value
were
of
instead
of place
notation
were
subtraction
early
additive
Roman
dot
a
of
idea
connecting
any
by placing
and
these
additatively
taken
without
indicated
in
be
to
other
plus
as*in the
in order
numbers
the
Results
numbers
processes
ol the
by
PROBLEMS
numerals.
given
during
introduction
indicated
written
the
first two
used
AND
sand
above
were
the
were
the
or
The
do.
we
in
abacus,
written
always
below
METHODS
are
sidered
con-
ort
of
slogan
at
the
very
little
nature.
The
early number
work
child's
ADDITION
be
must
AND
wholly
almost
very
little with
teacher's
task
in the
is twofold:
memorization
of
116.
Teacher's
The
is
taken
the
produce
least
the
in
the
in
the
upon
take
not
general
a
idea
advantageous
realize
various
because
a
particular
be
of
will
the
come
117.
be
lessons
for
follow
that
oUiers, this
in
seems
matter
the
it
make
he
will
all may
subject, and, further,
the teacher
text
this
of introducing
is
of arrangement
following lessons
fundamental
impossible
here
to
ment
arrange-
accordingly advised
teach
from
for
every
arrangement
vantageou
ad-
more
concerning the question of
and
those
to
this
point
stand-
topic
as
it
consideration.
in
Oombinations
arranged
order
not
presentation of the four
advantageous
up
of the
presentation
limitations
Space
order
attention
some
carefully the
study
of
order
detail, however,
into
in these
can
given
in
to the
aid
an
do
in each
logical than
they deal with
operations.
to
elements
addition, which
case
that
unification, and
of
of the
pends
subtraction, de-
as
As
texts
traction,
sub-
subtraction
arrangement
follow.
to
of
texts
is recommended
text
is
ticular
par-
before
of
all
or
The
on.
certain
use
new
and
will
like
fully the importance
more
the
to
plan
Most
all of addition
the
so
that
under
addition.
to
as
work
New
pupils.
preceded
up
and
in
gradation
such
the
of
Therefore, the
required
teachers
in
topic, in this
related
it.
preceded
as
ject
sub-
work
topic of the subject, such
any
teacher
difficulties of any
case
multiplication, and
work
various
has
multiplication,
before
book
text-
what
do
day
present
the
to
^as
"
The
confusion
upon
section,
go
and
Here.
singly and
up
generally based
as
it
that
shows
combinations
various
the
Duty
that
arranged
so
are
has
work
number
the
and
analysis of
An
**why."
the
early
'^how*'
the
application in computation.
their
the
with
concerned
**what." and
the
99
SUBTRACTION
one
Addition.
after
the
While
other
in
the
nine
362,880
digits
different
100
MODEBN
ARITHMETIC
combinations
45
only
ways,
digits by taking
45
only
7
learned;
two
of
sums
is
2
requisite for
equals
third
a
the
that
number
of letters
spontaneously
Teachers
children
to
; as
compared
definite
A
jective
of these
and
Combinations.
taken
is
up
memorized
thus
one
table
of
one
a
to
around
the
loss
nine
mechanically
he
4
and
of
in
the
7,
say
time
met
likely
was
+
but
addition
children
these
column
came
the
learned
readily
them
children
the
by
had
tables
the
or,
7
bination
com-
a
9
just
as
a
he
of
the
of
in
tables
greater
of words
words,
undertaking.
the
factors.
Addition
45
combinations
these
Not
are
ago
years
many
of
4
2, 7 +
than
unable
+
in
3,
in
7
+
the
turn
4.
any
apply
When
addition
the
1, 4 +2,
alert,
to
figures.
of
7
+
who
quickly repeat
columns
with
were
Children
order.
usually
wpre
o"b-
that
necessary
Up
could
if
is
tively
compara-
toward
the
that
might,
1, 7 +
a
that
I's, 2's, 3's, etc., which
4
+
fact
these
small
Taking
say
to
at
of
procedure
matter.
teachers
formation
very
are
thoroug^h
a
Good
the
in which
order
important
an
contained
texts
is
sums
Order
The
so
numbers
as
be
must
knowledge
with
Systematic
Advantageous
118.
2
+
pronunciation
the
45
this
part of the teacher
the
on
are
including
goal
naturally
daily demonstrating
acquire
letters
learned
be
combinations.
45
may
formation
the
the
children
age
tions.
combina-
45
necessity of
absolute
land
young
; 7
in
experience in teaching addition
had
the
with
word
the
throughout
from
a
bination.
com-
is cat.
have
realize
acquaintance
normal
spells
and
be
speed
single digit
two
any
inevitably
as
c-a-t
as
who
of
sum
one
and
these
with
are
must
as
combinations
these
that
treated
same
there
2, which
+
accuracy
acquaintance
these
that
means
7
as,
from
7 are, of course,
+
It is necessary
thoroughly
This
numbers
thorough
a
time.
PBOBLEMS
made
be
can
a
two
first
The
addition
at
2 and
+
AND
METHODS
of
etc., until
the
In
numbers
either
still older
he
way
case
of
ADDITION
counting
turned
learned
be
as
of
process
will
be
no
needed
of
of
suggestion
to
**run
for
for
will
as
+
have
been
selecting
at
In
the
the
above
1.
to
Draw
this
of
the
have
next
within
the
of
similar
to
left
diagonal
two
small
comer
and
learn
and
lines, and
numbers
first
so
and
on
in
all
the
sets
One
same.
learned
the
order
to
be
to
of
as
between
the
all
left-hand
the
arise,
reference
A
are
separate
definite
contained
sums
upper
to
the
column
as
some
order.
drawn,
the
manner,
be learned
combinations
and
there
until
manner
a
difficulties will
lack
a
that
contained
list.
teacher
show
Begin
1
is
a
up
the
words.
should
Otherwise
combination
namely,
the
the
in
read
into
table*'
going
In
in
itself
by
the
must
useful
be
to
learned
is
down
be
combinations
there
diagonals
then
letters
combining
follows:
the
will
the
order
is reached.
sums
table
7
+
combination
Each
in
fact
4
where
scheme
added.
If
it is well
above
of
isolated
an
selecting them.
cases
unit
addition.
Although
facts
each
finger being
a
"
combination
figures the
that
for
down
101
SUBTRACTION
fingers,thus, 8, 9, 10, 11,
the
upon
AND
is
as
learn
diagonal,
the
binations
com-
through
small
sums
102
MODEBN
used
are
ARITHMETIC
first and
at
5
-f-4,
and
all such
that
noted
be
included
are
scheme,
has
as
depend
There
are
learned
in
moving
used
as
well
as
an
fixing these
in
child
already
have
119.
Order
addition
1
this
work
those
Use
oral
from
will
cents
The
3
then
he
value
vim
mind,
written
work
to
him
the
pupils the
an
pay
extent
The
that
of
that
the
in the
in
work
2,
2 +
Make
mastered.
adapted
following
how
more,
1"
until
problems
cents
:
many
most
they
directly proportional
taken
oral
work
up,
but
in hand.
hand
new
trouble, without
order
The
teacher
which
become
is recommended
neglecting
been
taken
combinations
any
and
should
have
are
it
cease
oral
two,
to
Let
not
the
combinations
to those
it.
should
combinations
the
later
giving
in
teacher
this
is
all of
following
as
2
be
especial attention
Let
the
thoroughly
concrete
earns
will
learned, after
previously
VII.
Chapter
combinations
are
pupils,
and
progress
use
of the
mind
like 1 +
new
no
the
have?"
in
continue
of
drudgery
Begin
ous
Vari-
previously suggested.
as
the
cents
difficult.
in the
Addition.
simple
of
alertness
written, should
the
be
system
some
under
consideration
this work
of
and
borne
when
has
of
some
introduce
experiences of
''If John
be
and
large number
the
tion
combina-
to the more
considered
2, etc.,
+
already under
a
the
2
plans
each
that
combinations
in
a
which
simple combinations
few
a
1,
been
Work
of
with
3, 3 +
+
simpler
the
such
Such
that
and
fact
eliminating
drill work
by
combinations
providing always
isolated
from
for
devices
diagonals,
teacher.
plans of selecting the
will likely work
be
the
also
5, and
+
of
pair
one
simultaneously
up
ingenuity of
many
4
as
previously recommended.
been
the
upon
taken
be
It should
practiced.
any
taneously
simul-
increase
sums
combinations
dual
between
would
hence
and
more
grows
PROBLEMS
AND
numbers
both
child
the
as
METHODS
up.
giving
to
such
troublesome.
for written
work
:
104
MODERN
able
to
each
ten
ABITHMETIC
10
of
and
quickly.
and
accurately
use
in
tens
in connection
thoroughly
121.
Grouping
No
in
bear
mind
that
should
process
of the
order
of
lion
idea
The
hundred
each
with
the
Tens
at
of 10
introduction
in
ones
reviewed
be
must
this process
122.
Short
of the
add
to
in the
of
reading
short
cuts
by those
new
the
5
this
If
a
column
a
instance, if, in
as
6's, then
and
a
a
6 and
there
5.
are
There
of
number
the
the
the
in
4
seven
are,
few
the
good
times
it
of
times
second
+
occurs
column,
8 farther
6's
in
however,
it
that
some
up
one
rate.
accu-
is that
add
instance,
glancing
in
up
right hand
the
over),
4
52
count
the
up
then
by
occurs
2
+
experts
12
^
gether.
alto-
in
the
its value
4
is
as
besides
on
5,
multiply
is considered
column
+
times
several
and
5
umn
col-
gives
10 's make
is to
in
whose
groups
r^emaining
scheme
and
the
which
For
itself
the
of
used
most
(8 passed
6
oughly
thor-
with
a
following
additional
4
to
example
with
8
its
must
learned
have
column.
figure repeats
number
child
of numbers
found
5, 4 +
the
a
the
introduced
the
are
+
and
when
some
has
digit numiber
groups
bottom
7 ; the
+
addition
schemes
is 10
:
of
the
each
selec-
by
or
especially quick
in
the
sum
3
children
accompanying
from
10
of
are
out
exact*
by the
never
who
10
to
the
be
picking
up
in
the
of numbers.
each
of
teacher
five years
or
can
One
of
method
combining
by
used
sums
the
already found, they
sum
making
stims
When
the
Let
adding
first years,
not
Outs.
of
and
This
schemes.
place, but
learn
that
column
in the
of numbers
First.
first four
the
during
invariably be
groups
proper
by
numbers
similar
other
8
PROBLEMS
of carrying.
idea
for
AND
METHODS
;
sidered
con-
two
an
addition
ADDITION
who
do
can
right along rather
sums
of numbers.
is, of
123.
which
a
in
ployed
1. 10
ii. 10
8 from
is taken
from
13
13
from
and
iii. Subtraction
becomes,
124.
the
explanation
is
30
50
and
the
as
the
the
20
tens
make
the
30
to
28
to
to
is
result
the
but
when
the
there
units
carried
the
the
to
Hence
tens
below
8
which
to
3
of
an
be
be
is,
the
right in the
of the
5
to
this carried
line
the
and
writtai
was
added
the
can
question
3, the
be
case
should
Since
only the
over
tion
ques-
simple that
gives 13 T'
column
remainder.
which
In
above
addition
the
the
give
first digit to the
the
In
10
8
and
53 f"
so
one
make
to
to
"
the
em"
making
13
addition
Method.
nothing
is 5,
in
thus
making
quite carefully.
remainder.
and
the
usually
most
3
of
added
the
added
the
larger than
in
:
to the
Austrian
to
answer
and
is
ones
reverse
be
must
**What
above
subtraction
50.
superfluous,
8 in
of
case
in
subtrahend,
added
example
added
the
to
40.
considered
down
and
the
The
accompanying
best
individual
as,
the
are
and
much
Example.
minuend,
from
is treated
''How
subtrahend
minuend
from
20
case
this subtraction
both
to
that
methods
performing
is added
subtraction
is 8
in the
following three
this grouping
of
use
Analogous
is
in the
one
the
by carrying
fits the
method
addition,
digit number
corresponding
The
by making
of Subtraction.
in
work
follow.
to
one
Methods
^'carrying"
than
Whichever
the
course,
faster
and
accurate
more
105
SUBTRACTION
AND
are
the
10
20
and
to
be
106
added
to
3
make
' '
30
tests
make
to
the
other
and
should
be
other
with
of
relations
arithmetic.
have
learned
with
one
gradual
unifying
valid
of the
school
other
change,
be
The
addition
it connects
an
idea
of
processes
the
to
to
Austrian
children
who
advisability of requiring
here
already familiar
are
change
to
as
various
given
of
one
advantages
is that
who
system
however,
the
no
this number
many
objections
means
methods
use
giving the children
thus
process.
a
has
It is
through
of
reverse
big thing
other
of
since
learned
who
children
the
method
by all
children
all the
is the
give
to
it should
no
is
making
found
been
children
than
the
no
are
in
have
who
* *
method
method
natural
the
trian,
"^Aus-
change,
the
of
it have
Austrian
by
making
or
children
use
as
addition
There
and
method,
be
Subtraction
subtraction
number
the
methods
"
the
as
universally used
to
progress
the
' '
added
self-taught arithmeticians.
all
taught
so
While
the
over
better
methods.
relation.
it is
pupils who
those
to,
many
many
to
naturalness
and
for
method
simplest
and
adding
that
by almost
adopted
other,
"
also be said
It may
change.
the
fact
the
by
shown
"
simplicily
The
is referred
method
continental,
method.
it is
This
PROBLEMS
simply, ''How
more
or
AND
question becomes, ''How
the
50?'*
make
5?"
"
METHODS
50, and
make
to
added
to
ARITHMETIC
MODERN
is
A
questionable.
is not
suggested
only
possible,but highly practicable.
125.
Order
of
tyx"es,
pupils
types
i. A
are
or
SubtractioiL
examples
thoroughly
of addition
single digit
of
taken
each
subtraction
acquainted
previously
number
Introduce
with
outlined
from
as
the
of the
soon
as
lowing
folthe
corresponding
:
another
single digit nnmber:
ADDITION
ii. A
single digit nmnber
less
remainder
in
one
iii. Same
vi. Several
+
learn
6
than
6
which
in
that
the
time
"John
has
"How
many
less
another
the
ones
in subtrahend:
sponding
corre-
in
numbers,
is 6
time
problems
they
that
such
10
learn
is 4
that
following, of
the
as
dren
chil-
the
that
and
4
than
more
same
addition
with
daily, will be found
great aid
a
:
3 cents
now;
how
many
cents
must
John
earn
he
must
to have
earn
to have
8 cents, if he
7 cents
has
f"
5 cents
now?"
"Mary
is
7
years
old.
In
how
many
years
will
she
be
11
years
old?''
"John
ouffbles has
has
5
James
a
two
than
corresponding
than
digits
be given
should
at this
from
giving
number.
10
Simple
many
number
subtrahend:
less
three
each
at
10.
=
being
coordinating subtraction
will
4
with
digits in
thus
taken
number
subtrahend
zeros
minuend
in
(iv)
as
the
digit
two
a
minuend:
the
digits
Same
more
digit
from
digit
two
of
107
SUBTRACTION
taken
A
10.
(ii) with
as
iv. Some
By
than
each
digit number,
".
AND
marbles
than
and
Johnf
James
''^
9
marbles.
How
many
mojfe
108
With
themselves.
These
since
the
have
been
which
addition
And
thoroughly
untiring
drill
addition
of two
idea
tion
both
If these
**
is
test
by dividing
dividing
vary,
it does
the
upon
number
a
the
its
100a
can
of which
and
hence
remainder
from
in
addi-
of
used, that
as
useful
at
a
as
any.
later
time
there
certain
are
it is not
hence
same
as
digits by nine.
for"
that
the
cases
infallible.
the remainder
that
of
out
obtained
found from
Any
number
explanation,
of
be
can
by
represented
These
;
sufiicient
is
digits, which
pupils,
the
''casting
but
is the
nine
by
of
sum
test
fact
the
first two.
the
disclose
not
and
include
checks
addition
The
shorter
up.
subtraction.
mean
is
of
type
constant
to
commonly
third
Nines."
the
by
simple
no
those
is taken
disabuse
to
columns,
a
of
one
somewhat
is based
the
in difficulty,
subtraction^
in
well
always
most
one
down
confirm
which
errors
three
and
Casting
a
The
additions
generally
nines''
This
numbers
Unfortunately
up
two
127.
of
only
available.
are
adding
will
occasionally
Checks.
126.
suggest
Each
the next
It is also
numbers
two
and
be
to
course,
its inverse
before
day after day.
that
of
only be accomplished
can
up
will
learned.
with
understood
any
made
occasion
are,
being
are
that
gradually increase
used
together
thoroughness
of the
will
and
example,
be
must
problems
numbers
easily be
can
the
suit
will find
teacher
alert
to
time, examples
PROBLEMS
AND
simple problems'
of these
that, in
the
practice
little
a
number
METHODS
ARITHMETIC
MODEBN
[99a +
9b] +
number
in
the
the
as
into
rearranged
be
106
+
a
number
+
6 +
dividing a-\-b
the
+
form
the
b
[a +
+
c,]
first bracket
(1) divided
c
(1)
c.
divided
by
-^-c by nine
by
9.
is
(2)
is divisible
9 will
The
have
the
remainder
technically
by
9
same
found
called
tlie
**
If the
excess."
is added
(1) above
in
number
109
SUBTRACTION
AND
ADDITION
another
to
as
100m
the
will
sum
in
written
m)
+
form
the
b
[a +
which, again,
by nine.
+
m
(2)
n
'ft is
ft]-^
+
example
9
+
+
C
the
of the digitsis 23, giving
9.
The
right.
note
the
second,
sum
IS
4
-f*5 and
8, the
the addition
excesses
then
excess
of 9 's
presence
same
is
of the
have
as
sum.
the
the
likely
the
as
in
to
sum
be
been
numbers
been
of the
9
one
to
k]f
" 4-
+
cj
of
the
is divisible
c
+
as
short
the
digits of each
number
The
etc.
excesses
Had
correct.
9,
as
been
sum
divided
first,1 +
at
the
14, its
sum
excess
compared
by
the
we
8 in the
of
excess
added
+
m
A
placed
in the
of
and
e] -7-9,
first is 5
are
of the
have
6 +
of 5 when
third,
over
is **E"
also
5
+
c
[ai+
of the
numbers
4
(5)
[E +
oi
remainder
place of adding
In
+
n
excess
other
of the
excesses
as
a
n)] +
excess
same
The
suffice.
must
+
the first bracket
of
excess
the
be
+
m
within
part
**e,'' then
must
+9(6
m)
the
Suppose
+
n
the
(4)
ft),
(c +
n) +
10(fc +
+
[99(a +
of
(3)
ft,
+
be
100(a
or
lOn.
+
;
the
hence
of
the
would
with
the
"^
110
MODEEN
ARITHMETIC
128. Other
AND
METHODS
PROBLEMS
Checks.
be suggested that a form
It may
used by those liable to be interrupted
of addition much
while
adding,is to write
done above,and then
each
to the
right,as
sums.
partial
Subtraction can be verifiedby either casting
out the nines,
the
from
remainder
the
subtracting
minuend, or by adding
was
the remainder
down
sum
to add these
to the subtrahend.
129. Number
Relations. It is often possibleto recast
diflference can
or
given numbers so that their sum
easilybe found without the use of penciland paper. For
the
instance,
49
+ 37
34 + 25
28 + 13 + 46
54
The
etc.
number
use
"
"
"
"
"
56
=
26
"
2
"
=
30
2
"
similar schemes
of these and
87 ;
"
28;
"
for
bringing
within the range of mental processes gives
an
practicein seeingnumber
relations,
ability
work
excellent
which
+ 36
86;
35 + 25
1
59;
30 + 17 + 40
70 + 17
"
26
"
50
"
real arithmetical growth, as has previconstitutes;
ously
been
pointedout.
EXERCISES
XI.
1. Give
historical r^suin^
a
of
development of
the
addition
and
subtraction.
2. What
work?
time?
should
the
or
object of
the
first four
the art of arithmetic to
in number
years
predominate at
this
Illustrate.
3. Discuss
What
be
Is the scienco
method
fullythe
is the big essential?
4. Should
addition
be
of
teachingthe
addition combinations.
Illustrate.
"completed"
before
taking up
subtraction f
Why!
5.
Explainhow
to teach the
6. Discuss the nature
7. Discuss
or
U
the nature
three years.
8. Explainthe
carryingprocess
of the work
in addition
of the work
to
a
class.
duringthe earlyyears.
during the last two
in addition
of groupingthe digitsfor rapid addition.
principle
112
a
MODERN
ARITHMETIC
17.
Suggest
the
18.
Suggest
a
for
order
series
of
taking
four
AND
PROBLEMS
subtraction.
up
for
examples
subtraction
suitable
for
class.
grade
19.
METHODS
Suggest
least
at
three
problems
for
each,
addition
beginning
subtraction.
and
20.
Explain
21.
What
the
are
in
examples
three
of
casting
usual
checks
out
the
for
addition?
nines.
Apply
each
three
to
9.
What
22.
idea
the
are
in
examples
the
checks
usual
subtraction?
for
Apply
each
to
16.
BIBLIOGRAPHY
1.
Arnett,
American
3.
Processes,
The
of
C.
S.
Journal
A.
of
IV
of
of
:
the
of
Children.
and
X:
Teacher,
Arithmetical
Simple
XVII:
Growth
of
School
362.
Psychology,
Measurement
School
in
Memory
Psychology
The
Elementary
and
Growth
Psychology,
E.
American
Courtis,
4.
L.
Journal
Browne,
Journal
327.
T.
Bolton,
American
Adding.
"md
Counting
XVI:
Psychology,
2.
D.
L.
58,
1.
in
Efficiency
XI:
177,
metic.
Arith360
171,
528.
School
VIII:
Bimonthly,
7.
VIII:
8.
9.
Henry
Kelly,
Frederick
Educational
Arithmetic.
in
Educational
Tests.
C.
School
Stone,
An
Cliff
Bimonthly,
A
Study
Teacher,
W.
subtraction.
of
XIV:
Arithmetical
College,
comparing
Journal
the
of
94-97.
Their
Marks;
College,
Teachers
article
1915.
Teachers'
J.
L.
also
73-84,
pp.
Teachers
Phelps,
of
March
Efficiency
Courtis
The
B.
Howell,
Them,
12.
for
189.
Elementary
11.
Tests
C.
Standardization.
10.
Elementary
123.
E.
Hinkle,
Arithmetic,
in
127.
C.
E.
Hinkle,
Scores
Standard
XII:
Teacher,
6.
A.
S.
Courtis,
5.
New
Errors
York.
in
pp.
Tests
Variability
5-10
and
and
85-89.
Ability.
of Adding
29.
Columbia
Austrian
Some
and
Abilities
Factors
mining
Deter-
University.
and
Experimental
**
taking
Pedagogy
away*'
esses
proc-
(London),
CHAPTER
Xn
MULTIPLICATION
History.
130.
CLXXIV
by
until
idea
of
with
such
of
vent
here
place
appropriate
column
each
as
and
pair
in
the
of
digits
by
Often,
second
36
X
replaced
column.
for
a
column
113
digit
of
learned
the
simple
first
single
goes
the
out
ad-
process
15012.
"
are
a
the
With
the
became
417
finding
feat
a
the
carried
be
to
impossibility.
an
multiplication
of
comJbined
was
Yet
embodying
had
operations
almost
was
of
all
impossible.
as
up
number-system
the
MMDCXV
divide
or
given
Multiplication
abacus,
Products
later
value,
illustrated
be
today,
of
S3rmbols.
the
multiply
to
try
invention
division
while
To
would,
the
DIVISION
AND
down
set
in
counter
over
to
illustration.
and
the
the
next
These
114
MODERN
ARITHMETIC
combinations
column.
The
the
gubar
of
process
the
which
work
multiplication.
performed
as
numbers
later
the
and
additions
work
they
as
final result
that
so
remained
referred
usually
only few
to
the
the
as
**
upon
of the
were
the
only
at
change
not
were
today, but
are
a
tablets
partial products
progressed
the
are
processes
the
quite
sand
hold
could
each
for
counters
the
As
with
computations
also, instituted
numerals,
was
PROBLEMS
in
equal-valued
9
necessarily large figures
for
AND
of themselves
care
containing
abacus
an
in
take
METHODS
tained
rebined
com-
original
These
end.
scratch"
or
''galley'' method.
early
Another
as
the
form
the
was
in
shown
the
so-called
''latticed
accompanying
illustration
multiplication of
417
by 36.
The
Egyptians
left;
the
Greeks
and
Hindus
from
right
to
plication,"
multiof
multiplied
from
left
to
right.
Division, prior
number-system,
excluded
our
of this method
was
modem
the
to
a
coming
sort
idea
of division
of
of
the
repeated
of quotient.
is too
complete
subtraction
As
complicated
an
to
Hindu
which
explanation
be discussed
AND
MULTIPLICATION
in
these
referred
The
of
use
pencils
revolutionized
the
smallness
The
pencil
while
upon
time.
it
which
of
use
have
may
Arithmetical
divisor
factor,
below
and
division
by OuGHTRED
used
all of the
a
work,
consumed
in
aid
recting
cor-
matical
mathe-
Thus
by the
division
and
division
"
of
the
of the
reverse
132.
the
two
be
four
work
same
division
grounded
with
manner
as
What
Leibnitz
so
:
for
signs
the
clearly the
is the
each
that
them
as
1,
in
need
these
reading,
a
be
2, 1 X
1 X
becomes
d-o-g is dog.
to
as
the
last
addition
combinations
time
as
in
operations applies
well
as
no
said
was
fundamental
factor
a
so
Harriot
that
show
namely,
;
;
multiplication and
for
they
century
in 1659.
unfortunate
as
processes
multiplication, since
containing
by Rahn
Leibnitz
by
really only 36
are
17th
the
other.
and
There
by placing the
tiplication
day signs for mul-
present
It is very
MultiplicatioiL
regarding
-f-
accepted
not
were
relation
and
suggested
w
Our
division
indicated
product after
for
sign of multiplication and
the
this time.
at
Hindus
originated during
in 1631
as
*
they
dividend.
the
and
the
with
accurate
abbreviation
indicated
last
that
division.
made
in.
more
The
the
must
completely
great
a
crept
Symbolism.
the
r"
also
17.
p.
difficult and
made
be
be
retain
to
figures is
multiplication by writing
also
on
are
pencils.
131.
X
list
study
multiplication,and
possible
could
computations
its
computations
figures could
of all the
errors
any
in the
correspondingly
were
Eetention
of
which
made
paper
in
paper
processes
with
erasures
and
in
interested
histories
arithmetical
the
to
further
those
pages,
ll5
DIVISION
36
sort
so
to
taken
3,
etc.
lesson
tiplication
mul-
to
traction.
sub-
and
be
learned
up
with
The
in
those
pupils
combinations, however,
of reflex action
that
3 X
7 is 21
in much
as
taneously
spon-
116
ABITHMETIC
MODERN
In
order
from
achieve
to
combination
temptation
combinations
other
**run
to
down
is reached.
of the
the
combinations
numbers,
same
3 X
as
For
7.
for
as
be
be
added
this
of
fact
be
itself
to
J
common-sense
the
3
of
often
as
combinations,
such
teaching
bination
com-
inversion
of 7 X
use
no
command
a
the
by
is
needed
such
securing
taught
this
statement,
" of
or
as
an
abbreviation
of
fraction
will
multiplication by
that
with
accord
the
apart
be
will
there
until
children
on
fact
drills.
on
The
in
not
to
example,
Multiplication should
addition.
aid
suggestions
VII
Chapter
see
table"
the
by
that
so
the
One
skill, each
of
degree
single isolated
a
as
PROBLEMS
AND
desired
the
learned
be
must
all the
METHODS
as
time, in
a
of
a
number
a
no
invalidates
sense
multiplication
looking
at
the
early lessons
to
enter
manner
not
can
integers.
wise
It is not
in
these
discussion
this
Indeed,
As
need
with,
and
133.
not
abbreviation
should
texts
the
without
not
all teachers
and
because
in
of
approaching
multiplication
134.
addition
the
for
with
teacher
class.
being
dealt
vital importance
the
work,
be
of the
Because
such
matter
texts
of order
following general outline
the
four
therefore,
can,
using
Most
of the
teacher.
the
an
as
multiplication
work
the
to
as
the
for
suggested.
2's,
Step.
two
explain the
finding such
by
fortunate
so
Introductory
of two
the
addition.
in
step
integers, and
modification
are
is
of
Taught
in
step
new
corresponding
operations
followed
be
each
elementary arithmetic combine
fundamental
sophical
philo-
multiplier.
a^e
Multiplication.
in
addition
of
follow
on
Work
of
which
a
things.
concrete
Order
the
attention
the
engage
it is numbers
fact
of
abstractness
the
never
of
matter
a
of
into
sums.
3's, three
shorter
One
work
by
2's, etc., after
which
Introduce
or
process
two
the
"
the
let
^multiplication
multiplication
"
combina-
MUIiTIPLICATION
tions
be
can
ieamed
in
examples
such
simple problems,
''How
each
cents
and
of
Friday
did
cents
two
put
she
It might
much
as
child.
By
week
put
in
her
daily
thus
savings
none
if he
days
two
these
that
three
earns
and
a
real
last
and
and
the
of
cry
fresh
other
problems
in
all
that
deal
should
of
element
modern
interest
many
daily life of the
the
work,
How
days.
''
weekf
little
Wednesday,
Monday,
on
the
experiences
humanizing
hue
bank
on
bank
savings
possible with
Steps Accompanjdng
addition
accompanying
i. Combinations
3,
in
earn
teaching,
is infused
into
is
the
work.
135.
3 X
her
but
added,
.**motivation,'' the
introduced,
John
in
cents
last
be
as
numerous
"
dayf
"May
will
eeots
many
include
must
cises
exer-
following:
the
as
daily with
be mixed
daily work
The
which
first lesson, after
multiplication should
addition.
in
this
during
117
DIVISION
AND
4
X
ii. Numbers
2,
yielding
Addition.
The
is recommended
products
less
than
following
der
or-
:
10,
2
as:
X
2,
2
X
3,
etc.
of
digits
two
multiplier, requiring
no
in
carrying:
the
multiplicand
and
one
digit
in
the
118
iv.
AMTHMETIC
MODEEN
Simple
136.
teacher
of
With
to
two
METHODS
Two-Digit
explain the
at
Numbers.
meaning
It
of
this time.
so
on.
Multiplication by
partial product
This
paves
the
in
way
the
such
for
multiplication by
in
the
the
bers
num-
unit's
units, tens, hundreds,
ten's
tens, hundreds,
for
is necessary
Multiplication by
digit gives the first partial product
and
PROBLEMS
involving carrying:
examples
digits
AND
digit gives the
thousands,
multiplications
as
and
:
second
so
on.
120
MODERN
in
there
ABITHMETIC
12?"
4'sf
by
for
from
early
separation
the
question
many
twelve
for
the
each
showing
in
each
of
forms,
two
delight
in
divided
by
that
2
not
Order
for
they should
process
no
other
relationship
is the
learn
6
teachers
and
each
of
in both
drilled
will
of
and
they
learn
the
two;
the
other.
learn
to
teachers
apples
be
divided
by
forms
in
2,
sophical
philothe
to
be
in
be
can
Such
results
the
children
the
between
6
better
would
will
inches.
confusing
as
fully understood,
between
undoing
be
best
12
them
apples.
The
multiplication
way
2
by
to
can
from
there
Many
that
apples
liable
When
be
requiring
the
as
distinguish
to
tition.
par-
made
from
questionable.
by following
in division.
In
is
be
are
by
matters
can
formed
of
divided
be
it is
opportunity
of inches
able
wisdom
Division.
as
of
of division
such
number
be
that
or
to
of
obtained
combinations
close
formed
groups
division
which
each
to
are
are
with
children
the
into
forms
and
lengths
form
apples,
they
soon
the
each
can
as
as
division
groups
be
can
excellent
an
Two
the
equal
but
having
6
children
one
into
group
sticks
measivrement
children
discussions
be
each
13
aflfords
3 inches
3
the
of
names
will
duced
intro-
of remainder.
by
of
the
these
139.
in
divide
second, with
; the
for
but
sticks
is concerned
groups
It is well
the
three
happily
of sticks
number
Division.
of
former
of
be
be
can
in
be,
to
meaning
division
The
inches
a
also
group
Forms
recognized;
12
of
three
pupils
the
Two
number
division
there
are
sticks!"
in
138.
PROBLEMS
many
in
would
of
groups
Requesting
3 sticks
work
AND
**How
as
the
which
''How
often
as
The
METHODS
omitted.
division
cation,
multipli-
ing
by the correspondlearned
have
their
certain
simultaneously
to
see
for
inverse
so
namely,
after.
there-
clearly
that
the
each
MULTIPLICATION
i. Division
of
less
numbers
6-T-2
of
in
of
V.
-^
but
256
be
divisor
divisors
by
56-7-4
-f-
7
18-7-8
the
5
-f-
542-7-4
in
is
number
a
above
all of the
used
of
is
quotient
by
the
examples
digit.
one
The
above
placed
the
The
pupils
will
most
will
this
the
should
of
be
led
to
see
-j-
section
of
guidance
to
able
be
detail.
far
on
the
the
the
and
that
teacher.
to
carry
of
part
that
the
of
out
the
troublesome
part of the
the
volved
in-
reason
Selection
most
ducing
intro-
marks
the
the
see
:
25
for
division
all
on
576
in
drill
patience
16
work
must
by
remainder
a
that
they
found
in
time
unable
:
384-7-32
the
be
in
remainders
no
the
careful
require much
deal
-f-
is
also
may
steps
be
345
serious
most
various
trial quotient
great
is
it
successfully
processes
a
13
in
12
-H
digits resulting
As
pupils
the
though
and
digits resulting
384
two
-i-
division
require the
This
of
427
upon
in
two
Division.
Long
will
divisor
12
-f-
entrance
of
182-7-14
long
part.
remainder:
no
is recommended.
vii. Division
the
3
remainder:
a
567
which
182-7-13
Even
:
remainders:
should
in
vi. Division
which
-H
12
-7-3
the
4)356
140.
-r-
giving
giving
and
division
dividend
remainder
no
36-5-3
5
16-^5
whenever
form
60
4
-T-
65
5
-f-
742
and
digit, giving
one
36-^6
2
carrying
Carrying
Short
of
4-h2
digit requiring carrying, giving
one
54
14
remainder:
no
9-r-3
dividend:
Divisor
no
121
DIVISION
giving
46-7-2
40
Later
10
divisor
and
two
24-7-2
iv.
than
8-i-4
ii. Dividend
iii. Zero
AND
the
pupils
teacher.
first digit in
a
1 22
MODERN
number
ARITHMETIC
quotient,
while
increased
by
much
the
The
future
but
today,
needs
excesses
casting
the
in
the
of
57
that
and
by the
the
The
this product
is added
of the
dividend.
illustration, the
their
5, and
added
As
the
is also the
that
literal
in which
by its
out
-SfXwi,
as
equation
in
advantage
andZ"
"
is
work
this
sum
accompanying
is 8, of 23
is
this
is
or
To
1, making
be
law
5.
idend,
div-
to be
correct.
used
to
of
is merely
number
good
excesses,
replaced
the
operations carried
for the
original numbers.
and
indicated
of
likely
the
and
excess
of 824, the
can
stating
each
excess
19,
is
the
to
is 40.
excess
division
the
The
of
multiplied
quotient
of 35
product
is the
product
If the
in the
excess
excess
is
excess
Thus,
of
division, the
divisor
of the
57
excess
4845, the
For
85.
the
In
of
excess
is 4.
of the remainder.
equal that
of the
excess
excess
correct,
tion,
multiplica-
multiplicand.
of
of the
excess
be
can
(12) is 3, which
product
for
preserved
the
and
of 85
of
results
the
For
process.
cause
be-
results
division
and
illustration
as
is
work
multiplier
same
"
nines"
and
equals
product
this
P
the
completion
the
at
all of the
when
is 3 and
Thus,
to
out
numbers
original
the
must
be
often
testing their
in
multiplication
in
excess
of
**
successfully by this
very
product
487
left nothing
the
comparison,
the
Hindus
galley method
Even
checked
in
of
process
by the
computation
sought.
first digit
the
digits in
the
1.
used
their
PROBLEMS
AND
practically decide
the
Checks.
141.
was
will
613
as
METHODS
^X(i
+
B.
MULTIPLICATION
Cautions
142.
which,
3652
as
must
be
check
to
dividend.
be
The
made
the
division
will be
there
them
from
143.
in
division
In
the
order
of
order
and
combining
20
The
more
the
land
per
easily
same
at
acre,
or
the
X
of two
teacher
will
should
be
that
and
should
For
acre
be
stantly
con-
that
taught
numbers
can
work.
number
regular
more
or
tion
multiplica-
by
select
they should
For
the
the
instance, the
multiplied tpgether
first
of
5,
or
of
numbers
is divided
one
factor.
$45 per
wideawake
simplest multiplication.
713
if
cases
the
result
numbers
product
illustrative
a
the
under
few
pupils
three
affect the
X
in
the
finding the product
in
the
in
discussed
those
than
useful
more
even
are
Only
but
first place
third
error,
existing
relations
number
computations
the
the
missing
check, thus
division
and
of
copy
The
subtraction.
giving
first
to
in the
finding others
will not
in
given.
considered,
be
tial
par-
appear
Relations.
simplification
be
the
temptation
multiplication
here
never
the
illustration
and
the
multiplication, and
Number
addition
later
and
the
in the
as
then
cases
the
obtain
multiplier, since
will
in
the
and
quotient shoiild
products
both
divisor
to
awkward
teacher
applying
remainder
the
tion
Multiplica-
multiplier and
the
becomes
by multiplying
the
Checking.
reversing
The
in
division
by
adding
64.
X
by
123
DIVISION
in
however,
cautious
quotient
Observed
checked
be
multiplicand
such
Be
to
also
can
AND
and
price of
160
the
acres
X
at
of
$22^
much
multiplied by
other
that
8.
found
be
price of
the
as
same
315
X
often
can
example,
is the
25
80
per
160
acres
acre
acres
at
of
$90
is the
124
MODEBN
of 40
that
same
as
cases
the
mental
multiplication
gives
and
division
or
of
a
if the
remainder,
true
of
power
instance,
by
is
10
25
is
Similar
connection
of the
Use
of
the
above
inverse
144.
few
1.
above
The
a
third
43
the
25,
37
9
For
two
annex
discussed
be
fractions.
under
and
add
to
this
328, 546, 287, 654.
requires
comprehension
a
dexterity in seeing
practice.
The
following
are
of which
one
(40
9
much
as
from
it, comes
b^.
"
3)
is
3) =402
"
"
3^
1591.
=
between
10
and
20
is of
+
90
the form
a) (10 +
(10 +
Thus,
13
X
16
13
6)
(10 +
"
X
16
=
=
100
+
3) (10 +
118
a
:
numbers
two
will
are
{a"b)=a^
"
divisor
multiply by 4,
or
first by
is below
1600
first
parts
Devices.
(40 +
=
of
product
the
plication
Multi-
2.
exact
an
we
operations
numbers,
b)
by
relation.
divide
and
drill and
second
=
2. The
by
devices
of two
X
in
quotient
true
a
is
together with
of much
or
number
relations
(a +
Thus,
simple
a
simplified
which
number
processes
as
to
divided
be
multipliers
product
reduced
decimals,
multiply
time-saving
useful
both
70,
aliquot
Labor-Saying
Some
of these
In
be
-T-
Similar
results
factors, the
with
279,
by
itself.
times
30
100.
240
divide
or
since
multiply
also
useful
very
places
To
can
latter
point oflf two
in
been
in
or
further
thus
-=
zeros
^ of
PROBLEMS
acre.
per
number
a
a
multiply
to
35
-T-
quotient
true
a
AND
example,
120
which
has
Division
For
manner.
$90
at
acres
operation.
same
METHODS
AEITHMETIC
+
90
a6
+
10
6) =100
=
208.
(a+b).
+
18
MULTIPLICATION
3.
product of
The
5)
(10a +
for
(30 +
5) (60 +
65
X
5)
digit
left
and
"ts
+
25
+
50
1825
+
50
X
9
so
50
+
(3 +
6)
1825
+
=
6),
(a +
it to
all of the
the
450
2275.
=
writing down
by merely
adding
with
through
on
is effected
11
right then
the
to
25
+
1800
=
Multiplication by
the
lOOa*
==
digits ending in 5 is
becomes
=
4.
5)
125
DIVISION
of two
numbers
two
(106 +
which
35
AND
the
to
next
one
digits. It is the
same
adding in
5.
Multiplication by
98, is accomplished
times
the
number
a
by taking
just under
2 times
100
number
the
1000,
or
from
as
100
number.
.
6. If
the
divisor
first
divide
642
146.
by this factor.
30
-^
Order
is taken
with
than
It is also the
All
as
operations
placed
used
th^
to
next
denote
should
raised
2 X
rem.
6,
of
12.
or
interest
the
This
Signs.
connected
to
give it undue
call attention
to
The
importance.
the incorrectness
multiply first.'*
**You
is
factor,
common
practical application.
its
not
to
statement,
operation
takes
of
here
a
Arithmetical
of
urged
desire
21,
=
because
more
up
because
parentheses,
5
-f-
have
Thus,
Priority
is therefore
teacher
of the
107
=
of
question
it
dividend
and
within
be
to
position
first.
performed
some
:
signs of aggregation,
[3 +
multiplication
the
power,
5
"
comes
2
+
8 X
after
If
the
such
sign of
power-operation
8] .2
the
The
' *
raising
of
to
"
a
126
MODEEN
power
that
so
of
are
ABITHMETIC
the
12^2X3
f ^f 6 is 12
X
importance
equal
in
2) X
12
The
18.
or
I)ortance and
12
4+6-T-3X5+8-f-2"
difficulties
the
Give
2.
Apply
teacher
of
[6h-3X5]
keep
must
mind
combinations
Explain
5.
of
form
+4"5
and
the
early
2563
364
by
methods
division
and
while
giving
the
of
and
compare
multiplication.
meant
teaching
by
addition.
What
in
this
may
object
be
are
the
tions.
opera-
teaching
multiplication
objections
that
fundamental
object
How
Why?
ideas
general
four
the
predominating
f
is
what
5
"
today.
be' the
should
[8^2]
+
.10
CLXIV
of
multiplication
in
im-
also of equal
cases.
r6sum6
those
to
What
4.
two
(12-f-
EXERCISES
by
the
short
a
with
Compare
3.
in
is
13.
XII.
MMPCXV
formed
per-
of order:
scale
5=4+
"
Multiply
-"
Hence,
appear.
are
"
be
to
are
12-f-6, but
nor
and
signs X
"
"4+
1.
The
signs
the
and
in the
last
PROBLEMS
operations
3),
(2 X
signs, +
come
4.
-f-
which
-f-
AND
the
and
in
order
is not
3,
METHODS
the
tiplication
mul-
attained?
as
breviated
ab-
an
sometimes
raised?
Elucidate.
6. Discuss
7.
fully
relative
When,
multiplication
be
Illustrate
8.
the
last
the
to
taught?
the
paragraph
under
other
fundamental
four
Explain
introduction
182.
operations,
should
fully.
of
the
multiplication
to
process
a
class.
in
the
of
order
under
each
multiplier
Explain
11.
12.
Explain
13.
Show
this
be
the
fully
two
fully
that
taught
each
the
up
of
set
would
various
examples
work.
be
good
in
ways
Suggest
for
at
class
which
least
board
zeros
(i)-(vi)
under
suggested
six
examples
work.
enter
may
into
the
multiplicand.
of
number
a
which
case,
and
detail
taking
Illustrate
10.
by
in
Consider
9.
to
the
digits;
the
division
the
of
meaning
of
three
carrying
is the
children?
the
form
of
multiplication
digits.
process.
inverse
usual
of
Illustrate.
Illustrate.
multiplication.
Why
should
128
MODERN
27.
Suggest
When
and
what
extent
15f
by
last
the
Illustrate
then
11
taken
be
for
in
up
class.
grade
a
class
grade
a
carried?
divisibility by
9 f
is divisible
4
is divisible
three
number
are
by
4
by
last
the
by
to
the
2 f
by
3?
by
by
divisible
by 25;
or
if the
25
by
or
divisible
is
digits
might
be
involved
six
least
at
suitable
Explain
in
work
the
for
order
a
c,
5 f
by
125
125
by
i. 3
X42
8.
or
d-^
"
right
to
left
etc., is divisible
test.
in
simplest
the
by
and
process
each:
simplification
be
similar
examples
effective
made
in
those
to
in
grade
question
priority
of
V.
16-T-4
arithmetical
signs.
Apply
X2;
[3-|-2]2-f-5
4X5-h2
3;
"
X4-i-8
.
VI.
iii. 4
3
"
6-^9;
X
iv. 12-5-4
X
12-r-6
8;
vii.
[4 +
3
5
X
X
2]
3
"
--
5
+
32.
BIBLIOGRAPHY
Browne,
2.
3.
C.
E.
The
American
Jburnal
Psychology
of
Howell, Henry
B.
pp.
84-87
Alva
W.
pp.
92-96.
Stamper^
31
work.
7;
+
ii. 3 +
Processes,
8
or
following:
1.
6?
part made
by
d, etc., from
b -{-c
"
divisions
grade
of
h,
a,
if
11
and
following
this
can
by
Prove
zero.
principle
How
a
is divisible
equal
the
of
digits
Suggest
33.
the
made
Perform
workf
that
it be
number
a
number
explain
32.
simplification
for
test
digits
two
the
or
31.
in
suitable
examples
each.
If
the
by
work
similar
should
that
part
30.
six
is the
Prove
the
least
PROBLEMS
AND
METHODS
Illustrate.
29.
if
this
What
28.
at
should
to
by
ARITHMETIC
of
the
Psychology,
and
97-105.
Arithmetical
SMnple
XVII:
1.
to
CHAPTER
Xm
COMMON
146.
the
History.
of
men
just
of
schools
past
are
in
that
of
The
fractions
the
this
only
have
in
forms.
that
is, those
well
to
The
with
consider
Roman
divided
into
separate
resulting
from
The
12
a
used
copper
all
the
to
of
tions.
reduc-
such
other
than
of
coin
and
fractions,
will
Teachers
adopted
weighing
the
and
duodecimal
12.
this
of
unit
the
are,
sexagesimal
astronomers
both
nominator,
de-
constant
These
the
scheme
the
the
numerator
used
Romans
carefully
as,
by
denominator
a
In
Considerable
i-
had
fractions.
Hindus
The
constant
a
numerator
a
constant
a
favored
greatly
period.
of
papyrus.
reduced
be
tables
Babylonians
place
was
symbol.
separate
the
for
which
Ahmes
i +
to
J, with
of
have
to
i +
papyrus
sexagesimal
as
were
sexagesimal
the
a
of
known
early
first
=
was
unit-fractions.
as
had
fraction,
one
60,
fractions
in
the
work
devised
earliest
document
known
fractions
fractions
hence,
hence
the
and
the
tions
frac-
with
were
in
recorded
interesting
are
given
Ahmes
The
A
among
fratj-tions, as, |^
is
For
1, does
difficulties,
the
unit
attention
schemes
and
computations
of
Many
system,
in
computations
feat
to
children
the
to
complicated
as
counting
our
texts.
in
1
developing
recently,
a
Egyptians,
the
numerator
sum
Until
the
the
overcoming
in
stumblin^block
a
stumblingblock
a
considered
late
proved
ages
today.
were
placed
Fractions
the
they
as
FRACTIONS
by
do
them.
one
pound,
the
11
was
unciae.
name
the
was
division
given
of
129
to
this
each
unit.
of
These
names
subcoins
were
130
ARITHMETIC
MODERN
applied
also
the
through
also
also
was
fractions.
the
Hindus
but
without
Thus, according
been
placing
the
of the
in
Leonardo
reduction
down
to
147.
Informal
Note
that
that
of
fractions
no
of the
up
The
VIII
this
fear
a
Formal
on
said
was
that
judgment
get
89
be
of the
What
fractions.
entirely objective,
the
to
reduction
benefit
The
etc.
is the
they
done
nection,
con-
elementaryof
experienced
be
of
from.
away
the
on
we
which
well, in this
be
made
which
can
fractions
to
and
full
texts
introduction
confusion
Introduction.
arithmetical
is to
is
with
the
introduction
incidental
and
amount
in
He
As
in, division.
It will
the
Stifel
Michael
time
present
of
one
unit-fractions
into
denominators,
subject
be left to the
148.
the
tional
frac-
to
fractions.
of
with
incidental
common
by
was
computations
reference
formal
to
later
taking
the
Pisa
find
all of this early work
and
gained
at
what
regarding
work
devoted
we
times,
Introduction.
re-read
to
must
for the
hard
trying
are
were
divisor
the
modern
more
rules
complicated
of
notation.
modem
treatment
it.
over
denominator,
the
a
wrote
oval,
an
denominators.
common
inversion
the
suggested
in
to
a
double
a
Egyptians
or
is
op
with
present numerals,
above
abacus
scientific
the
dot,
a
writing
of
with
The
4".
separation of 'fractions
the
explained
we
yi? ^V
numerator
our
line, which
Roman
computations.
to
numerator
dividing
columns
forerunners
3' 4"
written
wrote
the
the
denominator
the
by
denominator,
The
come
tItj ^^^
"
decimals,
duo-
these
to
methods
the
in
first wrote
at
followed
have
J would
their
Vf
result
would
addition
5^, ^,
variety
great
twice.
used
Three
recognized
Greeks
The
single accent,
only
In
12.
which
names.
There
accent
fractions
of
denominator,
one
they
separate
forms
11
the
to
PROBLEMS
AND
METHODS
nation
elimi-
by children
wholly
are
familiar.
un-
incidentally-
thus
teacher.
was
said
language, applies with
in
Chapter
especial force
CX)MMON
fractions,
to
of
the
that
perhaps,
it is in
as
symbols.
this
work
topic
more
degenerates
to
attention
Special
of the
subject
secure
grasp
of
In
proper
the
individual, just
is smaller
measure
smaller
16
as
is the
larger
the
nothing
counting
by inches.
The
be
when
of
clearly
Linear
as
counting
such
to Fractions
square
just
as
he
There
is
by inches
than
in
inches
than
in
Objective.
In
fact
it is best
similar
ones,
relations
be
must
on
just
the
to
four
as
really
operations
effective, the
alert
This
integers.
by
teacher
work,
with
is
texts
most
supplemented
the
should
objective
some
in
mal,
for-
The
fractions
include
suggested
this
make
to
to
of
each
paper-folding
mentioned
the
16
These
squares.
quarter
informal, introduction
measure
does
by
large
a
square.
them
counting
of
unit
^
examples
the
larger
the
are
it exhibits
count
can
the
The
illustrations
to
as
useful;, but
number
in
introducing
fractions.
very
diflScult
feet; in
objective.
work
of
Introduction
well
as
child
number
more
by
149.
the
and
counting
fractions
individual
an
child
the
instance,
For
unfolded
much
as
may
its Embolism.
to
that
-^^ of the large
each
are
pupils
to
things, separate
only
way,
when
other,
any
and
home
isolated
integers.
being
same
or
new
their
that
so
one,
the
that
brought
be
with
each
squares
count
can
than
squares,
smaller
in
folded
be
can
square
manipulation
Jie did in integers; that
as
individuals
separate
mere
a
in order
that, in fractions, he is dealing with
and
in
thought-matter
first place, it must
the
than
therefore, be given
must,
the introduction
a
131
FRACTIONS
a
few
great
ments
suppleshow
must
as
supplies
excellent
illustrations
paper-folding example.
Eulers
sixteenths
fine
fourths, eighths, and
furnish
of
tions
frac-
graduated
material
for
132
ARITHMETIC
MODERN
familiarizing
METHODS
the
pupils with
the
Idea, the
AND
PROBLEMS
of
some
the
common
more
fractions.
After
150.
Fractions.
As
soon
let the
grasped,
writing
soon
of
symbols
The
each
follow, and
the
ideas,
of
Eeduction
vice
to
versa,
the
to
idea
the
to
1 of
the
time
in
ago
business
arithmetics
the
for
use
have
omitted
some
from
dropped
they
the
time
no
place
altogether.
a
arithmetic
the
texts.
number
2
J
by
grasped
brought
excellent
learned.
oncq
continually jogged.
Followed
they
are
objective
an
they have
Fractions.
with
terms
fractions
are
a
have
place in
not
just beginning
At
the
work
and
in
common
were
deserved
consequence,
but
be
these
found
an
numerator
may
will be
These
fraction
having
to
long
the
new
ideas
to
multiply
return
be
to
be
day.
now,
get the
J
there
to
without
ideas
the
with
as
in
time
to
fractions
of the
to
2
that
with
permitted
A
needs
Usagfe
the
see
These
to
retaining
time
to
fashion
process.
according
be
numerator,
and,
addition
changing
led
and
numbers
In
be
objectively.
in
Business
actual
the
mixed
to
combined
are
not
of children
memory
Some
being covered
latter
used.
must
these
from
aid to children
151.
with
simultaneously
of the
text
mechanical
of
back
demonstrations
The
ways
al-
should, however,
not, precede
should
children
the
facts.
of these
truth
presented
in the
that
They
purely
a
the
fractions
may
adopted
J.
add
7, in
writing
ideal
danger
fraction, pupils
make
3 and
or
in 2 and
J's
basic
improper
may,
order
improper
six
of
way
J^,etc.,
=
new
be
oughly
thor-
been
has
shorthand
the
in
Writing
former.
by the
up
J
recognize
is then
there
as
learn
to
never
of
Way
of fractions
practice
J;iof
=
they learn
as
idea
pupils
them
Give
them.
i
as
the
as
Shorthand
present
may
been
to
be
time,
well
be
COMMON
is
There
texts
As
half
they
which
the
and
Addition
and
Then,
its
to
same
In
order
naming
will
part
denominators.
introduced
and
encouraged
both
meaning
look
is the
the
have
tion
subtrac-
or
nominator
de-
same
that
the
denominator
and
the
numerator
to be
is
the
ing
great aid in* introduc-
a
equivalent
to
for
worked
the
with
numbers
different
fractions
resulting
to
this
at
multiply
fraction, of
each
will
be
pupils should
which
for
given
may
objectively
out
thoroughly
"
with
ones,
denominator,
common
fractions,
the
duced
re-
all
the
have
denominator.
same
It
is
advisable
is
Emphasis
fractions
c.
m.
fractions
not
thus
placed
reducing
To
with
refer
to
equivalent
to
of
process
Li.
found
its
to
4"
place here.
term,
that
so
"
in
The
of the
terms
course
be
this is understood
When
time.
is
fraction
denominator
e., must
fraction
of fractions
changing
ples
exam-
thus, f +
by addition
fact
the
the
will be
Paper-folding
the
alike; i.
of
of
through
one
the
tion
addi-
fraction, as, f -[-i, f -J-^.
review
part
which
combinations
be
must
A
numbering
other
permit
to
fractions
the
in
T^.
to
is used,
where
examples
equivalent
of the
that
as
those
come
may
32, 7f is
be
should
denominator
same
is
5^, ^
tions
frac-
denominator,
the
Introduction
fractions
of
the
is
many-
business
used
without
that
overlooked.
be
not
most
Subtraction.
and
which
the
are
Thus, 5^
subtraction
in
should
written
commonly
are
is 4 understood.
152.
be
quarters
of today
notation
which, however,
omitting
are
a
business
common
a
133
FRACTIONS
carry
out
a
and
denominators
plan
which
of
upon
this
at
m.
idea
not
common
this
c.
l.
the
upon
ones,
to
to
are
is
of
reduction
the
mechanical
denominator
it
time.
through
to
necessary
easily factorable
use
at
first.
After
with
a
the
common
real
meaning
denominator
of
changing
has
been
fractions
to
those
fully grasped, the
134
MODERN
term
c.
l.
its
and
m.
be
The
introduced.
be
may
PROBLEMS
AND
ing
follow-
easily comprehensible
found
to
children.
Hence,
As
l.
c.
m.
2X5X3X7
=
rule, it is best
a
it often
since
leads
learned,
permissible in
of
modes
permit
of
but
Contraction
as
multiplying
be
to
usual
the
therefore,
the
bolical
sym-
English
of
not
are
taken,
be
The
things
two
telegrams
must
* *
the
notation
of
inaccuracies.
correctly.
Care
statements
contraction
and
one
telegram
a
speech.
such
is
learned
be
must
confusion
which
210.
=
discourage
to
to
language,
is
use
will
form
common
the
METHODS
ARITHMETIC
to
not
fraction
J by 4
gives ^."
The
only variation
in addition, is the
no
fraction
in
the
been
It is
good
a
minuend,
If
plan
to
the
precede
**
taken
up
a
fraction
only
certain
names
kind
present
such
with
examples
has
diflSculty.
serious
no
than
integers
in
case
is
there
it is smaller
which
analogous
in
As
before
all of
It is well
completed."
of
in which
that
two
three
or
integers.
Multiplication.
be
in
from
subtraction
numbers
or
this will
in
examples
should
a
the
considered
well
153.
of
of mixed
case
subtrahend.
similar
been
in
in
encountered
by
the
a
whole
to
the
in addition
work
begin with
5 X
-ft
(sixteenths) is
to
has
tion
multiplica-
the
Since
number.
fraction, then
of number
multiplication
integers,
the
means
be
nator
denomithat
3
multiplied
136
MODERN
154.
ARITHMETIC
Mixed
to
such
making
Only
one
of
in
integer
an
the
out
give the
product
asking
of
the
7 X
than
the
the
average
product
as
in
approximations
illustration.
the
pupil
erroneously
example
only
are
placing
the
wrong
needed,
the
as
in
more
ducts
pro-
column,
be
may
mistake
to
than
the
less
figures found
numbers
in
the
reality much
The
but
suffice
error
give
is in
found.
given
by
ever,
rule, how-
of
This
must
of mixed
a
of
found
be
must
made
who
which
but
multiplication
in
right.
9, which
multiplication,
required
simple
by
illustration
arise, that
the
to
As
Products
multiplication without
fraction
a
computation
pointed
to
best
can
the
One
and
PROBLEMS
fraction.
perform
to
is liable
error
3J,
improper
reduction.
a"
AND
MultiplicatioiL
2J X
as
an
preferable
it is
in
numbers,
each
reducing
as
Numbers
mixed
small
METHODS
may
many
the
not
cases
be
in
as
close
accompanying
:
COMMON
Division.
155.
The
of
first
fraction
by
factor
is,
i
as,
is
see,
divisor
the
the
II
=
integers.
equivalent
3
divided
of
fraction
the
just
give 3-cent8,
easily be reduced
multiplying
to
the
|f-f-4.
as,
by 4 would
T^, which
=
of
numerator
factor,
a
division
be
gives 3-nineteenths
4
numerator
-T-
the
as
by
it may
divisor
3
-T-
reverse
of
case
should
class
number,
divided
If
of the
the
the
as
multiplication, similarly
the
to
whole
a
12-oranges
or
3-oranges.
or
given
containing
12-cents,
in
in
order
taught
integers.
12-nineteenths
This
be
done
was
the
examples
fraction
a
as
for
fashion
the
should
as
also to follow
It is well
to
Division
multiplication just
of
137
FRACTIONS
to
is not
which
one
pupils will
the
denominator
the
a
soon
the
by
integral divisor.
in
Division
of
order,
for
**How
In
order
of
magic
foot-rule
the
12-77!.
one-fourths
not
may
The
be
also
in 12, there
will
be
be
As
is, **How
48
an
many
As
3,
-=-
sort
a
used.
12?"
comes,
be-
foot?''
a
to
appear
from
made
in
there
question
be
can
the
to
objectively it
are
objects should
other
Applied
handle
can
really is,
question
12?"
in
quarter-inches
one-fourths
3
The
is next
fraction
unit
a
J.
-j-
pupils
many
comes
by
there
are
the
wand,
of
48
12
that
extension
are
example,
which
**How
groups
number
fourths
many
foot-rule
whole
a
16
or
there
such
groups.
Finally,
V"
as,
are
in
this way,
divisor
this
a
are
^^- one-thirds
-^ of two-thirds
2j=
examples
156.
There
fraction
one
by
in
another
hence
V-
tion
fracthere
,
^-r-
the
""
-^
of
division
comes
method
Division
division
denominator
and
multiply
of
if they
-f-
for
procedure
fractions
thus, f
^.
may
^ would
of
solving several
After
pupils should
by Comparison
both
;
the
in
directed
be
have
vert
in-
covered
already dis-
not
themselves.
Numerators.
be
to
Again, in
reduced
become
if
to
-7-
the
same
if which
is
138
MODERN
same
as
the
same
kind
in
in
those'
obtained
see
not
be
results
their
from
the
into
into
then
be
requiring all
As
better
would
divisor
shrewd
reduced
can
verified,
be
numbers
rule, mixed
be
3
with
teacher
to
3
of
compared
results
a
of
groups
The
observation.
of
12
groups
pupils if reasonably
integers.
division
in
occurring
divided
by
multiplication by the
own
in
with
done
divided
may
the
cases,
rigorous
too
was
as
The
through
of unit
divided
oranges
most
this
cents
12
or
each.
In
inverted.
12
unit;
PROBLEMS
AND
kind
certain
a
of
each,
oranges
will
of
10
the
cents
METHODS
ARITHMETIC
improper
to
fractions.
All
157.
pupils
in
indicate
be
used
be
encouraged
of
to
to
by preliminary
number
of
158.
can
cancellation
first and
add
effective
whenever
of
short
every
a
quite
mental
example
1236
a
process
X
the
58.
it
to
As
readily
for
them
it
suflScientlylarge
a
goes
teacher
cut
which
16f
a
has
on
been
in
which
direct
the
but
of
in
introduce
to
process,
use
use
the
from
dictated
aliquot
tion
multiplicabecomes
which
value
use
learned
aliquot parts,
applied
at
this work
make
insists
their
100
few
also, that he employs
is well
The
.
of
multiplication
only
To
the
by the
in
use
on.
It
long
aliquot parts
"
consider
to
admitting
problem
a
be
would
the
that
problems.
and
through
merely
time
time, numbers
examples
of
into
opportunity arises,and
to
parts
of
better
as
it is necessary
throughout
time
Some
is far
these
to
in
out
prepared
has
manner,
advisability
they will take
profitably be brought
It
can
examples.
Parts.
division.
fraction
a
The
pointed
solution
the
as
3-f-4, they should
=
division.
teacher
the
in
work
Aliquot
very
and
the
that
soon
operation in this
been
already
that
provided
enough,
f
as
every
actual
the
As
fact
the
enjoy cancellation
children
most
express
has
plan
a
with
division,
a
performing
such
Cancellation.
"
fully familiar
are
before
Step
One
of
as
in
aliquot
COMMON
parts is often
number
25
or
in
159.
is
725
those
that
have
clear
a
operations multiply
A
with
suflBciently
not
which
do
have
of
part
a
5451,
of
value
the
the
of
absolutely
in
work
following
:
fraction.
a
fraction.
a
fraction.
a
comprehension
thorough
is
completed
the
change
not
divide
operations
It
understanding
2. What
4.
in
Know.
who
pupils
operations
What
instances
Should
1. What
3.
does
aliquot part, as, 33^ in 533J X
Pupils
necessary
teacher
9348.
X
What
fractions
in
useful
a
the
because
its value
emphasize
one
lost
139
FRACTIONS
of
four
the
fundamental
operations
fractions.
5.
aU
That
showing
final
results
what
both
form
that
be
to
are
is and
placed
how
their
in
simplest form,
it is obtained
from
the
given
the
study
fraction.
XIII.
should
Why
1.
children
expect
we
EXERCISES
have
to
difficulties
in
of fractions?
Give
2.
modern
r6sum6
a
How
have
the
Analyze
these
fractions
in
before
use
recently,
very
handled
tions?
frac-
fractions
Illustrate
difficulties
should
how
the
this
children
the
knowledge
children
are
gain
be
may
likely
to
during
acquired.
in
experience
first years.
is
What
6.
' '
numbering
7.
Explain
8.
Show
a
of
texts, until
of
years?
first three
5.
to
kinds
fully.
knowledge
What
4.
modern
the
Discuss
* *
various
the
times.
3.
the
of
meant
and
that
the
use
calling
by
' '
the
naming
fractions
of
the
' '
parts
units.
are
paper-folding
and
numerator
in
of
a
fraction
denominator
?
the
Illustrate.
Illustrate.
presenting
the
idea
of
fractions
class.
9.
Show
of
10.
symbols
11.
arise?
to
How
rather
would
How
numbers
pupils
how
you
improper
do
business
are
than
liable
the
to
real
introduce
fractions
men
to
and
write
memorize
idea.
a
mechanical
a
How
may
class, the
vice
i, i,
this
tion
manipulabe
reduction
overcome?
of
mixed
versa,
and
f?
Where
would
these
140
MODERN
12.
to
METHODS
Ezplaiiii briefly,how
addition
AND
of
PROBLEMS
fractions
should
introduced
be
class.
a
State
13.
the
taken
be
to
are
ARITHMETIC
which
addition
subtraction
and
fractions
of
Illustrate.
up.
Suggest
14.
in
order
least
at
four
suitable
in
examples
for
each
grade
a
class.
15.
of
is very
It
fractions
changing
giving
about
Arrange
17.
Define
in
multiple
divisor.
and
of
28, 84, 21,
Perform
the
21.
Discuss
fully
22.
Discuss
to
folding
24.
taken
the
and
State
the
26.
Suggest
for
to
ig.
multiples
and
C.
H.
two
the
L.
M.
and
H.
C.
C.
D.
12.
the
L.
0.
M.
and
D.
of
two
L.
M.
C.
the
by
in
Austrian
addition,
and
method:
the
proper
of
contracted
notation
introduce
multiplication
in
early
work
would
you
in
in
this
connection
tions
frac-
of
paper-
multiplication
should
Illustrate.
objectsf
order
made
be
can
use
of
which
the
subject
of
Illustrate.
up.
Find
reduced
fj, and
two
subtractions
of
use
way
similar
25.
numbers
the
What
class.
a
fully.
Illustrate.
Explain
23.
go
product.
disadvantages
the
fractions.
you
it.
introduce
to
Find
D.
of
following
20.
time
C.
H.
product
their
Give
meaning
50.
the
that
equals
numbers
-
Jf, H,
42, 35, 63, 70;
Show
would
Explain
magnitude:
and
L.
How
of
M.
Define
the
understandingf
C.
both
understand
denominators.
common
and
of
19.
be
order
children
the
36
18.
in
to
that
them
16.
divisors
of
important fhat
the
a
following products
six
graded
grade
improper
:
in
examples
class.
fractions
When
f
should
Whyf
the
multiplication
the
mixed
numbers
of
mixed
not
be
COMMON
27.
Explain
28.
Explain
how
of
division
the
introduction
the
141
FRACTIONS
fractions
to
is the
class
grade
a
plication.
multi-
of
reverse
division
of
of
fractions.^
29.
Outline
30.
Explain
order
the
the
fully
taking
of
the
pupils
be
required
31.
What
part
of
his
income
will
much
How
In
32.
girls.
are
boysf
33.
A
sales
What
part
What
was
of
other
36.
A
later
sells
What
39.
533
40.
Water
pounds
feet
of
ice
A
half
he
remainder.
43.
clothes
of
save,
who
of
f
other
his
1
expenses
year?
per
equals i
boys
number
the
girls? boysf
are
^ of
spends
i for
and
$1600
number
How
many
sold
lost
retail
for
^,
$20.
f of
|J
of
expands
A
does
62}
A's
in
a
his
be
his
loss
the
pound?
per
for
$21,000.
part
did
he
he
gained
share
sellT
one
i and
how
much?
wholesaler^,
and
the
cost
of
manufacture?
of
his
land
of
sells
i
has
he
acres
itself
of
^
number?
foot
the
was
many
equals
buys
papers
at
cents
each
part
On
and
by
cubic
boy
Simplify:
cents
What
lose
or
If
of
a
ice
retailer^
problem
of
foot
and
38.
weighs
water
How
weigh?
land,
372?
with
cubic
on
left?
equals
Compare
freezing.
has
What
6
would
of
cubic
many
pounds?
money
3
at
own?
What
how
what
much
for
What
each.
section
a
how
sells
f
" of
sells
$120
gain
increased
number
f of
sugar
still
he
he
remainder;
what
of
for
Did
i.
owning
weigh
does
makes
the
customers
factory
a
horses
two
at
his
fraction
What
pound.
a
valuation?
total
he
for
pounds
factory
equals
62}
together,
man
number
total
cheat
f of
i of
38.
42.
he
owning
man
If
the
ounces
200
sold
A
41.
15^
manufacturer
article
37.
the
buying
the
man
the
an
of
a
salary
a
pupils
does
the
A
for
40
uses
man
What
35.
^
of
reason
fax
How
girls f
grocer
A
does
out
part
customer
34.
salary
divisor.
the
this
understand
to
board,
save
of
by weight
a
on
he
school
a
of
to
rent, i for
for
subject.
inverting
for
reason
should
this
up
} of
B's
and
money
they
have
The
first
$40
each?
of
the
the
rate
but
of
3
receives
buying
price
for
5
only
did
he
cents.
2
each
cents
gain
or
lose?
for
one-
the
142
MODERN
ARITHMETIC
METHODS
6*
Simplify:
44.
AND
PROBLEMS
7i
'
3i
3i
Criticize
45.
.4
3
"
the
following;
3
+
1
1
8
8"0+9
11
"
z=
0.
=
2
lU.
"
.
~
5
46.
2-f3
"
Suggest
expect
47.
before
Give
48.
and
37
of
49.
50.
in
fractions
in
~
3
"
12
which
3
would
you
stating
of
of
any
symbolical
of
any
which
all
the
in
operations
"
them.
of
statement
the
operations.
operation
each
problems 33, 34, 36,
Compare
is worked
this
out
form
with
the
take
vantage
ad-
reasoning.
the
Suggest
of
out
attempting
one
12
difficult.
advantages
complete
with
of
progress
find
21 +
problems
two
carrying
a
before
solution
to
the
Explain
problem
least
at
pupils
your
6
3
least
at
three
problems
which
in
one
may
this principle.
What
aliquot
are
multiplicationand
51.
Perform
52.
Suggest
53.
Show
54.
What
parts?
division
several
by
following
the
six
examples
Explain
in
the
aliquot parts
on
are
their
use
in
illustrations.
operations
aliquot parts
that
Illustrate.
based
simplest
suitable
upon
the
for
manner:
grade
principle
work.
of
ciprocal
re-
operations.
work
55.
should
the
in fractions?
What
kind
pupils
Illustrate
of
fractions
know
after
they
be
avoided?
and
Coffman.
"completed"
each.
should
BIBLIOGRAPHY
1. Brown
have
pp.
182-94.
Illustrate.
the
1 44
MODERN
from
France
old
the
ARITHMETIC
this
about
it
English
it is the
that
few
and
which
than
needs
no
it
English
rounded
length
the
nail, and
it
Romans
Romans
eU, the
end
of the
yard
the
Roman
the
present
length
of
the
of length
units
of
to
a
hand
the
outstretched;
outstretched
;
end
the
of
name
Thus
day.
foot.
of
The
pound
a
**inch,"
I decreed
of the
legal length
people, today,
chin,
or
nose,
to
arm.
have
the
been
the
span,
; pace,
a
the
inch, by
terms,
the
the
length
little
ref
hand,
from
finger when
finger-hreadth; fathom,
arms
of
some
from
yard
the
Henry
the
be
:
three
of
to
one-twelfth
unciea,
should
arm
and
use,
first joint to
English
the
considerably
Scandinavians
the
the
as
ounce
traced
of the outstretched
Other
thumb
to
length of his
measure
width
have
ounce,'' both
the
that
the
designated
also
elbow
the
length
one-twelfth
be
to
is
and
variously defined
and
that
20
penny.
a
universal
from
give
they
unit
the
French
of
ton
from
arm,
be
to
of
been
has
pounds
pounds.
almost
thumb
held
was
etymologists
and
inch
held
was
of the
the
tip of
**
The
100
Babylonians,
This
is in
barleycorns, by the
was
fore
the
finger.
foot, which
the
2240
Egyptians,
of
middle
explanation.
the
and
early
integral
The
origin in the weight
length
tip of the
the
the
the
the
was
older
the
of
the
of at least
weight
of
customhouse
was
hundredweight.
a
its
has
cubit
Hebrews
a
;
fact
the
schoolbooks
in
pounds,
therefore, contained
Pennyweight
it
called
hence,
hundredweight,
The
gave
only
meats
only in the
112
or
first
articles.
heavy
found
ton
did
as
At
grains.
interesting from
used
now
grains
weighing
all
is
long
Eight stones,
of stones
was,
for
weigh
to
is
but
ago,
7680
7000
now
used
was
origin of the
in mines.
number
is
stone, 14 pounds,
decades
and
by
which
used
was
PROBLEMS
AND
contained
and
avoirdupois pound
The
to
1300
pound,
statute
later, however,
a
METHODS
ering
to
the
the
the
meaning
tip of the
the
the
length
length
of
hand
of
a
is
the
step
;
WEIGHTS
AND
hair's-hreadtk, used
of
units
measuring
suiim),
length
thousand
nautical
is
of
the
the
Gill
gallon.
drinking
Peck
box.
meaning,
perhaps,
piece of
open
of
amount
For
161.
Choice
selected
time,
have
of
Land
order
connect
of
work
Yarn
per
of units
In
Ih
day.
are,
nations
of
of
was
of the
any
uniformity
of
been
the
day.
one
interesting
nections,
con-
as
an
40
be
and
of the
the
variety
service.
special
some
in
perches
work
buying
to
and
in
the
acre,
in
the
amount
with
could
spinner
in
as
day.
a
the
upon
accomplish
selections
question.
of
units
uniformity
of the
pose.
pur-
prices, merchants
accomplished
that
among
question.
effort toward
made
for
inventions, such
out
out
modern
more
tods, depending
a
units
The
size
of weight
modem
to
of
changing
which
approach
were
less
suit
paid for
could
of
obviously
units
an
in
plow
pack,
merely
have
to
those
to
into
sacks
course,
consideration
mentioned,
some
into
day
our
meant
convenient
of
wages
of work
amount
average
Acre
a
tain-sized
cer-
a
from
Peoples.
by
as
units
which
divided
was
gilla,
meant
come
are
made
divided
the
land
Latin,
have
of
quarter
one
have
could
Early
of
been
was
on
tion
deriva-
supposed
there
by
different
used
selling.
to
units
instance, instead
England
the
the
to
oxen
sometimes
were
For
of
Units
always
Selections
is
early peoples,
the
by
regarding
all.
at
of
earth's
to
carry.
but
other
none
to
pair
a
the
or
load
a
the
course,
supposed
is believed
ground
land
of
most
is
on
from
comes
Bushel
glass.
is, of
qu^rt
likely
mile
geographical
or
information
The
gallon.
(mille pas-
mile
great circle
a
some
feet.
chain, 66
accurate
no
of
arc
of
length
long;
teresting
in-
Other
space.
rod, the
furrow
a
(knot), the length of 1' of
There
the
paces;
surface; Gunter's
minute
a
are
furlong,
rod;
a
denote
to
145
MEASURES
have
been
the
early
establishing
early
as
of
Eegulations
William
a
degree
the
146
MODERN
ARITHMETIC
Conqueror,
this
in the
due
was
would
have
and
but
they
than
time
time
to
that
as
To
plicate
com-
community
the
days of the
the
of
merchants
the
with
after,
there-
Some
prices.
change
varied
changes
diflferent for
even
were
facts
weights
these
from
PROBLEMS
satisfactory results.
trivial
change
matters,
and
without
such
AND
made
be^
main
to
lather
METHODS
in the
week
community.
same
About
the
defined
Winchester
the
bushel
standard
in
the
cu.
in.
and
of
Secretary
unit
of
which
t^jVtt of
be
to
of
Hii
162.
the
the
to
inch
an
unit.
accepted
By
States
even
the
1893,
a
This
was
yard
yard
English
1855,
of
that
than
of
copy
found
formerly
the
defined
was
has
since
time
as
time.
gress
Conmeasures
The
United
that
century
the
has
been
units
have
measures
in
established
was
greatly modified.
been
international
no
to
Standards
last
and
weights
yet
the
Measures.
weights and
the
from
of
within
and
Weights
Bureau
scope
therefore, well
the
in
the
accepted
present
States.
governing
laws
National
of
of
standardization
1901, whose
The
1836
meter.
'enacted
their
of
act
an
United
shorter
Standardization
has
and
presented
was
containing
In
then
adopted
was
replaced
were
1824.
the
been
have
bushel
in
of the Union.
yard,
length, the
in.
cu.
chester
Win-
the
and
measures
and
gallon
copies of
states
in.
cu.
These
England
century,
^since 1834, but
2218.19
had
various
to the
sent
States
imperial
State
231
as
in.
cu.
United
by
England
eighteenth
gallon
2150.42
as
in the
277.274
of the
beginning
is,
It
tion
standardiza-
attained,
been
though
al-
decided
upon.
163.
development
It began
with
value,
north
Standards.
Monetary
of
moneys
direct
such
countries,
or
can
barter
as
cattle
wampum
The
hardly
fascinating
be
led up
to
in
Rome,
fish
the
of the
upon
here.
touched
that
among
story
a
medium
in
some
Indians,
and
of
of
versal
unithe
finally.
WEIGHTS
metals
precious
became
wondered
has
name
pound.
This
of
for
double
but
words,
in
reckoned
not
weight.
This
reference
already
coins
not
do
result
different
the
The
to
a
a
which
in
mill
sufficient
amount
these
the
by
sovereigns
the
a
late
its
to
as
add
of
arithmetics
article
in
Dec.
its
has
1912
gave
exchange,
on
until
is often
Cajori
the
that
This
importance
but
English
early colonists.
coins.
mark
and
Farthing"
of
use
the
own
undue
S
* *
use.
The
with
in the
and
in
penny.
day
Easterling
name
now
a
Tears.
All
first four
very
taken
be
to
very
clearly
issue
of
the
of
the
origin
is
that
work
in
denominate
the
should
the
informal.
be
work, however,
involving
years
of
years
of number
the
any
end
of these
It is
the
as
Easterlings, brought
the
name
of
Monthly,
the
enable
to
The
as.
common
easy
child
of this
units
and
marbles
and
as
or
who
is
period,
their
use
interesting
tops.
have
to
A
given
such
measures,
compelled
to
be
should
foot, inch, yard, quart, pint, gallon, pound,
school
as
U
Pour
for
ton,
peoples,
Roman
by
in
dollar.
numbers
during
was
original values,
given
received
an
Science
First
164.
the
to
origin of the dollar
of
out,
Spanish
lack
a
but
all
among
debased
of matter
has
The
Popular
as
until
deal
combination
pointed
to
coin)
the
upon
made
colony, also, had
great
subject
due
than
early times, money
these
was
fourthing
separate
recently.
a
which
prevailed
a
is not
traders, called
Sterling
to
the
from
rise to
been
been
having
coin
later modified
Each
of money
custom
possess
German
England
money
and
a
countries.
east
comes
weight
in
usual
having
today
that
found
have
word
the
schoolboy
a
not
(the stamp
the
was
their
of
fact
tale
of
of
use
the
to
units
Many
could
English
their
147
MEASURES
medium.
a
the
why
different
AND
and
ounce,
drop
gained
a
out
of
hension
compre-
in measurements.
to
add
Reductions
inches
should
or
be
quarts
made
148
MODEBN
only
from
ABITHMETIG
denomination
one
pints
never
thing
of the
handling
feet
by the
showing
or,
pints in
linear
of
fractional
over
5 feet
The
the
of
in the
the
bulk
An
is called
month,
day, week,
as
Simple problems
easily made
in
in
the
pints
feet
and
of
the
the
of
to
will
the
to
be
these
furnishes
fractions
as
and
its
and
pecks and
upon
foot
selection
of
division.
2
reciprocal
their
able
and
their
suggest
relations.
reduction
mentioned.
is of
division; by
a
little
of
to
the
the
3, of
gallons
to
of
number
relation
amples
ex-
ing
instance, dur-
the
simple introduction
previously
for
reciprocals; by
any
The
judgment
suitable
reciprocals. With
to
he very
can
but
For
in
and
dock.
reciprocal ; by 4, of quarts
bushels
a
the
in
capacity.
hour, minute,
reductions
multiplication by
and
pecks
teacher
based
of
quarts
yards
to
the
the
is the divisions
subtraction
and
difference
and
of the measurements
any
multiplication
study
to
quarts
from
up
is necessary
in addition
by
be
must
greatly
help
work
year,
second, including the telling of time
of
Weighing
weight
between
little
number
the
and
will
subject for this informal
time, such
the
object.
an
takea
a
children
the
weighing
of
a
directly, without
learned
that
in
on.
and
pound
substances
distinguish
to
important
of
different
of
them
teaching
weight
and
be
length
2
are
quarts
a
be
of
process
8
should
so
yards;
there
account
the
better
in
no
weighing, except
capacity
same
in
that
water
inches, and
9
over
might
with
the
between
that
ounces
ton
of
process
acquainted
So
little
a
number
pounds
measurements
actual
finding lengths in
gallon, and
a
good
a
the
longer lengths
or
in
quarts
parts.
or
sand
pints,
It is
in
time
scending
de-
or
gallons, or
pints.
to
example,
later
of
use
4
quart,
a
In
peck.
first,and
at
ascending
to
this
at
for
measures;
inches
and
gallons
or
PROBLEMS
next
quarts
practice
much
give
to
the
from
gallons,
to
AND
to
reduction
as, the
but
METHODS
8, of
practice
problems
inch
elements
to
WEIGHTS
Square
in the
a
of desks,
areas
of the
Much
in
elaboration
formal
the informal
the
teacher
not
have
Teach
the
idea, at most,
for
as,
of
an
good
be
880
boys,
and
be
not
in
width
referred
idea, are
of
arise
baseball
and
which
of the
block, dimensions
or
that
may
diamond.
measured
school
feet
or
300
100
yards.
the
to
measurement
of
way
knowing
be
may
the
spent.
average
feet, 220, 440, and
especially
events,
a
every
distances
5
in
feet.
this
so
on.
60
can
cord
heavy
that
Other
manner
lot, length of
building, and
the
to
also
do
as
These
ascertained
and
bushel, the
attention
no
by
is knotted
sidewalk, street,
of the
a
appeal
athletic
be
which
to, of which
will
in
elements
children
50, 100, and
these
slightest
No
has
life of the
''stepped off"
measurements
are
teacher
is perfectly
the
or
chief
pay
the
the
stretch
100
since
Many
feet
90
either
does
little
to
life.
who
only
gallon
community.
future
the
of
the
of
application in the
various
the
they often
as
has
in
frequently
yards.
often
a
child
a
may
fundamental
conception
their
length
plan, therefore,
neglected
has
person
the
or
the
and
instance, the size of
acre,
Let
in everyday
of the
meaning
in which
pupils who
Many
of
review
any
clear
a
tables
reductions
used
Distances
to
the
to
short
a
practicaluse
the
of the
where
or
further
no
introduction
work.
measuring.
with
measurements
how
of
art
complicated
must
already
recommended,
is
children
the
their
and
familiar
a
needs
and
numbers
preliminary
give
to
units
It is
has
measures
an
special attention
this
had
As
just suggested
pays
various
extent
in
work
denominate
work
object be
arise,
furnishes
on,
material.
Ideas.
of
study
most
so
rectangles.
here.
Fundamental
165.
first
measure
drawing
in
previous lesson
a
linear
floors,books, and
early number
into
gone
later
and
variety of usable
great
been
of squares
drawing
Measuring
directly from
arises
measure
149
MEASURES
AND
a
city
150
MODERN
166.
ARITHMETIC
Indirect
METHODS
Measurements.
The
areas
fundamental
be
rectangle
must
rectangle
is divided
is linear
units
of
units
many
dimension
of
is
fact
units
As
soon
linear
idea of
feet
square
to
child
that
It is
only
and
more
A
the
most
in
contents
Much
of which
is
What
which
It
the
of units
the
weighed
of
weight
gross
is well
bushels
value
in
only
to
for
your
state
such
problems.
within
the
of
to
linear
a
typical :
of
load
and
that
10
of
permit
to
length
is
by
the
units
of
This
and
of
contraction
being accepted
is that
to
the
solution
the
oats
34
at
tare
2780
cents
The
grains should
uniformity
in
of
per
finding
weighing.
problems
bushel
for
pounds, respectively!
large weights
pounds.
of
grain by
of
load
is
out
safe
nimiber
measurement
given
a
''reducing
above, in which
the
Europe
be
various
Lack
62.
States.
of
is 4640
point
to
See
this fundamental
over
seem
of breadth.
United
following
the
area
importance
puzzle
of the
contraction
should
attention
as
other
the
strips.
multiplying
by
area
indirect
common
be
multiplied by
it would
multiplication
in the
more
must
undue
he will not
readily enough
accepted
in the
of
be
not
the
rectangle
units
fully acquainted with
that
length by the number
is
area
giving
can
gets
a
that
a
strip contains
of linear
number
of
area
the
as
each
the
feet"
he
the
means
is
cubic
to
say
breadth.
so
ments.
measure-
area.
child
area
the
total
by
units
of
the
as
by
wasted
get units
finding
the
that, hence,
that
to
number
the
as
often
linear
strips
directly,
in-
considerable
studied; namely,
many
as
derived
are
two
direction, that
one
area
and
Energy
him
into
strip multiplied
one
the
in
thoroughly
ments,
measure-
our
receive
must
in
idea
of
weight,
from
found
are
PROBLEMS
most
and
phase of the subject
All
attention.
As
*
excepting time, length,
that
AND
like
these
standards
be
the
employed
various
are
used
in
in
all
states
152
MODERN
The
of
hay
8. One
9.
cubic
follows
as
feet
taken
the
to
512
cu.
ft.
=
1
ton
new
422
cu.
ft.
=s
1
ton
stack
5
months
342
cu.
ft.
=
1
ton
stack
1
year
216
cu.
ft.
=
1
ton
stack
3
or
e()uals one-half
cup
of
flour
A
measuring
cupful
of
com
A
measuring
cupful
of
butter
A
measuring
cupful
of
chopped
A
measuring
cupful
of
granulated
A
measuring
cupful
of
liquid vweighs
of
wood
board
thickness
a
is
a
pile
4
Lumber
is
or
A
bundle
of
of
lots
pound.
one-third
one-half
weighs
sugar
one-half
of
generally
sold
contains
250
shingles
one-half
pound.
pound.
surface
a
pound.
feet.
8
by
pound.
pound.
weighs
meat
board
old.
years
one-half
weighs
with
old.
4
weighs
by
4
old.
one-half
weighs
meal
less.
in M.
stated
is
foot
1 in.
of
age
pint.
cupful
cord
the
upon
stack.
measuring
A
depends
ton
:
measuring
A
PROBLEMS
AND
A
10.
feet
of
number
the
METHODS
ARITHMETIC
1
ft.
sq.
by
and
1000
the
a
board
shingles averaging
in. wide.
4
Measurement
11.
in sections
lands
If
advisable
fractions
and
the
city
in
thereof
extended
more
front
feet
be
must
study
or
entered
be
may
feet
square
into
and
thoroughly.
until
delayed
farm
the
last
years.
168.
than
Reductions.
at
omission
and, hence, their
is
more
of
arise
is advised.
numbers
and
Eeductions
complex.
given
ing
involv-
infrequently
very
following order
The
suggested:
i.
Descending
of
two
denominations,
as
""
""
(t
....
it
it
..
."
iii.
''
'*
iv.
*'
u.
V.
formerly
was
denominate
denominations
three
than
more
much
were
time
more
reductions
to
present
problems
the
Much
'*
Ascending
*'
vi.
Some
three
''
two
"
"
"
"
"
three
schematic
is advisable.
The
form
'
*
'
**
following
are
qt.
3
gal.
4
3
1
"
pt.
to
1
qt.
gal.
3
qt
gal.
1
pt.
"
"
"
"
pt.
to
1
to
10
qt.
to
gal.
13
pt.
to
gal." qt
is clear
and
*
which
3
suggestive
"
takes
:
pt.
pt.
to
pt.
pt.
qt.
"
^pt.
little time,
WEIGHTS
AND
153
MEASURES
Deseending
gal."
4
^
Aa^nding
qt." 1 pt.
13 pt.
2)
4
6 qt"
4)
19
qt.
1
gal."
1
2
pt
qt
1
"
pt.
2
39
169.
pt
Four
the
delay
to
numbers
four
until
in
all final results
to
the
reduction
no
operation
to
be
not
derstood
un-
performed.
quire
Re-
placed in the simplest form.
be
to
that
see
learned, but
been
have
necessary
denominate
of
operations
reductions
the
exercised
is needed
It is not
Operations.
fundamental
all of
be
must
care
Fundamental
1. Addition:
Use
the
gal."
3
3
4
2
2
11
$
0
10
3
form
common
1
qt."
pt.
(This auxiliary line
2
but
Subtraction
carried
and
introduce
Austrian
1 pt.
2
pt. ; 1 qt. and
but
at
firat
i"0B8ible.)
as
denominate
numbers
more
Austrian
a
than
practicable when
method
(2)
little
because
it
It
is
examples
a
addition
such
as
is required.
those
By
1
becomes:
plan
good
the
pt. and
qt. equals 3 qt. ; 2 gal. and
gal.; 1 gal.,
requires
is
2
Example
1
few
it.
(1) below
example
qt. and
a
with
borrowing
no
This
gal.
of
soon
up' shortly after
taken
with
which
in
gal. equals 5 gal.
2
be
simultaneously
on
method
equals
equals
6
should
subtraction
given below
3
as
used
be
may
diseontiiitted
is to be
Subtraction:
2.
to
given below.
2
once
gal., and
3
insight into
more
the
3 qt. and
becomes:
other
form
understood.
requires
a
better
3 qt.
gal. make
reduction
of borrowing
To
bar
the
knowledge
of
154
MODERN
ARITHMETIC
is like
reduction
the
METHODS
PROBLEMS
barring, multiplication because
by addition
work
AND
without
needing
we
learn
to
can
the
do
plication
multi-
tables.
3.
As
Multiplication:
in
addition, the
results
discontinued
4.
as
first.
at
soon
be
may
This
permitted
to
give preliminary
is to
practice, however,
be
possible.
as
Division:
Division,
understood
by
^would
better
grasped.
Care
reduction
partition
and
by
the
thoroughly
Mention
"
pupils
and
is suggested
of
these
be
terms
omitted,
must
be
borrowing
by
pupils
"
but
taken,
be
as
two
separate
^partitionand
the
also, that
perfectly elear.
all
must
processes
The
on
2, introduction
3f division by
partition requiring
of
those
measurement.
no
borrowing;
requiring borrowing;
be
of
following
:
1, examples
lems.
prob-
measurement
involved
idea
be
must
measurement,
WEIGHTS
As
dividend
rule, both
a
reduced
the
to
would
divisor
and
6
last.
like the
(2)
(1)
3)
be
better
examples
in
denomination
same
155
MEASURES
AND
3
gal."
15
qt
g
gal."
13
8)
i
2
2
qt
3
(3)
2
(4 gal."
qt.)
under
division
under
mensuration.
i.
ii
Linear
V.
of
Explain
What
the
such
5.
this
6.
of
later
Liquid
following:
the
Measure.
and
Liquid
Compare
viii.
Monej
of
Foreign
ix.
Money
of
the
Measures
Dry
States.
fully
should
fourth
arise
will
Dry
Measure.
United
the
3|.
=
division, especially
development
vii.
of
3.
4.
the
Measure.
2.
these
of
report
Measures.
in
V
=
EXERCISES
vi.
Variation
qt.
others
and
Measure.
Dtj
5
qt. h-
under
arise
Weights.
iii. Land
iv.
historical
a
18
=
by measurement,
XIV.
1. Give
qt.)
problems
excellent
Many
1
(l.gal."
-i-
how
each
year?
you
would
child
know
this
use
of
and
work
the
United
States.
Regulations.
in
denominate
should
what
Upon
Laws
x.
Countries.
the
grades.
numbers
emphasis
be
end
the
at
placed during
early years?
Explain,
the
as
What
early
What
as
you
would
to
elementary
an
class,
some
one
unit,
inch.
are
the
main
for
reasons
taking
up
denominate
numbers
f
should
be
the
main
of
object
the
study
of
denominate
the
main
numbers?
7.
What
of
a
8.
What
9.
Give
is the
approximate
city block,
part
the
of
of
the
an
acre
dimensions
length
width
is the
of
the
of
of
a
the
library, of
street?
average
baseball
city
lot?
diamond.
ing,
build-
156
Give
10.
i. A
horse
ii. A
march
iii. A
lake
i. A
ii. A
iii. A
"yards
700
yards
2500
's
race
course
per
per
minute.
miles:
in
of
of
of
30
miles
22
feet
hour
per
second
per
measurements?
indirect
What
13.
Explain
as
14.
Criticize
"12
ft. times
15.
Answer
the
following
would
you
hour,
per
leagues.
5
12.
are
knots
12
furlongs long,
6
with
minute
deep.
speed
speed
a
paces,
fathoms
ship
flight
Compare
400
of
following
the
high,
hands
15
4
PROBLEMS
AND
in feet:
following
the
Give
11.
METHODS
ARITHMETIC
MODERN
to
with
40
; 25
miles
Give
the
class
a
feet
second
per
per
;
with
hour
illustrations.
three
introduction
of
square
measure.
may
fractional
What
16.
How
17.
ft."
by
put
What
"How
pupil:
a
is the
are
here?
trouble
How
of
part
an
is
acre
a
garden
40
feetf
60
by
feetf
30
by
to
"
feetf
sq.
remedied?
it be
16
reduced
feet
square
question
cubic
72
gives
6 ft
acres
many
there
are
in
milef
by ^
^
pasture
a
20
by
rods?
60
the
18.
Explain
19.
Locate, by
of
number
How
of
acres
24.
25.
State
measuring
be
grain
in
eachf
for
a
grade
bushels
is the
cost
if the
gross
three
cup.
led
* '
"
gross,
"
' '
tare
'
and
'
'
net
in
process
finding
the
involving
the
in
terms
Ex.
20
plant
40
class.
what
35
weight
would
be
used
to
oats?
and
cents
weight
problems
the
see
load?
a
and
at
to
problems
three
clover, wheat,
com,
15,
class
least
many
What
Oct.
of
at
suitable
are
23.
in
by the expressions
a
bushels
Suggest
22.
sold
should
How
21.
:
6"19"25.
sec.
there
are
following
the
lines.
range
4"23"46.
sec.
1 of
and
township
Illustrate.
weight?
that
is meant
What
20.
1 of
acres
many
of
use
drawing,
a
1 of N-W
ii. N-E
How
of
means
} of N-W
i. S
and
meaning
per
is 4620
suitable
for
bushel
of
lb. and
the
a
grade
a
load
tare
class
of
2740
ear
eom,
Ib.f
involving
the
'
WEIGHTS
Give
26.
Give
27.
and
How
28.
for
suitable
explanation,
an"
weight
between
suitable
explanation,
an
157
MEASURES
AND
grade
a
for
class,
grade
a
of
class,
cubic
of
measure.
difference
the
capacity.
bushels
many
of
ear
wiU
com
crib
a
12
32
by
feet
10
by
holdf
29.
About
30.
Discuss
31.
State
32.
Explain
33.
State
applied
simple
to
as
and
Distinguish
36.
Suggest
under
168,
suitable
for
of
in
example
some
work.
reduction.
operations
fundamental
four
the
oral
as
numbers.
illustrating
example
grade
35.
class
reduction
holdf
barrel
a
suggested
in
each
does
water
reductions
grade
a
explain
an
a
of
examples
denominate
for
feet
of
order
the
Suggest
suitable
cubic
many
six
to
34.
in
how
would
be
class.
the
between
fully
examples
two
which
operation
each
suitable
classes
two
for
a
of
grade
division.
class
Illustrate.
under
each
case
35.
ex.
BIBLIOGRAPHY
1.
and
Cajori,
F.
A
History
of
Elementary
Mathematics,
pp.
176-79
215-17.
2.
Klapper,
3.
Myers,
Teacher,
Paul.
George
IV:
91.
pp.
W.
135-57.
Modernised
Arithmetic.
Elementary
School
OHAPTEBXV
DECIMAL
170.
History*
Hindu
Decimal
Simon
published
of
set
a
from
10
of
For
divisor.
the
invention
is
It
elapsed
is
idea
of-
before
the
invention,
half
had
receive
the
this
used
result
of
about
square
root
raised
10
the
and
when,
by
of
number
method
this, but
only
mathematicians
face
have
number-system
Not
extracted
the
should
450
by
to
puters
com-
years
ing
annex-
a
power
annexed.
ciphers
the
extracting
The
root
square
earlier.
Decimals
about
of
that
the
the
decimals
of
years
Hindu
the
the
in
they
dividing
and
of
fractions.
Europe
in
zeros
idea
1000
over
idea
figures
Eudolff.
to
staring
was
southern
their
to
simple
used
divided
many
were
fundamental
that
include
to
decimals
Hindus
much
ascribed
before
ciphers
equal
also
incredible
of
this
of
1584
before
years
as
there
as
use
almost
extended
was
number
the
actually
fifty
comma
a
of
in
mathematician,
off with
the
ited
cred-
who
he
About
of
generally
which
in
German
a
the
it is
mathematician,
decimals.
cutting
by
right
the
their
of
authorship
The
tables
Eudolpp,
of
powers
interest
idea
this, Christopf
Flemish
a
extension
an
although
doubt
Stevin,
present
our
in
are
fractions.
to
is
system
to
did
200
years
not
after
recognition
And
been
fractions
number-system
decimal
by
FRACTIONS
stubbornly
the
into
come
with
commensurate
opposing
and
States
has
adoption
the
158
even
their
for
a
of
until
however,
use,
invention
their
United
actual
beautiful
number
the
do
today
metric
of
not
plicity.
simyears
system
160
ARITHMETIC
MODERN
these
'
did
'
If
will
is the
What
These
only,
6
chairs
After
these
as,
February,
how
much
months?"
paid
is
if $8.40
chair,
one
is
followed
for
each
paid
for
4
f"
' '
chairs!
involving
problems
by
chair
cents
of
cost
6 books
15
at
$.15 each?"
at
involving
problems
come
each?
cents
the
of
use
the
as,
zero,
"What
is the
cost
"What
is the
cost
During
problems
require
money
found
very
familiar.
to
point,"
make
be
may
incidental
point
of
would
number
to
namely,
that
the
;
the
for
mentioned
the
new
subject
are
already
it takes
the
into
bring
to
left and
the
ten
to
the
of the
''decimal
name,
this
during
first time
be
will
well
do
The
former.
United
of
going
not
if
introduction.
Formal
two
one
children
the
teacher
the
a
quire
re-
cost
manner
it presents
this, although
between
decimal
right of the
172.
extent,
any
relations
the
latter
with
incidental
that
which
with
Along
to
in
later
handling
Much
this
in
problems
helpful
?'*
that
chairs
6
right
the
those
will
**What
for
arrange
to
or
introducing
Thus,
chair
one
will
left
the
bushel?"
per
teacher
to
$.05 each?"
at
$1.05
at
wise
before
it.
over
wheat
carrying
point
material
theory
of
6 bushels
each?
5 cents
at
that
States
through
pencils
early stage the
paid for
is
6
this
carrying
$2.35
of
of
decimal
the
from
in
$2.15
and
mal
deci-
the
over
simple problems
such
cost, if $2.15
of
carrying
no
January
PROBLEMS
as,
is the
out
in
two
be
may
"What
of
the
cost
be
may
AND
**
require
$1.35
during
earn
"What
*
They
earned
John
he
'
should
examples
point."
METHODS
Introduction.
quite different
in the
Decimals
points of view
writing of fractions
;
2,
as
be introduced
may
:
an
1,
as
a
new
extension
tion
nota-
of
the
DECIMAL
Hindu
number-system.
both
will
select
the
one
The
suited
First
Method.
expressing
Let
etc., for
its
has
order
the
should
the
used
be
not
advantages,
own
that
text
Considered
old
the
are
merely
ideas.
In
teacher
and
may
his
to
confused,
He
way
of
convenient
a
there
$2.45
own
however,
; a
2
are
new
and
fractions
fractions
J, J,
as
which
there
that
way
lars.
^^^^ dol-
denominators
explain
writing such
of writing
way
such
with
now
may
new
notation
new
a
fractions
to
of. 10.
as
first suggest
teacher
reduction
powers
in
to
pupils
fractions, decimals
of
each
of both.
use
173.
As
presented
best
inclination.
by the
be
here
161
FRACTIONS
with
is
are
a
more
point, thus,
a
.75, .4, etc.
Considerable
fractions
common
their
of
these
to
179.
The
Second
174.
but
given,
effective.
in
the
fractional
be.
the start
the
it
The
language
What
the
direct
is, that
left while
to the
right, and
of
the
digits
to
the
same
decimals
direct
process.
decimals
will
be
than
the
one
that
reason
as
more
just
more
pointed
was
it would
to understand
value
in
rate
increase
from
moved
as
when
and
new
wise
other-
moved
right of the decimal
of units, but
as
teaching of this
be made
see
process,
of
must
the
"
to
easier
the
difficult, than
at
general
reducing
more
10-fold
permit
as
in
of
must,
early work
notation,
makes
they decrease
only parts
the
for
and
digits increase
that
Care
fractions
with
decimal
resume,
pupils
ing
chang-
10.
comprehend
to
much
first
number-system
unfortunate
historical
by
introduction
Hindu
more
ing
chang-
really the
side
by
The
pupils
is
to the
represent
the
of
for
side
on
in
that
is
in
fractions
omitted
process,
Method.
extension
difficult
out
of
reverse
carried
be
of
Reduction
be
here
notation
new
powers
fractions, which
should
given
only such
better
would
be
select
to
reductions.
common
an
to
exercised
decimals
under
this
to
denominators
be
course,
should
practice
point
decrease
162
MODERN
when
ARITHMETIC
moved
United
States
for
readily
tV of
that
children
PROBLEMS
Thus
integers do.
as
be
drill.
A
do
is
Confusion
in
should
in
decimal
the
to
the
be
never
read,
but
hundredths,
much
and
ciphers
its
alters
value,
effective
An
both
way
ciphers
annexing
that
must
of
terms
a
10.
avoided
by permitting
numbers
in
only
of
the
hundred
always,
decimal
annexing
multiplying
to
of
two
by the fact
special attention
way
the
sion
confu-
of each
value
that
to
Much
occasioned
are
no
part
clear
minds.
is to show
powers
reading
and
lowed,
is fol-
plan
made
be
by the pupils.
also be
may
and
word
by
is
principle of place
their
receive
equivalent
fraction
common
integral
in
understand
decimal
a
dime
a
Whatever
must
decimals
must
of
is ^
fundamental
fixed
this matter
decimal
five
Methods.
corollary of this, namely,
of clearing up
a
in
of which
each
18.]
clearly recognized
be
of which
decimals
not
this
right of
the
in
pupils will
The
dimes,
4
are
the
firmly
pupils
hence
;
of
work
poor
the
also
extension
tration
satisfactory illus-
only
introduction.
under
Both
on
and
and
place
[See
fractions
to
the
5 cents, each
Remarks
value
of
$.45 there
in
dollar.
a
the
that
method
dollar, and
175.
the
left just
or
affords
money
this
see
a
T^T of
to
AND
tV of 1, .01 is tV of .1, etc.
.1 is
to
right
the
to
METHODS
order
number.,
join the
to
Thus,
254.65
sixty-
fifty-four
hundred
of
use
fifty-four and
and
two
the
and
sixty-fivehundredths.
At
be
time
some
taught
reads
to
in
last
decimals
read
For
them.
the
as
instance, 24.6532
sixty-five,thirty-two.
children
names
not
be
are
of the
This
introduced
busy
learning
various
decimal
;
the
the
idea
places.
be
of
should
scientific
read
man
twenty-
twenty-four,
or
method
during
or
would
two
children
the
business
a
four, point, six, five, three,
should
grades
two
point,
reading, however,
early years
of place
value
while
the
and
the
DECIMAL
Addition
and
of decimals
need
176.
in
which
.16
.43.
+
the
that
with
.5
to
up
such
in
will
examples
result
numbers
to
.5,
+
tenths,
even
in
with
be
These
such
examples
either
to
18.146
14.06
side
X
7,
etc.
proceed
by
easy
etc., which
units
and
arrived
at
in
a
or
the product
cents.
may
decimals,
as, 4
of
that
the
; to
serious
no
as
as
general
with
way,
X
examples
not
such
point.
is carrying
it, as
over
64.305
decimal
.5 X
2,
X
$58.32 X
much
location
from
the
give
and
8, or
drill
if the
the
product
multiplication, these
-05, etc.
will be
the
problems
product
multiplicand
is
152
each
must
tain
con-
multiplication by integers
reversed, and
-3, 8 X
the
multiplier integral,just
$62.47 X
in
7.
X
point is
of the decimal
that
5"
2, .4 X
5, etc., which
fact
or
should
point
as
will
there
there
as
3, .7 X
thousandths
or
be
zeros
the
over
to the
thousandths
of
3, in which
point but
.6 X
tion
multiplica-
begin
to
finally, to the general case, 3.426
; and
After
the process
know
the
addition, just
in which
decimal
some
units
hundredths
hundredths
as
the
stages from
tenths
be
must
of
understanding
An
in
right of the decimal
the
to
by examples
Carrying
give
wise
be
$65.23 X
as
figures
with
3, including
X
point
point
give
of
reverse
It will
be followed
may
will
in
integers.
carrying
no
Practice
decimal
the
as
which
hundredths
decimal
the
this
units,
to
up
pupils that
the
as
Follow
.05, etc.
+
Subtraction
added.
Multiplication.
177.
.08
+
see
9-hundredths.
and
of
etc. ; then
under
comes
the
add
as,
children
point.
which
the
diflBculty if presented
outlined
decimal
first;
used
make
to
tion
subtrac-
Combinations
be
is 5-tenths
.07
as
show
always
be
should
matter
tenths
and
diflSculties.
no
decimals
these
.6
Addition
carrying
simple
sums
the
present
no
a
.3, .9 +
+
Subtraction.
position of the
the
few
.2
+
is
of
sum
a
add
in
It
fixes
This
is
there
163
FRACTIONS
integers be
Inasmuch
same
may
multiplied by
as
whatever
be
children
the
the
changed
order
to
the
164
multiplication
of
position of
decimal
the
the
through
fractions
have
re-reduced
to
establish
178.
may
that
employed
in
in which
there
examples
point, '-^
as,
92.64
division
comes
25, 3.25
-^
the
as
general
most
verified
be
not
with
the
decimal
law
The
the
this
the
to
accomplished
dividend
question
to
follow
A
by
all
thorough
integers by
Because
powers
this
work
in
in
of
is
10
so
the
the
divisor,
decimal
is
decimal
point in the
arrived
be
an
of
of
integer
;
at
this to be
fact,
In
is not
ductivel
in-
all such
divisor
both
10.
of procedure
and
it is
good
a
a
one
decimals.
division
multiplication and
be
final step.
is the
first to reduce
at
of
familiar
result.
may
divisor
should
thoroughly
are
the
simple it
rowing
bor-
obtained
results
placing
in
case
should
this
**
is treated
quotient by
the
power
the
decimal
giving remainders
division
of
cases
drill
the
and
multiplication
this mode
by
gradually
up
division
If
etc.
pupils
the
manner
3, 5.6-^-4, 38.4 -=-12,
-r-
the
necessary
whether
in
4.2
position of
the
the
the
by
will
After
etc.
leads
decimal
in which
others
3,
by another
general
most
a
the
over
h-
which
pupils
point
by requiring
cases
as,
in
integer requiring
an
experienced
decimal
a
governing
for
result
by
Examples
before
of
factors
introduced
5, 36.12
point,"
be
attempted
Division
of
number
two
borrowing
multiplication
division.
placing
-r-
case,
should
after
**
multiplication of
by the
little trouble
point
of
is
product
the
the
common
multiplication; namely,
5, 3.25 -r-25,
-~
reverse
be
no
decimal
a
decimal
the
over
the
of
is
2, 60.75
-r-
the
sought.
Division
to
in
and
product
the law
of
Comparison
DivisioiL
similar
32.5
decimal.
a
plied
multi-
decimals
After
the
inductively
at
tc^ether, that
multiplied
been
of
law
The
of two
each
fractions.
common
PROBLEMS
arrived
be
can
of reducing
to
AND
integers.
by
point
places in the
decimal
to
decimals
process
together
soon
METHODS
ARITHMETIC
MODERN
included
is
on
no
at
this
account
of
time.
to
be
DECIMAL
in
work
urged
arithmetic.
179.
some
would
of decimals.
about
the
this
when
it
in
manner
four
future
as
to
fractions
common
decimal
a
f
then
to
will
called
consideration.
Such
be
to the
be
may
matter
work
no
in the
in the
or
sidered
con-
delay will
a
fraction; while
common
a
exclude
decimals
with
inquiring
decimal, before
a
,
it will
becomes
Pupils
18.
to
division
after
reductions
such
;
denominator
a
until
denominators
whose
establish
to
under
tion
reduc-
for
selected
ciently
suffi-
a
tion
reduc-
easily be expressed
can
a
to be
fractions
will be
and
taken
under
decimals
as
of the
study
in connection
to
act
may
three
with
forms
the
as
a
of
percentage,
193.
a
would
previously
translated
This
such.
Decimals.
becomes
be
and
common
problems
may
as
later
up
between
simple
extended
discussed
Circulating
forms
solved
more
hence
the
fractions
Circulating
fractions
of
common
of
forerunner
relation
the
some
notation
decimal
such
more
fractions
attention
work
to
of
by
of fractions
fractions
180.
the
of 10.
decimal
as
|f
their
operations
decimal
order
solved
of
for
affect the
decimals
powers
In
in
as
with
postponed
that
and
large number
to
is strongly
common
others
fraction, as
a
later
up
any
those
to
14.0000
told
fundamental
of
of
of
be
comes
be
reduction
reduction
in the
no
better
may
for
changed
division
time,
teacher
well
as
10, the reduction
The
the
merely
this
Except
of
power
decimals
in
the
future
all
to
of decimals.
readily be
can
this reason,
drill
Reductions.
which
importance
utmost
For
give much
to
difficult forms
of
it is of
since
neglected,
165
FEACTIONS
The
division
of
part
better
", f ^^^ etc., which
general
be
reduction
of
decimals.
introduced
produce
of
through
simple repetends.
,
Approach
etc., should
In
case
to
be
the
fractions
made
text
producing
long repetends
as
fV, A"
gradually.
does
not
give
the
language
of
circulating
166
MODERN
METHODS
ARITHMETIC
decimals, the teacher
which
reduces
would
better
do
is
.214285171428517
to
decimal
decimals
dot
the
Decimals
decimals
depends
to
the
upon
reduction
instance, the
For
to
means
+
Tjit
The
find
Ttv +
the
^*c., for
ttAttt+
a
last
figure of
of
Reduction
the
of
.6
an
infinite
to
culating
cir-
fairly difficult and
is
infinite
an
.21428517,
a
geometric series.
fraction
common
of
sum
of such
sum
of
the
.6,f by .83, etc.
fractions
summation
of
all
written
Fractions.
common
ing
circulat-
pure
which
and
be expressed
Oommon
to
first
the
mixed
a
.,
is commonly
over
thus
f would
repetend.
181.
placed
.
called
the
in
.
-^^
example,
.,
.666
or
former
The
being
},
as
.
it from
distinguish
to
such
figures repeat.
For
so.
.
circulating
PROBLEMS
AND
series
is found
number
of
terms.
b^
to
a
1
in which
a
following
1
In
the
of .83 the
case
+
A
series
The
and
hence
arbitrary
which
may
age
would
law
be
T*ir +
better
iihn +
ratio
between
Thus
omitted
upon
the
equals
etc.
etc., is summed
decimals
be
depending
to
(here ^).
This
at which
given
is the
fraction
common
is added.
^
of the
children
one
^
TU
tAtf +
i^t +
which
to
one
(here A)"
first term
the
and
term
any
is the
r
"
the
children
work
are
as
previous
the
difficult
is too
for
usually considered,
except
for
following
the
above
mentioned
if the
teacher
series
thinks
it
wise.
The
whose
repetend
denominator
is written
contains
as
as
the
many
numerator
9's
as
of
there
a
are
fraction
figares
168
first two
will have
value
in
with
begun
With
a
with
the
the
little
places
discarded
1
increased
of
place
figures
the
416
would
is
start
right which
the
preceding
this
figure
produces
second
into
way
which
until
"When
out.
the
first figure in
was
not
removed
in
happen
the
right gives
4, it will
not
affect the
illustration
number
to
carry,
result.
to
come
without
after
tell the
will
cation
multipli-
partial product
the
in
plier
multi-
multiplicand
of
3 X
the
the second
to
finding
multiplication
a
unit
to
above
;
carry,
since
is it above
Multiplication
figures fall
case,
-02
is
or
to
continued
usable
a
This
.1 X
4, hence
.06
of
or
partial product.
given
nor
partial
soon
prdcess
Unless
partial product.
the
the
the
present
figure
by each
the
total
first
the
using
right that
the
to
the
final
the
will
practice, the
multiplication gives only
the
practice
to
in
no
the
with
farthest
figure above
in
that
in
in
figure
multiplication
found,
case
so
carried
the
is the
result,
never
will
farthest
is the
in
multiplicand
the
part
a
have
written
in
some
Thus
little
A
of
been
off from
preserved.
off.
cut
final
been
be
have
played
not
figure
first
the
the
two
figure would
next
would
cut
in
partial
but
since
down,
practice
to
have
is to
which
affect the
is the
It
which
be
operator
.32
:
left
only
.31416,
Had
above, the
or
that
so
.31.
that
product
5,
follows
as
this
of
giving
preserved.
approximation
of
process
has
been
is
result.
written
are
be
to
are
final
3.1416
.1 X
figures of which
decimal
the
it is
beginning
as
much
are
multiplier.
the
process
as
will
as
first figure to the
down
the
of
natural
as
The
by the
in
left
be
places
tracted
con-
multiplications
such
the
to
will
this
appear.
case
first two
been
All
write
and
will
as
this
be
result.
multiplicand
the
product
to
many
figure
practice
multiplier
In
first
is,therefore,
work
decimal
as
first figure to the right.
multiply
the
final
the
PROBLEMS
AND
The
value.
any
give only
to
as
so
have
METHODS
ARITHMETIC
MODEBN
-02
the
down
go
in
decimal
multiplication
this
places
in
the
DECIMAL
illustration.
of 32.14
product
32.14
32.14
5.26
32.1
5-26
X
X
5
=
X
.2
=
must
sufSce.
decimal
places.
illustration
Another
169
FRACTIONS
to
160.70
two
"
6.43
"
the
first
the
product
Jmq
third
2*03
169.06
32.
X
.06
of
figure
next
from
the
same
in
1.93
="=
the
partial product.
three
given
IQQvn
Find
2
.2 X
and
places
to
it
8
was
would
.14
have
the
as
the
changed
3.
manner
last
the
as
product.
Division
can
be
approximated
even
cutting off figures
multiplication by merely
simply
more
to
the
than
right of
3562)48276(13.554
3562
12656
10686
1970
(2
cut
off from
divisor).
(6
cut
off from
divisor).
1780
190
175
15
(5
cut
off from
divisor).
12
the
The
contained
had
been
carried
figures only and
of
illustration
accompanying
divisor
have
place of annexing
in the
divisor
ciphers.
to
we
is
only three
but
would
two
have
ciphers
the
dividend.
self-explanatory.
digits, the
decimal
had
to
division
If
the
could
places by cutting off
to resort
to the
annexing
1 70
ARITHMETIC
MODERN
METHODS
EXERCISES
XV.
1.
Show
how
2.
Show
the
advantage
Simon
Stevin
by
decimals
PROBLEMS
AND
are
of
in
of
extension
an
decimal
a
which
tem.
number-sys-
similar
system
unit's
the
Hindu
the
holds
figure
that
to
gested
sug-
separate
a
position.
Give
3.
brief
a
r6sum6
of
the
effect
upon
one
figure
to
the
to
.8? to
the
and
development
of
status
present
decimals.
What
4.
decimal
point
of
effect
annexing
Discuss
5.
first
is
fully
through
years,
How
6.
would
Hindu
the
7.
zero
a
the
right?
leftf
the
to
the
moving
What
is the
8.0?
States
introduce
you
two
of
number
a
introduction
informal
United
of
value
the
of
decimals
during
the
addition
to
form
of
money.
decimals
formally
decimals
formally
as
an
number-system?
would
How
introduce
you
a
as
new
fractions?
8.
Compare
9.
What
the
of
methods
fractions
common
i. e., which
repeat;
two
introduction.
reduced
be
can
have
for
their
solve
in
decimal
decimals
to
denominators
which
some
do
of
power
not
10?
Illustrate.
10.
under
Restate
and,
common
fractions:
notation
32, 33, 34, 35,
and
the
problems
below
36.
m
11.
to
would
you
go
about
making
the
idea
of
place
value
clear
class?
a
12.
various
13.
in
How
Discuss
the
of
numbers
to
be
taught
would
Find
teach
you
a
class
the
location
of
decimal
the
the
Subtract
Give
16.
point
Illustrate.
67.8^, 7i, 9.6f, 15|, 7.0^,
of
sum
and
4^
correct
.001.
15.
the
throughout
grades.
How
addition?
14.
reading
by
results
Compare
Austrian
the
correct
finding
the
to
sum
method:
83.432
12^
.001.
19i
31
3.0]^
of
23^
15^
67.081
23.83
with
finding
the
sum
of
15.29
19.13
3.40
to
Outline
17.
in
is
by
How
60
at
cents
If
.25
times.
is the
Outline
21.
in
by
is
division?
this
that
of
power
of
10; by
24.
Of
25.
The
law
is .6 what
have
been
class
30
at
cents
75
culties
diffi-
the
up
led
be
from
that
multiplying
of
is
this
discover
to
decimal
the
of
point
Show
fact?
multiplication by
of
25.56
divisor
any
quotient?
and
quotient
and
or
etc.
divisor
both
10
dividing by
or
J, by .66", by .125, by .061,
the
taking
of
order
process.
method
of
is 414.06.
If
the
divisor
dividend?
is the
11
Criticize
26.
lb.
subtracted
be
may
location
derived
inverse
the
number
sum
15
the
the
governing
a
.33
what
by freight
fact?
350
weighing
box
than
point
this
discover
to
a
number
examples
should
short
a
send
decimal
the
decimals.
of
could
Give
23.
certkin
a
of
means
How
law
led
be
to
cost
of
number?
the
application
the
to
division
the
What
22.
added
be
What
class
a
it
location
the
hundredweight
per
culties
diffi-
the
up
f
hundredweight
20.
will
taking
decimals.
of
should
more
of
order
the
examples
governing
How
much
express
per
in
law
the
multiplication?
19.
of
means
multiplication
the
What
18.
in
by
171
FRACTIONS
DECIMAL
:
.3
.6
X
-
3.jB0
X
"
=
.75.
4
B0
6
=
"
4
27.
If
28.
544.932
29.
6165
30.
Express
in
31.
Explain
how
decimals
32.
of
.75
to
is what
11.25,
decimal
is the
what
number?
part of 612.3?
?
=
decimal
a
the
ratio
would
you
take
of
243
with
up
540.
to
class
a
the
of
reduction
fractions.
common
What
decimals?
.831
X
is
number
a
two
would
When
in
arise
cases
the
take
you
reduction
of
each?
up
fractions
common
Explain
your
to
reasons
fully.
Explain
33.
of
each
these
Change
35.
What
36.
What
the
are
Zt
e"
would
you
forms
two
34.
^f if i) it tf
how
of
6y
decimal
tf
to
decimals
corresponding
%f
tf
part
with
up
fractions
common
following
the
take
Ytst
ro"
of
ton
is 8278
a
to
TSt
class
to
forms
"nrj
the
reduction
of
decimals.
correct
decimal
t" %f
a
.001:
of
the
|, ^,
6^,
following:
iJ, tV" ih'
lb.?
6234
lb. 7
12345
Ib.f
172
MODERN
ARITHMETIC
Change
37.
the
METHODS
following
to
AND
PROBLEMS
fractions:
common
.325, .03125,
8.52,
.124.
What
38.
2.43156f
by
meant
39.
Qiange
40.
Explain
an
example.
by
.6f
decimals
the
in
.3f
Do
same
for
division.
42.
Approximate
the
following
J
iii. 8969
the
ii. 47.5
the
fractions.
common
results
243;
"+-
in
multiplication
vi.
1.28;
X
3.1416
v.
of
circumference
and
area
is
.01:
to
.7638
What
46.78
circle
a
9.43;
X
3.472.
X
radius
whose
is
inches.
8.52
freight
The
44.
for
863.4
ship
45.
To
found
in
is $100
price
the
46.
of
To
how
To
48.
to
how
decimal
a
per
it would
much
be
be
of
area
ton
cost
farm
a
computed
be
to
cents,
X
V2
if
acre?
per
is
diamond
inches
in
places
of
90
in
computed
order
feet.
to
give
f
must
be
ir
given
radius
circle, whose
in
order
to
8
feet,
rectly
cor-
order
to
at
the
is
inches?
decimal
many
should
places
circumference
the
compute
xTS
should
in feet?
circumference
feet?
to
places
$15
baseball
the
the
may
acre?
per
through
many
price
the
$25
diagonal
the
the
compute
acre?
decimal
many
length
47.
per
how
should
places
that
$.0047
averages
miles.
586
decimal
order
diagonal
The
how
the
in
States
events by approximation
to
many
acres
United
the
by freight
tons
how
in
rate
Find
mile.
each
To
iv.
1.32;
-*-
Find
43.
6735
-*-
to
approximate
41.
f
.125f
38
ex.
of
finding
decimals
"recurring"
or
.0617325f
the
i. 637.45
to
"repeating"
are
of
the
correctly
earth
in
given
be
ir
to
inches
equator?
Suggest
49.
six
in
problems
approximations
for
suitable
grade
work.
BIBLIOGRAPHY
1.
Cajori, F.
2.
Stamper,
3.
White,
Court
Open
4.
White,
Science
and
A
Alva
William
History
W.
F.
Publishing
William
F.
Mathematics,
Mathematics,
of Elementary
pp.
A
.
pp.
150-54.
97-101.
of Elementary
Scrap-Booh
Co., Chicago,
The
VIII:
pp.
Decimalization
409.
Mathematics.
59-65.
of Arithmetic,
School
CHAPTER
XVI
SYSTEM
METJIIC
183.
History.
and
weights
towards
f
decimal
and
of
matter
earth's
quadrant
of
unit
units
based
were
this
1799
made
not
Scientific
United
made
metric
in
have
in
to
of
metric
weights
The
system,
international
were
made
1894.
173
but
was
the
and
Since
its
John
by
suggested
have
attempts
adoption.
In
1866,
legalized
electrical
the
leading
system.
first
measures
all
England,
unsuccessful
exclusive
and
ber,
Decem-
France,
and
metric
was
several
its
world
Russia,
the
States
1821,
secure
obligatory.
the
the
adopted
other
1837.
Japan,
United
the
system
made
China,
In
in
the
selected,
units.
law
the
as
the
arbitrarily
a
as
of
All
and
of
Paris
through
reliable
before
throughout
Adams
Quincy
upon
compulsory
men
by
of
the
weights
(10)-''
the
thus
passed
was
of
meter.
one
system
a
report
States,
adoption
this
upon
excepting
nations,
called
Con-
investigate
system
meridian
they
establishing
thereby
been
which
length
makes
Borda,
to
arbitrarily
the
along
Monqe
suitable
selected
They
appointed
and
more
a
which
easy.
1790
Laplace,
selecting
measures.
in
numerals
tem
number-sys-
simplicity,
comparatively
France
Lagrange,
DORCET,
was
of
XVI
it
mere
directed
Hindu
unified
a
its
by
up
the
of
system
investigation
out
of
only
sprung
with
rounds
because
of
understanding
Louis
Taken
it
useful
not
scientific
a
purpose.
^'actions
universally
an
of
result
definite
a
has
the
is
system
which
measures
is the
but
chance,
metric
The
legal
but
units,
electrical
the
not
based
units
174
MODERN
ABITHMETIC
United
The
METHODS
States
weights,
quarter-dollar,
of
the
Of
late
years,
in
urging
active
the
metric
From
on.
goes
in the
the
United
to the
yard
units
metric
upon
in
half-dollar,
the
is defined
in
terms
Philippines the
the
and
metric
the
it
using
statistics
as
and
more
it may
is
systems
a
tem
sys-
universal
as
more
time
as
inferred
be
least,the adoption of
of all other
of
value
and
simple
Some
system.
the
to
been
have
pharmacists
awakened
are
at
exclusion
the
as
above
States
silver
of
adoption
and
based
of
and
men
measures
system
metric
the
obliga^ry.
well, have
and
PROBLEMS
use
are
Rico
Porto
is made
as
of weights
rates
even
medical
manufacturers,
the
in
system
Navy
amount
dime,
and
while
meter,
metric
total
postal
is also the
as
and
Army
exclusively, foreign
AND
that,
metric
the
only
tem
sys-
of
matter
a
time.
Fundamental
184.
metric
the
is
system
teacher
simple
upon
tables
and
only
of
to
their
in
use
be
should, however,
placed
As
system.
of
the
The
arising from
as
i. The
the
metric
The
of
the
pupils
to
see
English
this
phase
and
advantages
in the
change
the
emphasis
the old
over
be
recall
main
the
this
may
to
presenting
in
r"ume
a
able
be
system
teacher
resulting from
Disadvantages.
advanced
and
the
of
metric
system
prepared.
been
185.
to
following
subject the
disadvantages
has
aid
an
will
the
behoove
elements
educating
new
well
pupil who
a
of
adoption
the
minutes.
upon
of this
great advantages
the
future
few
a
with
that
so
it in the
the
of time, it may
pupils
the
measures,
use
Since
matter
a
acquaint
to
system
called
Problem.
generation
present
of these
use
tables.
system, however,
anything
ii. This
but
a
the
serious
change
as
following
adoption
of the
mtist
learn
The
extreme
will be
disadvantages
a
new
metric
set
are
system
of
tcAles
simplicity of
explained later, makes
the
this
matter.
of units
will
produce
confusion.
:
Com-
176
ARITHMETIC
MODERN
the
less
should
forefathers
our
desire
we
impossible
be
weights
For
and
to
measures
231.
It
United
is
States
measures
is to
ratio
of 277.274
from
change
to
the
Britain
in the
English
quart
the
the
to
as
just because
equal
not
are
English speaking peoples.
States
the
to
of
system
common
no
even
it would
sentiment,
troublesome
as
But
carts.
ox
of
is
PROBLEMS
gallon of Great
imperial
quart
bit
the
among
fully
in
there
gallon of the United
standard
Two
this
since
so
AND
rode
who
respect
do
to
instance, the
to
METHODS
liter.
have
they
the
name.
same
of
metric
The
V.
duodecimal
a
apparent
over
but
;
purposely
made
cha/nts
will
be
and
beams
the
converting
the
people
m
be
remembered
permitted
so
nearly
scales.
for two
gaUon
and
be required
would
the
to
show
yard
necessary
and
to
advertisement
expense
would
meter
replace
be
in
mal
duodeci-
a
16^, 5^, 3, etc.,
of the
brads
and
the
stick
as
the
the
in his
stamp
changing
need
two
weights
one
the
In
used
for
be
only
measures
are
manufacture
counter
parts
addition
to
the
re-
could
quart
eliminate
its fractional
with
expense
of
The
themselves
would
dry goods merchant
parts.
this
discontinued.
be
thus
The
real
only
liter and
time
would
the
scales
mer-
answering
In
the
the
upon
of recasting the
one
could
its fractional
yard
was
Hindu
the
from
expense
that
this
process.
change
the
very
system
to
as
general.
years
during
measures
by the wearing-out
great
use
measures
gallon
conform
factors
Spring
three
or
equal, and
and
course,
metric
further
be
small
Interchangeable
be
quart
a
comparatively
graduated.
quart
entail
the
is, of
The
to
superiority
great
?'
mill
must
of
order
such
upon
objection, it
issue.
anything
chcmge
and
in
The
system
Can
English system
The
the
decimal
sfystem than
vi.
decimal
a
is not
that
number-system.
of the
is decimal.
system
so
in
that
place of
it would
manufacture
meter.
manufacturer's
they
The
be
the
real
patterns,
METRIC
177
SYSTEM
.
which
would
about
be
the
as
done
without
of
result
a
invention.
new
plants of weights
both
of
To
systems.
them
if it
murmur
and
the
Again,
measures
change
brought
were
facturing
manu-
many
for
patterns
carry
would
be
would
suffer
saving
actual
an
expense.
vii.
various
The
occasioned
manufacturers
the
by
lay
out-
heavy
a
of patterns, casts, machines,
changing
"
In
etc.
first place, all
the
railroad
gauges,
ardent,
rational
for
clamor
systenoL
A
patterns,
and
time
long
so
on,
in
made
the
metric
of
wholly
the
metric
system
be
using these
it
wholly
manufacture
avoided
for
the
this
become
some
loss
change
to
loss is bound
expended
produce
the
upon
destruction
a
and
be
made
to
only
only, why
may
change
time.
not
?
a
better
it may
not
is true,
It
which
While
have
This
change.
a
few
the
as
much
$12,000,000
Moltke,
dreadnaught
constructive
If
once.
time.
would
though
such
through
occur
would
we
such
very
long
a
view
present
up,
be
there.
the
years,
the
at
even
for
operative
with
motorcycles,
soon,
the
rapid
our
even
1890
sprung
it would
of
manufacturing
system
twenty-five
since
Hence,
however,
come,
upon
have
amount
existing patterns
on
to, and
exclusively
automobiles
completely
will
in
be
to
improvement
in
in
all
that
invalidating
old
adopted
new
plans,
upon
limited
limit
referred
the
not
required.
the
Except
system.
operative
change.
for
most
of the
time
a
without
removed
been
which
of
be
time
fact, the
measurements
industries
Many
could
Even
eliminate
soon
making
on,
seldom
are
would
of
so
same
permitted
metric
the
plants, dimensions
changes
be
In
be
advantages
Had
and
the
at
use,
placed
as
does
system
adoption
be
measurements.
existing machines.
could
could
in
now
abrupt
too
limit
designs might
new
a
the
Moreover,
metric
the
such
standards,
altered.
of
or
designs, patterns,
new
of
hasty
be
to
advocate
too
a
not
are
universal
could
be
was
able
to
dollars
be
pended
ex-
such
losses
are
178
borne
could
distributed
be
it would
of the
said
before, and
for
be
the
the
merely
since
Wise
standpoint,
in
and
necessary
The
time,
the
will
they
administered
laws
which
much
desired
by
from
elbow
it will
cheaper
fewer
have
all true
grown
common-sense
a
is
room
been
has
as
thus
so
adoption
they will after their plants have
than
materially.
of
case
tax
revenue
particular.
accepted
this
in
expense
matter
a
it is
sooner
manufacturers,
to make
changes
in
one
no
is
system
the
slight national
a
felt by
be
metric
by
PROBLEMS
AJTD
individual,
by the
generally
that
METHODS
ABITHMETIC
MODERN
permitted,
of
advocates
the
are
metric
system.
viii. The
in land
change
It is very
of
system
time
the
that
The
begun.
land
the
these
and
difficulties
it would
result
place of the
many
reduction
acres
get
of
equal
acres
of
being
65
the
as
serviceable
in
the
together
only
one
of
our
variation
of
acre
as
are
with
small
measurements,
changed
only
as
land
out
closer
complete
is sold.
a
hektares
metric
land
scale.
there
the
city lots,
this
is not
The
change.
one
than
65
in
we
160
.3 of
over
divisions
as
the
making
.22
as
the
places
While
on
in
carry
noticeable
most
are
this
no
section.
is not
measure
need
thus
little
a
made
originally laid
only slightly by
we
it is still usable.
160
irregular pieces, as
If
lacks, only
of
quarter
a
affected
be
This
however,
six decimal
to
Even
relation
hektares,
surveys
was
to
out
was
metric
importance.
system.
at
with
of the
adoption
inconvenient
English
country
connection
insurmountable,
hektares.
a
the
decimal
a
Territory
in
arise
of any
.404858
to
number
a
by
hektares
to
contents
Its conversion
all
in
of the
southwest
Again,
from
hektares,
to
true
far
that
Northwest
will
venience.
incon-
cause
in this
adopted
ones
are
64.78
Most
cent.
per
to
only
would
regretted
the
which
the
equal
acre
one
of
in the west
vital
are
been
not
diflSculties
be
to
survey
measurements
system
much
had
measures
measurements
now.
east,
will
Records
be
METRIC
Advantapges.
186.
by
of
men
universal
is the
keeping
long
so
i.
best
for
as
it is easy
in
service
simplicity of
The
by comparing
length, with
barley
the
corns
12
m.
3 ft.
51 yd.
320
rd.
any
for
one,
learn
can
use
States
first part
=
1cm
=
1
ft.
10cm
=
1dm
=
1yd.
10dm
=
Im
=
1 rd.
10m
=
IDm
=
1 mi.
lODm
=
IHm
been
the
stubbornly clinging
in
tables
of the
a
metric
which
to
=
the
all
cent
10mm
=
1cm
10
cents
=
1
dime
10cm
=
1dm
10
dimes
=
1
dollar
10dm
=
Im
aid
of
this
unfamiliar
any
The
tables
in about
of them
analogy after
children
tables.
the United
and
the
tables.
1
give
customs
Excellent
familiar,
are
=
to
old
between
mills
able
IKm
minutes.
few
analogy existing
table, with
wholly
table
table.
10
Students
the
10mm
metric
of the
can
in.
the
monetary
tables
1
is not
be made
can
system.
=
who
one
efl"ciency
and
lOHm
Any
not
are
example,
English
corresponding
prehension,
com-
applica^
we
metric
the
of
in
nation,
a
thought
modem
Tables.
be shown
3
As
meaning.
plicity
Sim-
essentials
neglect the adoption of the metric
we
Simple
in
;
of
system.
the
extension, serviceable
to
as
with
pace
metric
simplicity it embodies
its
men
thought.
the
measures
tem
sys-
for
all
are
of
principle of
and
metric
recommendation
modes
simple
of weights
definite
the
scientists
great
fundamental
compact
of
use
sufficient
a
of
and
good system
tion, and
exclusive
Our
its beautiful
Through
a
is
adoption.
simple expression
of
The
science
179
SYSTEM
of
one-tenth
our
with
with
the
little
having
schools
of the
time
metric
or
heard
could
that
no
tables
hesitation
any
one
learn
they
have
of the
these
now
by
new
require
180
MODERN
the
learn
to
ABITHMETIC
and
accurately
more
meaning
than
there
no
is
refuse
order
have
few
a
tables.
the
English
the
English
minutes
in
posterity might
that
them
surely
And
learning
their
of
would
tables, who
this
have
know
also
comprehension
clearer
of
knowing
one,
would
They
a
they have
sacrifice
to
tables.
English
PBOBLEMS
AND
METHODS
in
ones
new
simple, time-saving
device.
ii. Compactness
cubic
while
.those for
These
tables
hektare,
metric
each
For
Kg.
will
and
in
larger
Thus
of
ton
and
such
is used
tables,
a
linear
measure
are
in
jewelry
and
in
kilograms;
in
the
ranging
host
two
alone
and
metric
from
a
"
practical
an
we
have
there
ton.
hair
to
a
It
in parvo,
and
to
our
just
is,indeed,
and
are
one
an
it is this
many
fifteen
for
collection
of
over
built
decimal
constant
admirable
quality
mated.
overestiof
system
only five tables
conforms
terms,
that
upon
number-
together
tion
illustra-
enhances
value.
Definiteness of
and
English
innumerable
^and
which
of its powers.
"
be
hardly
can
the
terms
constants,
only fourteen
of m/ultwm
definite
with
of unrelated
single principle
system,
compactness
comparison
heterogeneous
iii.
drugs,
for weights
this
of
value
In
its
certain
coal.
The
with
like
hay,
be
range
unnecessary.
and
coal
as
The
tables
weights
general
the
will
terms
in all.
ordinary, drugs
; more
wood,
several
amounts,
milligrams
table
one
make
to
as
delicate
amounts,
only fourteen
its
land, and
list of
the
of
two
measuring
measuring
included,
comprise
groceries
grams,-
in
used
length,
duplicates.
10, and
in
used
for
exact
are
and
square
that
constant,
one
meter
are
is such
instance,
weighed
only
hektometer
square
table
capacity
and
stere, cubic
ton, 1000
complete
a
weight
for
tables
slightly from
only
contain
If
powers.
of
differ
measure
The
of Tables,
means
the
the
same
Units.
thing
Four
the
liters
world
is
over.
perfectly
Pour
METBIC
quarts
may
ounce,
and
have
so
simplify
to
As
metric
the
bushel,
be
must
metric
system.
States, by
the
of
terms
should
has
system
articles
bushel, such
foreign trade,
States
United
sold
by
The
great
will
weight,
no
other
metric
system.
of
adoption
the
discrepancies existing
automatically
be
the
between
definitions of bushel
varied
thus
the
to
usually sold by the
are
after
weight
of their
reason
which
order
corresponding
unit
grain,
as
for
if
the
adopt
no
In
meanings.
different
have
increasing
our
the
reason,
also
on,
Ton,
meanings.
different
three
least
at
181
SYSTEM
in
eliminated.
"
Weight,
such
bulk, is the
not
United
the
and
com,
iv.
so
the
reduction
to meters,
Gunter
154
rd.
4
which
been
has
it makes
reason,
**the
it is
the
work
being
by
relations.
tables
also
are
selected
was
depend
upon
various
units
more
and
V.
Metric
in
of
in
many
important.
this
an
one.
computations
System
and
as
of
Only
one
unit
meaning
the
aid in such
and
this
facturing
manu-
time.
units
The
because
For
the
of
inches.
chain,
years
professions
all
mm.
based
arithmetic."
and
inestimable
to
or
measures
many
present
existing between
yet
reduction
measuring
his
for
the
at
of
5
cm.
the
rods,
to
system
a
easy
arbitrarily
is
of
9
with
feet,
surveyors
adopted
very
8 dm.
m.
but
learn
of
by comparing
shown
to kilometers
or
simpleness
to
easy
3
construction
establishments,
The
7 Dm.
value
the
used
clearly
lb. of
each?
The
only
2 ft. 7 in. to
yd.
in
decimals
is
lb. of
in
more
complicate
Why
Tables.
not
5 Hm.
centimeters,
to
recognized
upon
This
of 6 Km.
100
upon
and
lb. of oats, 56
32
in
quantity
more
west.
upon
them
of
measure
practice
of the
makes
apply.
to
easy
of
Application
tables
more
price
instead
on,
Simple
of 3 mi.
into
put
the
quoting
metric
the
being
true
States, especially in the
by
matters
is
as
cases,
only
"
others
thus
various
the
^length
"
made
given
scientific
to
to
the
ments
measure-
specificgravity.
DecimM
Fractions.
The
weights
182
MODERN
and
ARITHMETIC
of
measures
decimals
bring
fractions
of
the
It
here.
over
is
of
will
187.
of
Development
not
out
the
with
another
mean
the
common
advantages
be
into
gone
fact
that
the
of
adoption
saving
great
the
time
of
as
number-system.
our
Work.
the
The
need
point
to
simplification of
great
a
of fractions.
fractions
common
system
as
desired
will, therefore,
will make
and
fractions
common
merely
elimination
metric
form
its number-
of
system
foreground
uncommon
PROBLEMS
part
a
metric
the
the
into
decimal^
well
of
AND
form
country
any
Adoption
system.
METHODS
In
introducing
the
ft
metric
to
system
similarity of the
first day, the
is to
be
reduction
under
For
English
the
second
suggested
should
the
should
be
texts.
Several
and
naturally
the
be
may
5
pleted
com-
examples.
like
problems
rd.
and
the
yd. 2 ft. 7 in.
to
cubic
to
tables
measure
reference
without
capacity
relations
various
the
weight
reference
without
between
text.
liter and
of
and
giving practice in these
problems
the
to
of the
relation
of
class,again
or
clear
matter
500
that
side and
one
interesting
to
note
that
well
as
to
the
reductions
will
tables
weight
with
the
our
or
possess
set
a
the metric
English
5c
on
interesting
as
school
the
gram
stick,preferably
on
34
explained the
measure,
this
100
meter
few
a
suggested
difl"cult
of square
class
the
it is necessary
a
more
table
in
follow.
make
class
a
tables
has
the
some
only,
problems
that
to
The
parts,
simple
linear
the
S. money.
fractional
The
reduce
:
it upon
rods.
given by the
showing
To
by
of U.
similar
comparison,
come
teacher
cubic
to
gram
; to
out
fashion
a
with
day
that
several
to
base
teacher
involving
system.
are
worked
After
in
of
feet
to
applied
purpose
; to
Next
tables
table
and
the
following
be
linear
graduated
the
inches
metric
learned^
the
class, let
a
the
of
liter
a
of
other.
and
ments
measure-
It may
2
the
ure,
meas-
weights,
system
piece is nearly
to
mm.
be
thick,
184
MODERN
also
should
scale
METHODS
ABITHMETIG
called
be
of the
that
over
f ahrenheit
brief
a
Compare
2.
3.
Define
4.
What
States
of
that
that
What
f
system
metric
the
should
How
State
result
of
system
weights
and
yard, quart, pound.
of
metric
the
may
How
8.
in
system
United
the
to
occur
i. 3 mi.
125
Km
5
Explain
metric
the
Hm
7
Dm
f to
kilometers
of
use
in
metric
the
teaching
f
mentioned
and
system,
7 in. to
ft.
2
which
would
additional
any
8
dm
2
m
ones
5
3
cm
to
mm
f to
yards
to
milesf
millimetersf
to
f
United
the
inchesf
States
table
money
price
iii. The
of
of
price
6D1
various
mile,
cents
5
in
teaching
7
cents
cents
per
with
yard
of
that
meter,
per
$7.50
qt.
per
meanings
with
ton
per
7
at
that
kg
2640
of
cents
per
with
quart
that
of
liter.
attached
be
may
fathom,
pace,
finding prices:
ton.
3
gal.
45
at
lb. at
metric
per
1 at
5
52
5460
of
2 ft.
yd.
4
at
$7.90
at
in
following computations
7 dm
m
ii. The
ounce,
yd.
price
5
What
and
reduce
to
you
4
the
i. The
11.
metric
the
numerals
tables.
Compare
10.
aim
disadvantages
it take
rd.
meters
9.
the
of
Hindu
the
you.
does
long
chief
accomplished
be
adoption
with
number-system.
teacher's
discuss'
and
ii. 4
complete
the
this
taken
system,
a
be
can
the
from
which
ton,
metric
system.
system.
status
present
metric
the
the
other
any
fractions, form
6.
the
of
of
liter,gram,
is the
scale.
f
decimal
7.
review
development
meter,
Show
5.
the
with
-measures
historical
centigrade
the
EXERCISES
XVI.
1. Give
PROBLEMS
simplicity of
the
to
AND
bushel
quart,
to
of
following:
the
oats,
gram,
liter,
meter?
12.
532.5
7.3
Find
lb.
the
Find
specific gravity
the
of
specific gravity
Ii
of
ft.
cu.
1000
cc
of
of
cast
cast
iron
weighing
iron
weighing
kg.
13.
copper,
What
is the
cork, and
weight
brass
f
of
a
quart
and
of
a
liter,each,
of
mercury,
METBIC
of
1
yd.;
cu.
What
15.
3
of
6
qt.;
1
is
cu
Find
.92.
of
weight
the
1
ft.
eu.
;
m.
metric
approximate
of
equivalents
lb.
15
gal.
4
;
ton?
discuss
and
of
adoption
ice
of
dm;
cu
the
4^
yd.;
State
16.
1
are
of
gravity
specific
The
14.
185
SYSTEM
the
metric
from
result
additional
Consider
system.
would
which
advantages
the
than
ones
the
those
mentioned.
Change,
17.
"
by
4*"FtoC;
5"
"
What
18.
is
of
use
the
68"
figure,
a
CtoF;
F
12''
to
of
superiorly
P
to
C;
90*
C;
5**
C
to
F.
C
centigrade
the
the
over
C
16*
F;
to
F;
to
fahrenheit
scale?
BIBLIOGRAPHY
1.
Collins,
Beview,
2.
3.
B.
School
P.
Science
International
The
Bureau
of
Metric
in
Beform
the
Staies,
United
tional
Educa-
265.
52:
Williams,
System.
V.
Joseph
Standards,
Universally
Teachers
Why
and
Metric
Dept.
Mathematics,
System
of
of
Commerce
VI:
Weights
and
Favor
the
Metric
351.
and
Measwes.
Labor
of
Bui.
the
U.
S.
[Free.]
4.
A
British
Decimal
Coinage.
Literary
Digest,
Sept.
22,
1917
:
26.
XVn
CHAPTEB
PERCENTAGE
189.
History.
He
language.
increased
of
speak
are
from
which
of
terminology
the
of
simpler
need
to
be
not
time,
him
in
24
as
text
examples,
the
by
which
new
speaking
of
incidental
the
connection
is
easier
or
hundredths,
versal
uni-
has
given
in
place
a
does
teacher
or
of
of
business
the
an
45
ample
them
in
the
the
hence
refers
In
of
such
easily, either
very
pupils
such
etc.
supply
text
let
man
cent,
per
the
little
containing
percentage,
186
Some
problems
cent,
per
of
words
new
is undertaken
that
used
Most
the
of
many
study
formulate
can
numbers
cent
in
formal
contain
not
ones.
per
8
eliminate
introduction.
with
cent,
per
the
changing
entirely
it
been
that
over
percentage
can
lurk
.24, .08, .45, etc.,
as
the
case
teacher
before
week,
mention
these
Whether
thirteen
rightly
has
The
simple
a
a
say
decimals
to
it
and
**pj'*
wrote
improvement
Since
here.
C"
arisen.
whether
is
century-
**per
they
or
it has
that
15th
the
symbols
an
his
hundredths
by
likely
cent
but
year,
of
decimals,
into
hobgoblins
by
of
the
study.
dreadful
subject
is
per
Introduction.
190.
the
of
course
output
last
has
%
and
usage
that
century
percentage
entered
metical
arith-
the
17th
thirteen
say
business
the
the
fractions
common
man's
counting
which
universal
our
business
arithmetics
for
By
used.
**C"
of
of
custom
cento"
**per
say
that
Italian
The
one.
the
not
i of
This
25%.
old
an
does
increased
has
factory
is
Percentage
or
have
these
by
making
heard
terms
men
will
187
PEECENTAGE
be
not
wholly
thus
mentioned
the
will
Interest
use.
these
engaged
in
actual
men
they
in
pupils
days,
be
may
asked
100,
work
In
order
to make
the teacher
must
more
later.
the
the
in
of
and
20
up
the
cent
the
of these
teacher's
and
diflSiculties,
Most
is Latin
teachers
to
is
arisen
facts
mode
suggest
of
thwart
not
be.
may
The
is
as
beginner
attack,
to
ties
difficul-
of
the
before
in
the
forms
It is
the
foretell
only
those
are
truly
of
i, .2
the
pupil taking
a
that
simpler
it much
arise
which
All
difficulties
makes
effective
percentage,
The
to
new
which
the
considered
expression.
His
to
be
wholly
decimals.
language,
evolve
difficulties
have
and
into
to
as
this
almost
of
inator
denom-
all.
at
are
percentage.
as
terms
equivalent expressions.
are
of
to
the
clearly and
The
terminology, which
him
Recognition
the
Object.
fractions
in this^iew
to
business
difficulties
"
common
study
thinking
in
made
tempting
principles involved
per
language,
however
with
a
common
introduction
technical
such
percentage
language,
arithmetical
study
used
of
study
of
on,
Teacher's
The
191.
so
is not
base
informal
in
decimal
simple
some
nature
same
the
nology,
termi-
the
under
be
recognize its purpose
and
'base,rate,
should
of
use
percentage
fractions
the
this
by introducing
same
term
of
the
to reduce
This
terms.
decimals; and,
solving
are
common
time.
this
at
to
formal
the
they
decimals
and
in
re-state
questing
re-
sions
expres-
these
make
to
their
by
any
containing
of
pupils
and
day
to
encouraged
also be well
It may
notation.
heard
the
them
of
part
percentage
day
speaking
to
problems
some
fractions
when
in
from
or
be
business
with
aroused
report
can
cent
per
be
read
they
Furthermore,
few
to
have
may
expression
also
when
form
terms
familiar
becoming
may
the
bewilderment
no
As
interested
be
create
incidentally.
of
language
should
and
new
the
new
of
guage
lan-
subject.
to
ulate
form-
pupil's
remedies.
recognize, in
a
sort
of way,
that
the difficul-
188
ties of percentage
either
are
standpoint,
presents
the
chief
no
language
PROBLEMS
;
of
method
but
from
school
subjects in the whole
misdirection
of
this
Though
so.
centage
per-
principles, it is
arithmetical
the
rule, they
a
as
approach
do
to
attempt
new
being
reason
a
no
taught
poorest
find
to
make
or
of
those
are
unable
AND
METHODS
AEITHMETIO
MODERN
of
one
the
course,
emphasis
the
by
teacher.
192.
the
language, then, is
This
percentage.
of the
to
stages
correct
One
forms
percentage
be first to reduce
reduced
after
which
be
may
such
common
these
taken
of
the
to
discussion
will
be
the
A
number.
supplying
can
hy introducing
having
be
made
numbers
the
their
ment
establish-
good
to decimals
decimals
first should
easy
fractional, decimal,
and
and
to
and
to
the
be such
will
then
to
may
pei*centage,
notation
tions.
frac-
common
missing parts in
as
denominator
readily, an
which
and
way
Decimals
following, will be found
at
duction
intro-
pupils by
expressed in the percentage
fractions
reductions
study of
expressions and
fractions
common
in
used
the
language.
percentage
Daily drill work
the
a
fractions
common
to
as
the
expressing
of
lead
work
between
the
informal
of the
the
phase of this
in turn
numbers
will
which
in
Learning
etymological
an
continuation
above
number
a
reduced
columns
The
in
them
be
be
to
a
connection
of the
next
is not
Percentage.
of
first requirement
understanding
an
use.
re-state
the
used, but
terms
outlined
to
Language
Teachixig the
will
very
can
be
helpful :
reduced
When
100.
advance
result
step
in
is
frac-
PERCENTAGE
tional per
the
at
the
in
the
of these
not
the
all at
come
be
should
of
the
those
as
once
distributed
The
study.
same
numbers
into
entering
relations
of
At
percentage.
is
process
notation
that
the
a
reduction
2. Goods
a
reduction
of
3. Goods
a
of
Even
after
$1200,
.25.
For
at
$1200,
25
of
per
will
the
part
for
explanation,
and
be
to
should
the
always
language
the
centage
per-
in
recognize
to
following,
the
as
statements
of
fire,were
sold
fire, were
sold
fire, were
sold
only different
come
after
and
the
after
damaged
being
For
what
new
all
help the
sold?
damaged
by
the
were
goods
sold?
will be
language
eternal
far
examples
child
necessary
to
from
vigilance
object of problems
facility in its
teaching of percentage, it is
by
goods
only through
The
by
sold?
goods
the
were
the
damaged
being
what
teacher.
of
being
were
cent.
this work,
This
mastered.
what
at
valued
reduction
For
h
valued
after
$1200,
at
of
in
:
valued
1. Goods
but
as
quantity of
pupils
such
problems,
problems
problem
same
the
solved
problems
as
sufficient
A
enable
to
of three
sets
different
not
are
done
be
be
stated
solved
and
of
such.
as
problems
decimals.
and
should
;
re-stated
are
fractions
this work
reversed
under
language
not
stated
and
lessons
the
need
be
duced,
intro-
be
may
previously
into
examples
merely
but
now
solved
second
application
informal
in the
translation
first these
problems,
percentage
off-hand
for
This
problems.
to
A
is the
percentage
problems
decimals
and
common
in
Language.
the
previously mentioned,
fractions
The
of
work
assignment
the
by
new
be
Phase
language
number
as
at
must
and
early periods
Application
of the
phase
at
however,
drills must
these
work
are
problems.
193.
at
This
etc.
beginning,
throughout
used
1|,
as,
which
finallynumbers
cents, as, i, i, etc., and
100%,
over
189
"
on
selected
for
drill at
the
acquisition of this
To
use.
that
this
succeed
the
teacher
time,
in
the
never
190
MODERN
lose
of
ARITHMETIC
sight of the
fact
He
must
language.
of the
"!learness
that
be
is not
work
PBOBLEMS
difficulties
the
ever
AND
alert
the
on
to
because
clogged
those
mostly
are
that
see
the
the
is
language
understood.
not
194.
Fundamental
The
in
Attention.
into
the
of
problems
only that
not
Their
the
language.
fall
may
just because
the
it necessary
makes
also
solved, but
be
careful
teacher
attention
of
material
arithmetical
that
lems
probcent
per
excellent
importance
number
great
a
the
receive
them
new
relaxing the
simple.
are
given
a
should
makes
however,
danger,
error
finding
this
learn
to
of
Most
point
simplicity
which
is grave
There
with
this
hence,
Their
through
Problem.
deal
percentage
number;
4Some
be
METHODS
that
they
thoroughly understood.
Variety
necessary,
in
the
so
that
below, and
here
are
the
special
but
What
a
connected
with
most
important
nevertheless
is
of
20%
a
the
of
exclusion
of
salary
a
given
(7)
(4) which
all others.
of
is
We
language
subject of
the
who
man
absolutely
or
form
as
general form
more
one
(6),
(5),
freely
as
is
problem
as
to the
encountering
than
4.
used
standard
the
the
forms
such
others, be
made
often
of
statement
son
lescentage,
per-
one.
receives
$1500
per
jeart
5.
How
much
it is $1500
Two
6.
per
7.
of
a
of
How
many
total
assignment
$1500
195.
has
one
they
be
15
of
man's
Variety
of
for the
a
same
miles
apart
does
his
place
and
go
hour
and
the
salary,
when
at
per
end
the
pupil
solve
of
who
in
the
other
same
20%
6 hours?
fails
to
solve
20%
problems?
yearly
increased
been
above
of
spends 80%
the
25
of
rate
problems
natural
seems
last
will
a
from
start
the
far
will
who
save
year!
at
one
How
8. What
man
a
automobiles
"direction,
faster.
does
20%
Form
pupils
by finding: the
income
be
after
his
present
income
f
of
Statement
to solve
amount
Important.
problems
similar
of increase
or
It
to
the
decrease
192
MODERN
The
solution
present
the
ARITHMETIC
cient
of the
of
inverse
similar
given
.15.
for
instrument
percentage,
in
the
of
terms
with
will
found
be
For
first.
of
cases
A
had
unless
numbers,
useful
with
relation
isolated
this
to
to
make
of
the
be
and
numbers
the
old
has
for
texts
can
of the
understanding
is
commend
understood
also be retained
The
use
be
found
the
We
have
here
all of
place
subject
under
of the
to
the
dots
of
much
in the
cases
base,
be
not
can
be
tinued.
discon-
by
expressed
process
it
the
three
finding percentage,
listing of problems
it
in
involved,
takes
number
proper
will
and
the
connecting
relation
gested,
sug-
pupils,
of solution.
method
everything
ing
for show-
multiplication.
idea
any
pupils
question this
by the
reciprocal
simple
the
form, in which
correct
a
find
previously
as
in
supplied
simple unification
cases;
quickly,
the
the
to
certain
problem
the
are
clear
The
one
by
excellent
For
ones.
the
numbers
rate.
between
other
an
equation the scheme
accompanying
one
and
the
and
is
suffi-
is the
$325
is necessary
the
natural
isolated
is .15
taught
given
the
them
equation
given
two
the literal
between
at
literal
operation
what
relation
especially
been
will
as
unacquainted
the
been
dividing the product
by
.The
To
expressing the relation
showing
number
have
have
of which
one
they find
number,
for
PROBLEMS
problems, however,
and
processes
numbers,
two
which
unknown
and
and
practice in their application.
product
in
this
of
AND
special diflScultyto pupils who
no
idea
METHODS
over
the
several
clearly
more
much
memory
and
more
easily.
199.
and
for
Problems
football
were
Socializing
activities
Percentage.
mentioned
as
useful
Baseball
aids
in
193
PERCENTAGE
socializing arithmetic.
batting
averages
are
but
always
referred
and
decimals
chances
of
is spoken
in
are
fielder
baseball
a
possible
erroneously
of
spoken
as
computations
involve
of
older
somewhat
both
batting
national
to
read
computation
or
a
than
so-called
the
itself
**box
and
that
of
versally
uni-
are
sports
such
interest
for
the
decimals.
The
excellent
most
boy will be interested
as
well
A
successful
be
teacher
a
and
use
of
use
this
regarding
arid
knowledge
this purpose
the
in
individual
informed
for
to
his
as
general
is necessary
same
but
in
the
averages.
Unitary
hundred
course,
because
furnish
Every
score"
various
as
for
above
only
averages
26
This
.923.
is suggested
work
average.
Not
of the
Percentage
the
that
game
reasoning
stance,
in-
of
out
or
further
and
ranking,
game.
in the
ability
200.
team's
requires
interest
rates.
fielding
and
material
our
finding
his
computing
rather
24
of ""
these
this
mentioned
problems
for
the
For
cent,''but is,of
per
cents
figured
cents.
per
average
*923
children
topic of percentage
material
per
as
ing
field-
three
as
accepted
Because
cent.
per
to
has
an
*
as
individual
expressed
who
have
would
reality 92.3
three
of teams,
Ranking
per
ibialysis. Unitary
cent
method,
is often
sis,
analyused
194
MODERN
in the
the
ARITHMETIC
of the
solution
This
15%
of
a
man
1%
of
a
of
a
100%
in
introducing
salary
=
$325,
man's
salary
=
3^
man's
salary
=
$21"
when
explanations,
the
is
process
to
and
multiplication,
the
applying
of
a
Common
Avoid
201.
Care
to
and
the
.'. 15%
=
$325,
.-.
=
$21",
=
$2166$.
1%
.M00%
meaning
or
its
it
202.
direct
as
was
is
or
must
ideas
be
different
many
language
study.
at
the
Moreover
never
us
notation,
mere
the
to
of
the
in percentage
of
of
case
varieties
back
manner
be used
thinking.
in
or
cent,
separate,
a
cannot
of
one
a
fractional
a
as
'%
As
salary,
of
sense
of true
forms
this
first statement
give it.* Per
to
is not
work
to
unitary
by
man's
the
Hence
brings
This
used.
pupils
salary,
and
; one
the
pupils.
the
to
Analyses.
Excessive
proposition of looking
a
quantity,
question involved
problem,
in
Analysis.
Unitary
superfluous
in the
used
always
of fundamental
Avoid
sion
divi-
done
was
solution
intended
never
number
The
After
separate
a
permit
for
symbol
a
thing existing by itself.
above.
one
%,
some
intrinsic
but
which
symbol
of
part
as
is used
of
a
total
=
%
plete
com-
abbreviated
be
by .15, as
the
of
misleading
100%
Let
sign
making
necessary.
or
containing
to
it is incorrect
The
use,
corruption
addition
In
$2166".
in
on
place of
Corruption
never
following
analysis.
or
able
quite service-
it should
division
to
or
$21",
reciprocity.
taken
be
must
the
use,
idea
later
^^ the
or
made
be
however,
multiplication by W^"
one
100,
X
become
these
understood,
$325,
of
may
subject and
th6
this treatment,
By
become,
unitary analysis
of
form
would
's
PROBLEMS
AND
problems.
inverse
problem
preceding
Hence
METHODS
the
ment
state-
original
largely
as
treating the
195
PERCfENTAGE
subject
all later
lessons.
careful
not
On
of
operations
are
work.
The
should
without
soon
as
literal
just
as
soon
and
Explain
fractional
of
in
What
5.
order
that
Change
each
of
devices,
their
do
work
each
percentage
are
fractions.
What
incidental
introduction.
placed
be
a
class
introduction
What
tioned
men-
its strong
are
in
to
other
in
as
with
least
at
the
table
a
others
to
Suggest
following
the
denominators
upon
reduced
be
may
results
list
a
table
a
of
tions
frac-
denominators
fractions.
such
ten
of
forms
two
tions,
frac-
i, .145, 12^%,
192:
.56}, 31, .126, 245%,
these
What
solved
of
similar
Also
driUs.
further
taons
the
of
must
i, 33i%,
percentage.
8.
similar
each?
numerators?
the
Suggest
of
be
of number
form
can
of
forms
16.5%,
2.45, 15|%,
5",
.06i.
24.8%,
7.
and
common
are
use
form
they
integral
.1345, 145%,
should
as
criticize
restrictions
placing
6.
and
made
points f'
and
100
decimals
truly
States
other
some
its weak
what
in
of
be
190.
Suggest
4.
pupils
understand
EXERCISES
that
as
and
fully
under
and
just
United
the
Explain
3.
illustrate
notation
people
2.
other
other
the
explain
should
language
be
them.
XVII.
1.
and
they
problems
and
as
learn
to
in any
as
should
teacher
whicb
the
as
equation,
discarded
the
pupils
of inverse
understood,
be
hand,
language
Solutions
thoroughly.
solution
the
require
to
applied throughout
be
must
other
the
questions
minutely
mental
here, but
end
not
must
be
the
used
in
as
and
percentage.
solve, using
an
decimals
Make
the
the
list
for
the
of
difficult
more
for
the
study
forms
When
each.
tion
forma-
and
of
for
how
classroom?
in
the
percentage
fractions?
common
list
a
for
of
notation.
for
should
notation
always
Illustrate.
problems
translation
translation
percentage
kind
each
forms
two
arithmetic
suitable
of
by pupils beginning
192
similar
a
the
in
problems
from
in
other
stated
problems
Select
that
suggest
Give
fractions
three
to
at
into
least
in
the
six
common
language
problems
fraeof
and
196
MODERN
ABITHMETIC
Select
9.
them
from
Show
10.
fraction
the
11.
found
and
7
that
in
all
is always
What
six
text
some
5
forms
to
METHODS
PROBLEMS
the
of
type
4
and
change
194.
problems
recognition
of
problems
in
used.
AND
decimal
the
percentage
on
form
of
teaching
the
Illustrate.
take
teachers
must
this
of
in
subject?
12.
Solve
13.
A
that
the
bought
man
What
price.
used
being
If
14.
of
the
average
five
years?
business
the
successful
are
many
years?
in
for
men
of
town
a
198.
194, 196,
automobile
an
was
six
95%
under
problems
and
$750
it for
sold
of
40%
yearly depreciation
if
of
failures,
this
country
men?
business
80
are
of
sold
56?
after
of
how
124?
364?
of
15.
The
the
second,
the
output
the
decrease
16.
for
A
of
and
had
years.
600?
is what
What
is meant
19.
What
is
25%
his
The
1910
of
25.
as
32%.
The
of
Solve
g
Solve
means
idea
of
a
a
per
Suggest
and
360?
more
was
third
the
if
made
year?
each
20%
year
the
end
of
that
time?
of
36?
of
180?
of
explain
cent
a
explain
of
as
for
a
per
list of
each.
the
another
1800?
of
the
by
not
the
in
1900
to
was
a
a
1910
purpose
of
per
problems
of
the
the
two
of
cent
what
9
to
the
of
in
that
class, problem
conveying
1900.
cent
of
Henry
assignment
from
per
cent
and
total
town
per
in
years?
of
was
in
year
what
owes
certain
what
grade
of
it.
three
cent
in
first
using
assignment
per
15%
mentioned
factory
the
16
of
24%?
of
trate.
Illus-
65%?
of
end
cent?
per
16S%
problem
population
in
of
that
population
population
diagram
of
What
in
The
and
the
an
many.
increase
24.
of
year
$30,000;
in
90%
The
1900?
the
solves
solve?
was
cent
per
at
Henry
23.
owe
at
.2 of
third
the
indebtedness
only
does
it have
increase
15%?
of
75%?
mentioned
man
John
solves
for
by using
original
22.
What
indebtedness
cent
per
6%
" of
what
was
Solve
21.
he
3600?
some
45%?
60%?
output
15
cases?
by
of
^
of
The
problem
of
cent
per
18.
20.
does
much
second.
would
the
his
decreases
15%
60?
of
.81%?
in
How
and
first year,
the
difference
second
$1400
the
than
year
What
the
come
third
the
year?
owing
man
540
less
third
is $30,000
factory
a
15%
the
three
17.
from
output
in
to
that
1910?
196
pupils
by
the
cent.
problems
similar
to
Ex.
22
for
grade
work.
197
PERCENTAGE
26.
A
butter
fat
butter
will
will
gives
milk
produces
li
cow
100
butter?
A
factory
27.
efficiency 15%
bill 6%.
Find
28.
12,
won
29.
Gompute
in
appear
the
simply
31.
"Of
terms
them
32.
Suggest
for
33.
Explain
what
ball
its
fuel
f
A,
teams:
73.
the
players
be
solved
of
percentage
who
can
more
Illustrate.
13
problems
17
through
eo
to
as
fractions.
common
three
is
lost
milk
the
the
saved
was
averages
fractions?
least
at
in
in
of
199.
problems
numbers
problems
fielding
pounds
decreased
bill
96,
of
increased
engines
following
won
pound
of
cent
per
its
fuel
the
G,
14;
under
common
the
Ghange
make
of
What
many
consumers,
original
and
How
on
smoke
of
8, lost
score"
class
in
the
batting
"box
What
30.
of
won
the
f
produce
"percentages"
8; B,
butter.
One
fat.
butter
improvements
cent
per
3.8%
of
by installing
the
lost
milk
through
and
What
is
pounds
of
pounds
make
which
such
other
direct
by
meant
problems
for
and
classes.
grade
inverse
problems.
Illustrate.
it
what
weeks
in
half
is
In
and
home,
should
be
A
38.
the
can
A
received
is
in
of
What
be
of
four
had
which
has
465
votes
and
changed
of
for
the
his
farm
cent
per
How
at
in
a
test
of
cases
on
scale
a
if
percentage
in
much
water
40%?
the
cultivation
pasture?
The
The
cultivated
land
of
How
10.
number
of
15
questions.
8, 12, and
is in
land?
A
remained
have
medicine"?
80%?
to
for
voting
those
his
elected?
been
question
each
75%
cultivated
the
and
the
pasture
tions
ques-
remainder
is what
per
is what
per
cent
60%;
by
80%.
of
pasture?
problem
40.
Solve
41.
Suggest
previously
to
over
and
could
cent
per
it 45%?
make
Illustrate
has
man
pasture.
cent
the
10?
what
of
cent
per
"90%
by
marks
total
and
B,
still have
to
teacher
sum
A
month
a
received
A
What
390.
meant
added
is not
39.
for
would
What
37.
20%
of
year?
for
pupils
36
of
election, candidate
voted
have
last
attendance
enrollment
an
increase
an
receive
he
of
percentage
with
town
a
competitor, B,
could
did
$1800,
absentees?
day
36.
the
is
year
What
year.
school
a
this
salary
last
was
What
35.
42
A's
Mr.
34.
find
class.
found
difficult.
a
list
39
of
after
six
difficult
Let
these
replacing
problems
in
75%
by
percentage
or
which
be
tabula4;ed,solved,
you
expect
your
and
which
you
future
discussed
have
pupils
by the
198
MODERN
ARITHMETIC
METHODS
AND
PROBLEMS
"
the
is
Why
42.
three
it
of
cases
good
a
plan
finding
the
Explain
43.
not
percentage,
the
of
use
divide
to
and
base,
literal
into
problems
percentage
rate?
equation
in
solving
the
idea
in
problems
percentage.
Show
44.
how
45.
Explain
the
with
Solve
46.
literal
the
problem
47.
Discuss
48.
Illustrate
by
Select
49.
for
is
in
data
problems
to
be
used.
this
inverse
the
of
of
for
classes
quainted
unac-
Illustrate.
Show
analysis.
the
of
percentage
for
suggested
the
advantages
method.
means
from
forms
unitary
by
criticism
the
the
equation.
34
of
pupil
of
use
disadvantages
average
each
embodies
equation
Illustrate.
process.
and
literal
the
problems
are
the
grade
under
given
form
that
difficulties
World's
classes.
of
Almanac
Tell
the
difficulties
of
the
language.
or
in
201.
what
other
part
able
suit-
sources,
of
the
work
200
MODERN
AEITHMETIC
to
the
at
this time.
the
pupils
and
It is not,
under
stock
asked
to
get
a
business
city. The
the
are
1.
Buying
or
2.
Buying
or
3.
Selling
4.
Selling hogs
5.
Selling hay
6.
Selling
fruit
7.
Selling
real
3%
and
cheaper
selling grain by
cattle
owner
agrees
purchaser
60c
:
and
to
rarely
8.
Securing
9.
Selling
at
take
a
commission
a
certain
The
against.
:
value.
par
bushel.
^c per
$1
to
of
$15
carload.
per
of
in
states
cities
the
for
for
selling
property.
Sometimes
acre.
per
sum
charges
2%
about
at
expensive
more
$1
"net"
rate.
many
some
of
ton.
per
uniform
no
In
that
above
are
in
carload.
law
by
of
be
from
found
not
charges
maximum
a
produce:
by agreement.
lots
sold
50c
Fixed
estate:
to
per
carload:
other
carload:
up
$10
sheep:
the
by
head
per
and
the
also
places, if they
guard
^7o
of
markets
may
procedures
stocks:
and
An
subjects both
these
to
for
workings
commission
commission
common
bonds
selling
vacant
is often
thing
most
increased
Land
chief
the
of the
visits to
and
arising
plentiful.
are
these
of data
course.
illustrations
children
The
on
from
use
practice is
following
to
and
in
this, and
grade
a
Apt
trade, and
information
encyclopedias
live in
of
in
drills
Problems
teacher
interesting.
more
in
use
subject
of the
number
strictly for convenience.
for
the
board
of
topics found
Brokerage.
part
and
PEOBLEMS
basis
quite simple.
are
exchange,
useful
be
and
the
on
the
suggestion
introducing
explanation
to
of
a
topic
successfully
the
division
AND
lessons, is made
two
however,
this
will
made
Commission
204.
the
be
may
The
following
METHODS
his
the
The
property.
commission.
pays
teacher's
5%
positions:
in
merchandise
of
first
always
stores:
year's salary.
low
where
given,
1%
3%.
Selling merchandise
10.
ing 33^%
205.
to
Ooods
may
agent: generally high,
sometimes
reach*
40%.
Goods
Discount.
market
as
price
be
out
;
there
of
season
often
are
are
a
or
exceptionally large, payment
sold
number
damaged,
may
below
of
reasons
the
be
their
for
purchase
made
inal
origthis.
be
may-
within
a
APPLICATION
certain
wish
time,
keep
prices
to
short
able
be
of
classes
Selling price
second
special interest
206.
Profit
losses in per
otherwise
a
Gain
price X
marking
and
Loss.
stated.
The
loss
are
loss
the
always upon
fundamental
buying
price +
gain,
Selling price
=
buying
price
X
(100%
=
buying
price
X
(100%
Insurance.
of
banded
and
arise
together
will
be
mutual
which
may
each
In cities and
special fire
several
be
live
rate
to
for
and
sale
a
with
at
contained
in
the
and
towns
can
usually less than
as,
each
be
three
(1)
(2)
or
(3)
taken
advantage
it should
though
Years
ago,
the
this
insured
years.
principle of
a
following forms
fire,marine,
building
for only
The
a
has
short
to
nado,
tor-
accident.
its
own
term
depends
rate
construction, proximity
spirit
It
plate glass, and
business
by
present.
at
:
men
insurance
mutual
of the
text
the
losses caused
From
on.
included
be
even
other's
so
any
and
lo88%).
"
explain the
stock, burglary,
most
things
of
are
gain%)
communities
rural
together
not
loss,
"
+
sprung
teacher
the
company
crop,
years,
for
has
in
or
simple.
very
help share
to
prevalent
well
use
more
buying price, unless
side of arithmetic
are
kindness
neighborly
companies
or
profits
(or los8)%,
Hence,
sickness, fire,loss of live stock, and
of
their
is, today,
poor.
business
the
which
problems
gain
X
Insurance
rich
men,
part
(1)
:
=
a
two
equations
Selling price
as
of
calculate
price
by all
those
are
disc.)
"
and
Merchants
buying
of
Two
:
systems
=
207.
also
and
in this connection.
based
or
first
series
a
may
cheaply.
The
(1.00
arising from
cent
profit or
private
discount.
following relation
Private
discounts.
in
arise
those
are
issuing catalogs
prices easily and
marked
=
firm
retail merchant
the
these
the
upon
201
PERCENTAGE
wholesale
a
to
.problems
depending
The
or
change
to
OF
of
upon
inflammable
202
MODERN
ARITHMETIC
buildings,
so
PROBLEMS
on.
Farm
anii five
Stocks
one
dwelling
and
three
because
the
as
fact
It may
be
well
reinsure
such
lose
opportunity
the
in
1915
the
fire
$172,033,200,
cents
should
As
the
which
be
1.
**
taken
value.
do
not
our
of fires.
States
to
against about
as
In
amounted
33
fire protection
of
necessity
insurance
and
also
out
nearly
not
are
conditions
the
as
greatly,
vary
Life
and
most
under
these
can
Sick
benefits
not
so
end
keeps
cancel
his
on.
him
thinks
the
best
insured
:
makes
regular,
life.
Life,*' in which
Payment
payments
of that
of life insurance
forms
common
throughout
regular
the
Insurance.
if the teacher
fifteen,and
policy** which
face
The
must
connection
matter
capita
per
teacher
this
in the
United
the
insurance
fire
explained
makes
may
in
of property
Benefits
grpecified.At
he
nation
$1.71
forms
payments
2. **Ten,
The
out
Straight Life,'* in which
definite
insured
a
in
following three
also be
may
solution
the
here.
Sick
the
in
and
carry
carefully.
noted
are
into
208.
and
as
they
gone
as
about
other
common
so
point
to
lasses
or
be
to
mended
recom-
building alone, but
one
any
on
capita for Europe.
per
two-thirds
rule, companies
a
as
companies.
other
carelessness
extreme
that,
note
to
large fire risks
carry
than
more
insurance.
on
also
year.
a
is generally
consideration
into
taken
be
for
insurance
of
amount
for
rate
About
valuation
actual
proper
should
of problems
not
of the
the
values.
varying
against fire for
insured
rarely
are
of their
insured
are
.4 times
and
2^
at
years
three-fourths
that
properties
merchandise
of
year
or
or
AND
building, its private fire protection, and
of
use
METHODS
time
insured
policy and
for
he
the
number
will
have
without
receive
of
a
further
about
the
years
**paid
up
payment
50%
of
its
APPLICATION
**
3.
in
Endowment,
(2), except
is canceled
time
which
209.
face
insured
of
subject
funds
for
is also
well
carrying
tax, inheritance
personal
better
would
personal
As
to
of
second, that
certain
finding
'*
the
2. Give
the
list
a
this
making
3. Give
conditions
business
(sometimes
4. Find
of
of
all
to
and
found
following
the
service.
It
the
by
real
income
of
problem
the
of tax
to
and
is to
find
the
tax,"
and
the
be
paid
a
upon
in
of
applications
percentage.
think
can
you
of
for
pupils.
your
(b)
the
each
of
those
not
the
problems
(a)
conforming
to
conforming
business
to
tions,
condi-
texts).
products:
98.
X
233*.
b.
618
X
516J.
c.
562
d. 916
X
9i.
e.
642
X
lOi.
f.
624
X
19*.
g.
160
X
22*.
h.
160
X
H*.
i.
612
X
16|.
j.
125
X
816.
k.
632
X
16*.
1.
412
X
47*.
5.
Suggest
grade
work.
6. Outline
state,
estate
412
a.
and
problems
real
first
national
estate
revenue
and
possible agencies
the
illustrations
two
of
EXERCISES
importance
real
work
need
authorities
spreading
this
valuation.
assessed
Explain
at
age
revenues,
on
rule,
a
amount
XVIII.
1.
and
customs,
The
rate, technically called
tax
local
more
taxes.
property
levied
tax
the
to
collected
confined
be
The
inti*oduce
well
may
customs
municipality.
and
county,
which
at
the
upon
internal
the
by
policy-
insured.
the
between
direct
the
property
paid
made,
public improvement
on
tax, and
and
been
calling attention
distinguish
to
government,
teacher
by
if the
and
policy.
his
out
The
be
like those
are
policy depends
a
took
revenues
have
will
on
Revenues.
higher
are
payments
value
premiums
the
the"conditions
which
all the
203
PERCENTAGE
premiums
that
full
of
amount
"in
after
the
OF
six
similar
an
introductory
examples
in
lesson
short
on
cuts
for
commission
X
and
brokerage.
204
MODERN
ARITHMETIC
7. Describe
the
would
What
8.
is
price
98f
did
What
9.
commission
estate
What
of
house
a
the
market
receive
seller
the
for
of
cent
per
if
wheat
itf
for
the
What
$3550.
sales
his
was
did
the
owner
net
the
owner
price
'
real
A
For
$4500.
the
dealer
estate
what
of
remits
8%
sell it to
What
will
commission
produce
A
sell
to
agrees
he
must
selling price f
11.
work
the
f
receive
sold
of
did
What
man
man
3%f
at
receive!
on
commission
the
and
trade,
bushels
3000
f
bushel
per
of
PROBLEMS
men.
for
pay
you
AND
board
exchange,
commission
cents
real
A
10.
etock
and
brokers
various
METHODS
himself
give
his
commission
of
5%
bef
after
his
deducting
consignee.
the
to
to
a
commission
merchant
$289.80
house
a
What
sion
commisthe
was
total
sale?
Suggest
12.
local
a
13.
Outline
an
14.
Discuss
the
15.
What
order
must
of
be
dining-room
chairs
bought
dealer
give
a
commercial
to
sell
for
$5
each,
be
marked
of
25%
and
yet
marked
20%
to
in
discounts
given?
the
a
discount.
on
giving
clothes
been
has
having
f
actions.
trans-
$30
when
16S%f
at
discount
in
make
f
of
kind
18.
in
205
19.
20.
problem
Discuss
the
21.
23.
relation
discount
equivalent
following:
the
to
10%.
b.
20%,
10%,
30%.
c.
10%,
20%,
5%.
d.
25%,
20%,
10%.
did
selling
for
below
the
buying
the
is the
them
asking
illustrate
and
at
$60
at
them
if
marked
costing $2000
below
What
get
25%
house
upon
goods
some
he
in
equation
problems.
of
bought
each
in 205.
discount
fimdamental
the
of
found
20%,
Explain
loss
illustration
an
30%,
16"%
22.
of
class
discount
the
upon
use
grade
a
a.
man
A
the
single
the
price f
marked
by
depending
solution
Fin^
A
solution
fully
What
cost.
for
Suggest
17.
and
suit
a
25%
permit
to
20%
and
of
lesson
for
reasons
brokerage,
or
class.
grade
a
commission
on
introductory
must
^ow
16.
was
for
application,
discount
^
problems
two
marked
and
them
sold
he
them
at
100%
above
below
50%
the
price f
sold
was
price.
at
What
is meant
what
advance
an
was
the
of
25%,
asking
by computing
which
price f
the
profit
price.
in
gain
per
cent
10c
per
quartf
buying
peas
at
$1.90
per
bushel
APPLICATION
A
24.
house
in
cent
25.
the
is
of
SolutionB
26.
Two
A
27.
weighs
scale
merchant's
the
on
illegal gain
which
What
loss
or
per
gain
is
and
loss.
the
on
lossf
or
is
the
loss
per
the
fall
What
customer's
the
in
use
gained
was
pound.
the
to
their
profit
total
the
What
in
20%
was
ounces
centf
per
arise
each.
$2400
15
gain
illustrate
and
equations
other.
the
f
$3200
for
only
is
What
problems
sold
were
lost
was
various
the
houses
20%
one,
f for
$2800
205
PERCENTAGE
$3000.
fundamental
the
Explain
for
bought
it for
selling
OF
cent!
28.
A
decides
but
shrinks
His
bought
yearly
Mr.
thefeby
his
of
31.
At
3%
or
$1
acre
be
10%
below
10%
he
for
pay
$13,460.
of
the
original asking
the
the
year
ment?
invest-
the
on
price,
original asking
the
above
did
will
acre
what
and
house
was
real
a
commission
dealer's
estate
equal?
gain
cent
per
bought
are
sold
goods
on
retail
the
by
which
fall
should
the
34.
How
35.
Explain
A
in
a
Explain
38.
The
his
the
man
at
dealer
wholesale
at
at
catalog
discount
a
second
company.
of
a
fire
1%
loss
of
f
20%
the
mentioned
under
introduced
insurance
for
in
$2000
insurance
of
kinds
various
206.
to
class?
a
company.
a
ance
fire insur-
mutual
will
What
$850,000.
be
$1200?
forms
of
per
of
his
for
factory
$15,000
What
should
$1200?
property
dwelling
a
annum
for
of
of
class
be
mutual
a
total
of
grade
insurance
insured
house
a
a
equations
of
of
various
insured
reinsured
company
subject
valuation
actual
for
the
workings
fire loss
premium
A
within
carrying
37.
the
the
has
man
share
suitable
illustrations
problems
39.
case
per
per
company
be
for
for
end
sales?
the
on
the
at
th"
During
1.
goods
sold
and
profit
house
for
price
is the
Give
36.
his
of
not
interest.)
invoice
the
corn
order
35%?
33.
of
cent
What
$600.
what
which
prices
later
it
and
the
in
later
January
$9600
If
cent?
per
What
32.
ef
sold
of
value
a
of
in
corn
spring.
the
months
$4500
at
his
for
in
it 6
account
$2300
per
bought
gaining
gain
his
was
Jones
and
price
the
were
expenses
take
for
invoiced
to
bushel
per
receive
not
stock
goods
What
$4845.
30.
(Do
cents
higher price
a
he
must
dealer's
80
it for
lose?
a
A
he
year
what
13%
29.
hold
to
sustain
to
is offered
farmer
in
3
years?
against
another
each
house
of
insurance
is $1400.
for
fire
company
the
three
5
for
mentioned,
will
What
years?
This
$50,000.
and
$10,000
companies
pay
in
a
in
206
MODERN
ARITHMETIC
40.
Explain
41.
How
42.
Explain
43.
What
the
briefly
should
the
of
subject
is meant
forms
three
various
the
AND
METHODS
mentioned.
tnsurance
introduced!
be
obtaining
of
means
by
life
of
revenue
assessed
PROBLEMS
valuation
revenues.
of
property
f
is it
How
determined?
If
44.
the
actual
the
How
much
the
amount
49.
the
and
$60
is .0341, what
does
the
and
for
who
schools
when
is the
rate
on
six
owning
his
at
problems
if
collected
this
on
$4500
district
taxes
are
valued
in
at
of
is
tax
your
in
the
above
state?
pay
towards
mills?
2}
What
is the
time?
revenue
suitable
for
a
grade
class.
BIBLIOGRAPHY
Paine,
1.
Elementary
2.
N.
's
-'
E.
Cassie
L.
School
Teacher,
Stitt, Edward
A.,
1900:
W.
566.
--
"
I
"
A
1
Strong
XIII:
School
and
Motivation
for
Arithmetic
Work.
379.
Business
f
$75001
property
that
t#
yearly salary $90
living
man
a
property
owns
of
teacher's
the
$220,000.
schoolhouse
the
improve
to
is
district
school
a
will
tax
man
non-payment
Suggest
increase
to
a
in
increased
be
additional
much
How
valuation
tax
pay
How
|)enalty
must
district
maintaining
48.
property
much
mentioned
47.
valuation
60%
a
total
of
How
46.
on
valuation?
The
45.
rate
Arithmetic.
Proceedings
208
ABITHMETIC
MODEBN
for
teacher
the
above
the
with
children
institution
of
class
most
of the
days
month
before
only
paid
common
such
on
interest-paying
due
each
deposit Jan.
withdrawn
i. e.,
for
6 mo.,
$60
are
more
and
mo.
for 4
in
As
of
means
a
banks
At
the
children
at
frugality
and
213.
treatment
Interest.
of the
an
from
the
$50
has
1, then
Mar.
July 1
not
mo.;
in
postal
many
time
a
30-
on
on
on
$40
$10,
$50
these
the
and
formalities
school
schools
savings
throughout
the
instill
into
innovations
impressionable
early and
demand,
on
60-day notice.
or
people with
young
banking,
same
of interest
deposits
out
pay
require
may
being installed
are
country.
of
of
is
mo.
acquainting
technicalities
and
the
they
states
most
terest
in-
specified
the
account
4
no
Interest
deducted
$60 for
is
first few
draw
amount
deposited
on
Ordinarily, savings banks
but
an
in
method
the
"
on
will be computed
interest
6
$20
the
are
Suppose
for
and
deposit
on
deposit.
April 23,
($50-40)
is
all withdrawals
1, that
after
day
time
varies
following
5th
determining
In
date.
on
as
the
The
accounts
made
from
pay
following month.
the
money
half-year
longest
sums
first of
the
branch
obtained
usually
The
usually after
"
a
as
money.
savings
Deposits
:
in it
semi-annually.
on
the
to
business
a
also, been
banks
communities.
and
states
of the
one
interest
the
impressing
as
interest
Savings
talk
a
of
bank
play
well
of
any
as
way
have,
with
bank
well
as
the
their
results
Banks.
computing
different
of
be
absolutely certain.
happy
a
realness
about
is not
interest, compounded
4%
in
used
be
stimulating
school
a
Savings
to
will
Excellent
by operating
S%
the
of
and
arithmetic.
212.
by the
employee
It will
bank
local
bank
a
local
he
bank
PBOBLEMS
Bankmg.
concerning which
matters
class by
the
to consult
visit to the
A
About
Directly
Leaming
211.
AND
METHODS
age
the
habits
thrift.
Interest
is
a
form
subject of interest
of
rent.
should
A
include:
thorough
BANKING
1.
A
2.
The
3.
The
tor
need
The
Securities
7.
General
of
methods
of
to
rate
in
the
of
be
found
1.
An
exact
2.
A
with
years,
3.
Any
4.
Days,
5.
Any
of
notes
real
penalty
the
of
cases
where
dispute
of
during
a
of
months,
they represent;
months,
will be
excellent
a
*c
2
as
months,
of
to
any
years,
use
as
are
est
Inter-
general
a
equations
which
a
cause
may
is that
T
Interest
year.
varying
must
for
rates
:
6, 4, 3, 2, reducing
then,
combination
them
of
to
months
months.
number
Notei'-^AB
as,
years
10, 15, 20, giving simple
as,
grade
parts of
at
vantage
ad-
years.
of
4
6th
equations
principals
with
Percentage.
the
above
the
fractional
of
years,
niunber
the
several
items, which
only thing
the
and
various
year
of
through
or
number
2
notes,
mortgages.
devoted
above
the
Application
number
as
time
as
learning
be
may
an
thing
in
spent
interest
introduced
on
of
such
chattel
and
estate
subject.
in years,
children
name
forms
of the
as
be
certain
fractions
MtX
all
consideration
a
Interest
the
the
in
usually
computing
important
There
and
state
*
time
vital part
well
form
customs.
the
be expressed
may
interest.
paying
notes.
difficultyin handling
any
the
charged
various
the
application of percentage
The
in
permitted
the
or
of
local
Much
can
of
custom
interest.
paying
interest
coupon
6.
214.
of
the
rate.
and
a
of
history
justice
making
demand
the
209
INTEBEST
specified.
The
such
and
legal rate,
is
rate
5.
of
maximum
higher
a
4.
no
knowledge
AND
and
practice
the
student
papers.
7
fractions
months.
of
a
month.
days.
opportunity
cancellation
precaution
and
and
in
this
they
is warned
connection
for
should
en-
be
against signing his
210
ARITHMETIC
MODERN
couraged
or
time
of
finding
at
7%
$245
do
to
be
may
without
once
these
the
may
for
6%
by
realize $210
on,
the
pupils
What
interest
the
7th
is based
the
follows
at
introduce
be
may
the
interest
followed
will
be
taken
finding
$2500
on
immediately
be
to
necessary
From
shortly after
up
the
of
each
of
various
of
forms
abbreviated
some
that
The
interest
is recommended
for
2
ing
Dur-
of
form
puting
com-
following,
which
months
6%
at
is
;
for
6%
at
Interest.
Computing
introduced.
fact
$1250
on
in
problems.
grade,
principal,
Interest
to
principal of $3000.
a
may
Forms
8th
or
upon
plan
found
practice
be
to
on
with
direct
the
interest
y^T of
year
per
Abbreviated
215.
be
good
of interest
rate
familiar
are
in
used
statement:
may
find the
to
this problem
;
reciprocal problems
this
time
required
be
question,
the
T
had'considerable
have
pupils
at
a
by the
$1500
incidentally.
interest, they
1 year
which
difficulty. It is also
problems
When
is indicated
T, from
.07 X
X
reciprocal
the
as
during which
time
the
Thus,
interest
give $245
$1500
successfully
most
interest.
the
will
=
treated
finding the principal, rate,
for
Problems
so.
PROBLEMS
AND
METHODS
2
yr.
3
10
mo.
is
da.
as
:
Interest
2
''
$150.00
yr
2
mo.
''
1
mo.
"
10
12.50
days
(i of
2
mo.)
6.25
of
1
mo.)
2.08
(i
$170.83
At
be
7%
interest
subtracted,
banks
should
Until
States
more
\ would
be added
so
The
and
be
recently,
explained
exact
Government
by
the
on.
banks.
at
In
tables
this
been
it is
making
5%
interest
in
\ would
common
use
by
time.
has
interest
only, but
; at
a
used
by the United
being adopted
90-day
note
more
they
and
count
BANKING
the
number
exact
216.
of days, not
of money
in
thus
in 50
placed
Partial
interest
the
be
218.
to
few
a
United
in
partial payments
Notes.
banks
As
loan
of
most
considerable
considered
written
Absolute
gauging
his bank
to
over
well
as
arithmetic,
exchange
coins
and
a
and
recently.
out
in
which
he
of
the
to himself
borrows
standard
only
a
must
a
upon
his bank.
be
to
the
early
given
was
of
years
to
used
in
in
of uniformity
had
hundred
About
a
this
the
study
of
exchange.
complication arising from
lack
a
made
favor
attention
material, which
fifty
note
which
in
case
computation.
In
much
was
in
is
interest
Exchange.
a
that
as
by himself
accuracy
any
The
attention.
their
should
by discounting notes, this phase of banking
219.
be
should
computations,
problems
it
payments
interest, the
such
exact
short
three
most
tion
atten-
However,
ignored.
at
or
of
form
a
moderate
only
be
computing
is
payments
not
two,
calls for
Only
turns
very
compound
solved.
receive
of
In
Discounting
money
this
amount
$1146.70
to
The
For
r]^
receive
it must
restricted
be
rule, which
followed.
of
will
years,
years.
Partial
should
time,
problem.
one
note
which
same
wisely
P[l +
=
Payments.
problem
; at
need
in 100
$13,149.24
to
item.
interest
in 25
$338.63
amounts
enormous
important
an
pound
com-
equation is
217.
be
three
of
use
the
compound
to
4
States
this
5%
at
years,
and
years,
interest
man
date
same
common
savings banks, and
in 10
$162.89
The
loaned, make
example, $100
in
to the
merely
Interest.
Compound
interest
may
211
INTEBEST
hence.
months
to
AND
the
This
multiplicity
coinage.
Much
of
legitimate place in the arithmetics
years
ago,
all that
has
needs
been
to
be
retained
included
until
for
212
MODERN
ARITHMETIC
present 'needs is
METHODS
comparison
a
AND
cost, safety,and expedience
to
as
PROBLEMS
in
sending money
by registeredletter,postal order,
private check, draft,and telegraph.
SufScient practicein convertingUnited States money
into
in foreign trade may
English, French, and German
money
be given the pupils to afford them a notion of international
exchange and equivalencein values. For example,
money
English money
for one
pound,
is
quoted
as
as,
$4.72.
French
quoted similarly,as
19c
so
as
franc
per
or
dollars
many
for
marks
franc
per
mark.
per
and
Qerman
and
cents
many
24^c
or
francs
many
so
so
cents
is
money
mark,
or
as
It
is, also, quoted
5.21 francs,or 4.15
$1, as
marks.
Collecting Agencies. Drawing upon
a
another citythrough a bank is an important business
220. Banks
in
debtor
as
Mr.
transaction.
Mr. B. for payment
upon
B. resides.
where
Mr.
B. and
the
if
paid
this
use
their home
upon
they
may
be
banks,
of
a
by
an
on
a
book
authorized
may
of the
one
this is sent to
latter bank
him
a
few cents
securing
through
bank
a
demand
in the town
receipt for the
a
of
means
a
a
presents the demand
A. 's bank, less
Checks.
journey
ting
remit-
by drawing
city in which
in the
bank
same
to
for collection.
money
convenient
A
is
of
of checks
person
from
These
varying amounts,
In
at
checks
railroads.
whom
a
they
were
obtaining the
bank
or
a
of carrying
way
by travelers'
companies, and
by the holder of them.
by these checks as,
holder
one
;
his bank
stopping.
express
small
leaves with
leaves
bank
221. Travelers'
money
The
to Mr.
sum
Travelers
also
A.
each
sold
by
consist
signed
bought and
money
sented
repre-
hotel, where
the
stopping,he signs the check in the presence
this signature with the
paying it who compares
be
given above.
BANKING
AND
XIX.
2.
Give
3.
Distinguish
historical
short
a
EXERCISES
function
each
fully
1. Discuss
between
213
INTEBE8T
of
mentioned
banks
of
account
the
210.
under
institution.
banking
and
private, state, national,
federal
reserve
banks.
4.
Make
write
deposit slip;
a
of
one
each
the
of
forms
three
of
checks.
5.
Mr.
made
A
Whose
loss
itf
was
6. Discuss
the
7. Outline
fully
What
are
9. Find
out
8.
nine
savings
of
banking
lost
who
B,
got it cashed
(3)
deposits
school
a
by
it.
A
Mr.
0.
due
of
date
(1)
paying
interest.
July
Mar.
1, $40;
210.
bankf
computing
4%
at
under
bank.
bank:
savings
Jan.
"
mentioned
savings
a
of
mode
interest
the
of
local
1
following
the
on
June
10, $15;
1, $20.
2, $10.
postal
11.
Describe
12.
Explain
13.
Discuss, briefly, the
the
to
as
fetnd school
class
grade
a
savings
of
of
meaning
the
history
bank.
interest;
the
interest.
of
reasonableness
interest.
charging
is meant
What
state?
in your
Mr.
it and
to
of
your
Withdrawals-rMay
14.
of
name
management
functions
from
account:
's
elements
the
the
Compute
10.
B
order
Whyf
(2) rates;
interest;
the
to
Mr.
it,forged
found
tramp
check
a
by legal
is the
What
maximum
and
for
penalty
f
rates
charging
above
What
they
are
maximum
the
rate?
15.
Write
16.
Outline
17.
What
(1)
the
How
Discuss
19.
What
20.
In
21.
Discuss
these
22.
5
be
overcome?
4%;
time
the
the
interest
will
value
$2300
of
amount
giving
What
interest.
the
short
this
at
$2500
to
class
cuts
at
trate.
Illus-
Whyf
Illustrate.
6%?
different
several
are
grade.
6th
Illustrate.
interest?
in
the
timef
equations.
reciprocal problems
the
are
of
during
useful?
methods
When
should
used?
be
Find
the
20
Find
at
value
interest
dif"culty
this
note.
coupon
a
in
work
cause
what
months,
23.
the
to
computing
of
of
object
the
(2)
note;
is liable
can
18.
demand
a
7%;
interest, by
days,
the
at
at
6%;
the
at
interest
of
8%
three
(by
short
7%;
$1450
cut
at
5%.
for
2
abbreviated
of
given,
years,
7
$2450
months,
methods).
for
6
1
year,
days,
at
214
MODERN
ARITHMETIC
Find
24.
interest
the
exact
$1500
on
from
7%
at
Feb.
10, 1916,
July
to
1916.
15,
25.
Find
26.
Make
for
6%
1
27.
what
lot
is
A
Solve
30.
A
4
if
tax
the
investment?
for
$30
3
$2000,
up
Show
the
is
in
years
at
$9000
to
investor
an
$12.50
order
which
for
month,
per
could
What
is
Find
A
1916, $400.
A
the
$2000
of
use
such
per
the
and
taxes
afford
investor
the
that
for
year,
upkeep
if
to
pay
of
interest.
money
How
note
for
Dec.
on
1, 1914
Oct.
note
36.
What
37.
A
1,
June
terms
cash
6
a
when
note
mo.
4
6%.
at
years
plain
Ex-
regarding
interest, made
Oct.
it:
May
Mar.
Mar.
on
15, 1916, had
$200;
Dec.
15,
1917?
15,
and
it:
upon
May
1, 1916,
1, 1914,
indorsed
$600
partial payments.
6%
drawing
$750
1, 1915.
Oct.
on
What
interest,
15, 1914;
is the
value
interest
at
of
1915?
is bank
discount?
dated
note
of
for
$250
on
8%
due
dated
and
the
paying
by
save
state
your
on
was
following payments
he
which
for
6%?
at
rule
made
$2500,
goods
equation.
drawing
much
will
interest
States
payments
What
account
12%?
at
interest
$700
of
in
value
compound
United
worth
bank
the
at
for
for
note
cash.
legal
the
following
35.
for
compound
34.
$1250
the
the
205, taking
28, page
buys
written
State
due
the
on
borrowed
33.
$350
If
$600.
for
interest
5%
or
the
the
mos.
of
year.
What
32.
bad
12
to
up
end
merchant
mo.
31.
the
on
the
problem
money
due,
on
$1000,
6%f
29.
with
so
problem.
above
at
rents
per
is worth
are
and
sold
6%
house
$160
are
the
interest
the
bought
it be
realize
28.
for
table.
must
may
of
2 mos.,
mo.,
A
interest
table
a
interest
an
the
PROBLEMS
AND
METHODS
June
is
1, 1917,
1, 1916,
$1200, drawing
on
discounted
Oct
1, 1916,
Find
8%.
at
6%,
the
proceeds.
38.
A
interest,
39.
in
on
Find
What
order
41.
the
A
that
Find
Jan.
discounts
the
be
the
following
his
Nov.
1, 1916,
1, 1917,
8%
note
Find
7%.
at
$1400,
on
$2500,
on
the
in
due
drawing
5%
proceeds.
3
mos.,
at
the
proceeds.
must
the
dated
note
is discounted
Mr.
bank.
40.
six-month
face
the
proceeds
date
of
dates:
of
be
may
maturity
Jan.
a
of
3
note
mos.
drawing
8%
interest
for
90
days
July
5,
1917.
$750?
loans
12, 1917;
made
Feb.
at
a
bank
23, 1916;
CHAPTER
PARTNERSHIPS
222.
AND
Partnerships.
contained
ago
several
of
be
to
investments
taken
each
from
been
has
Much
of
the
and
in
place
in
withdrawals
the
time-element
are
salary
amount
is withdrawn
the
regular
is
risk
the
by
each
of
that
of
made.
and
the
firm
interest,
by
invested,
estimated
is
the
that
will
of
amount
taken
of
form
of
If
large
a
member
ing
bear-
other
similar
according
vary
capital
productive
of
length
some
some
or
nership
part-
partnerships
firm.
for
Arrangements
capital
partner,
member
one
given
the
by
whole
prominent
a
take
they
partner
by
rate
the
often
so
in
on
such
of
textbooks.
account
no
ferent
dif-
had
the
later
carried
in
rule,
a
of
were
that
deserves
reason,
small
each
often
is
arrangement
to
this
problems
is
a
investment
the
country
As
so
paid
note
a
the
involved;
regular
time,
of
arithmetics.
the
in
made
account
practice
abandoned
topic, for
the
Such
were
all
every
present-day
almost
business
the
business.
actual
time,
to
strict
which
years
there
withdrawals
problems
during
the
time
several
these
time
was
removed
topic
the
of
all
some
which
in
from
whom,
and
In
partner
far
of
published
partnership
on
each
amounts.
CORPORATIONS
Arithmetics
problems
partners,
number
XX
power
invested
of
each
member.
Treatment
223.
Any
questions
problems
assignment.
which
of
on
may
This
the
Interpretation
interpretation
arise
of
should
question
be
of
216
the
settled
Statements.
of
in
statements
at
the
interpretation,
time
the
of
however,
its
PABTNEB8HIPS
be
should
it is upon
One
wMeh
A
of
profits
$2100
for
the
This
division
of the
two
The
the
could
such
interpretation
week,
$1000
of which
B
give
balance,
after
this
his
that
will depend.
respectivelj,
and
the
B
into
enter
How
$3000.
a
the
are
agreement
possibilities
:
are
bearing
$2000
of
rate
current
divided
paid, being
is
the
upon
following
of
note
a
possible since
as
divided?
be
to
far
:
per
invests
year
as
profits will depend
partners,
firm
interest;
of
$18
work
problems
sufSce
B
in
partnership,
of the
must
and
$30
earning
1.
solution
illustration
A
pupil's
ability to make
his
in the
success
of the
part
a
217
CORPORATIONS
AND
in. the
ratio
offj.
A
2.
might
willing
be
The
3.
to
be
divide
224.
that
the
that
It is true
a
should
and
Europe
number
in the
systems
last
in
twelve
and
in
bonds
of the
The
small
shares
many
States
in
and
investors
has
universal
of the
corporations
Several
**baby"
United
of the
of
increased
$100
denominations
of
**baby'' bonds, for
of limited
means.
the
it
the
pletely
com-
have,
in
masses
largest railroad
and
the
during
of
value
existed
$500
it has
in this country.
however,
now,
bonds.
bonds
124%
par
arithmetics
are
for
the
the
topic, as
popular
one
when
cisms
criti-
and
and
at
evenly.
abandon
to
Stocks
more
drew
advocated
stocks
investments
popular
taken
be
Several
have
essential; but
considered.
be
bonds
and
would
he
divided
Bonds.
modification
is
mig^t
be
to
study, of
stockholders
years.
stocks
only.
of
profits
and
daily proving
are
that
salary they
curriculum
radical
long time, been
business
the
at
men
further
arithmetic
not
the
invested
money
Stocks
taught,
been
hitherto
The
the
all
arithmetic
from
elimination,
entering
the
of
Oorporatioxu.
of
a
of
partnership,
the
of
profits evenly.
power
and
rate;
entering
for
the
earning
current
desirous
so
in
shares
the
texts
issuing
stocks
$100, called
accommodation
Some
corporations
218
MODERN
ARITHMETIC
shares
issuing
are
their
secure
growing
of
In
form
of
as
a
topic
in
employees
cooperation.
stocks
this
give
to
PROBLEMS
AND
the
to
hearty
more
us
stock
of
popularity
behooves
METHODS
view
to
of
this
it
investment,
thorough
a
order
common-sense
treatment.
Stock
225.
formed
today
way
the
and
as
take
to
number
partnership
while
the
is liable
generally
Local
enterprises
in
order
to
thing
common
making
or
the
owned
by
226.
for
the
to
under
following
relation
between
detail
**
which
Taking
may
local
a
and
interest.
few
in the
In
country
story of
and
Capital
a
by
only $25
of shares
to
be
the
should
include
an
dividends
the
par
;
;
pany
com-
begin
;
of
lems
prob-
with
market
preferred
and
the
of
the
valine ;
ization
capital-
and
common
for
the
variation
and
any
other
reasons
value
stock
solution
value
of shares, par
;
plan
explanation
value
; par
good
a
supposed
a
account
stock
stock'';
price from
quite
shareholders
It is
they attempt
number
watered
It is
number
Company.
This
sales of shares
;
market
give the
head.
:
companies
low, sometimes
shares
Stock
a
organization
terms
of
owned.
stock
them.
number
limiting the
pupils before
this
for
reason
to
in
is
individual.
one
teacher
the
of the
then
Dramatizing
the
stock;
value
and
any
interest
in
corporation
a
as
tions,
condi-
by the firm,
of the stock
conducted
often
are
any
partner
incurred
in
to
such
Each
this
desired
as
increased
be
debt
value
par
increase
to
par
$10,
even
to the
large
as
shareholder
a
general
arouse
made
easily.
any
In
partnerships.
can
more
often
are
partnership, under
in
for
liability of
"
be
can
made
limited,
of
invested
sum
also, be
companies
place
the
Changes
amount.
can,
Stock
of partners
result, the
a
desired
the
Companies.
of
the
interesting
arise.
stock
the
as
company
city
such
as
a
basis
companies
adds
local
abound,
creameries, grain elevators, and
color
but
the
stores
AND
PARTNERSHIPS
illustrations
better
are
with
them
the
on
of
is available, the
The
example.
corporation
a
Bonds
227.
Defined.
secured
are
by
They
are
Some
items
of
forms
the
of
variation
3^
in
98
"
228.
interest
for
of Child's
is
of
with
the
given
by
a
rule, they
the
ordinary
private
and
taxes
bonds;
on
Schools,
from
1919
"
vestor's
in-
the
income
constant
a
the
on
briefly:
least
at
bonds
bring
of stock.
shares
C.
and
and
borrowed
market;
problems
other
any
include
means,
percentage
as
upon
lack
of
of
bonds
bank
to
above
far
is
as
by
the
of
the
comprehension
the
mystery
and
shares
clear
and
easy
of the
may
This
class.
up
this matter
of
nical
tech-
of stocks
the
of the
firm
a
to
to
pay
or
and
and
comprehension.
subject should,
of
some
corporation
be insolvent
the
possible. This
worthy
is, in reality,
points.
it is said
(assets)
Bonds.
difiicult
as
show
will
rational
When
its debts
them
a
these
Bankruptcy.
pay
a
and
'bonds
satisfactory treatment
a
Stocks
and
looked
because
from
outlined
as
the
of
stocks
Cancelled
all property
pay
as
as
K.
stocks
in
often
because
used.
Furthermore,
to
same
Difficulty with
involved
but
treatment
over
of
interest.
terms
to
just
as
less
principles involved
229.
As
municipal,
statements
much
child
by all
notes
issuing bonds;
fluctuate
make
an
as
direction
and
studied
be
bonds
simple
thus
should
that
very
a
the
sold
standpoint,
arithmetic
often
taken
connection
in
of mortgage
government,
as
interest;
The
be
be
individual.
an
and
bought
reasons
Cause
average
tion
illustra-
other
no
merely
are
between
of
points
can
difference
;
hence
and
form
bonds
corporations;
railroad
developed
of by
some
note.
If
Bonds
instead
corporation
a
be
pupils.
organization
the
familiarity
wider
the
solution.
for
given
problems
of
also
may
the
nearest
story
of
because
part
219
C70RP0RATI0NS
debts
form
study.
and
must
is
able
un-
turn
(liabilities),
or
of
This
application
phase
of
220
MODERN
business
ARITHMETIC
should,
least, be
at
required
be
should
METHODS
XX.
Criticize
1.
tions
State
Divide
3.
under
terms.
this
of
What
modifiea-
topicf
which
under
agreements
partnerships
formed.
be
maj
general
pupils who
partnership problems.
modernize
varieties
several
the
EXERCISES
of
form
to
necessary
are
2.
old
the
the
PROBLEMS
before
placed
learn
to
AND
A
profits between
$2100
in
B
and
the
three
named
ways
223.
4.
Suggest
three
5.
Criticize
the
6.
Why
similar
old
form
should
stocks
many
$100
of
stocks
teaching
^onds
and
partnership
in
problems
class.
grade
a
bonds.
and
retained
be
for
as
metic
arith-
of
topic
a
f
7.
How
capitalization
8.
must
What
did
What
income
2%
dividend
Compare
11.
What
12.
the
of
4%,
13.
14.
during
the
were
sold
at
Find
the
rate
if
they
Discuss
or
of
a
of
Explain
as
15.
Explain
the
16.
Give
17.
Explain
the
organization
18.
Explain
would
you
of
end
to
95?
class
a
a
was
loss,
at
the
when
87i,
at
dividend
yearly
over
if
761?
for
89^;
at
year
a
this?
mortgage?
or
bought
declare
82i;
84|.
partnership.
a
organization
the
declared
farm
82^?
at
brokerage?
interest
gain
for
stocks
the
the
6%
a
corporation
a
a
of
of
rate
on
for
quoted
corporation
of
which
6%,
the
advantages
the
loss
is
at
that
stock
was
rate
for
year
of
What
this
What
with
or
interest
sold
are
bring
loss, and
gain
of
shares
if
issue
company
$50,000?
owner?
income
end
the
of
$1000
year?
the
gain
of
rate
stocks
of
rate
the
was
uniform
the
five
former
the
net
do
twice
10.
stocks
they
stock
a
$350,000?
for
paid
be
must
f of
$200,000
What
87if
9.
of
shares
of
a
stock
company.
in
can
be
19.
Account
20.
grade
Look
for
Suggest
how
how
in
up
the
the
of
the
the
up
organization
daily
stock
local
stock
a
play
with
226.
upon
of
school
connection
great
three
based
formation
the
under
used
terms
illustrations
some
taken
class.
various
corporations.
can
company
be
shown
bank.
direction
and
the
solution
quotations
of
of
in
a
corporation
problems.
your
newspapers.
variations.
problems
from
these
quotations
suitable
for
a
PABTNEESHIPS
What
21.
AND
bonds
are
f
How
221
CORPORATIONS
do
they
differ
from
given
notes
by
individuals?
22.
with
the
Compare
5i%
a
income
from
5%
a
bond
quoted
9/2i
at
mortgage.
does
Why
23.
of
rate
of
price
the
stocks
fluctuate
than
more
of
that
bonds?
24.
Compare
25.
Discuss
26.
Consider
27.
What
bonds?
assets
Ezs.
What
Explain
29.
the
investments.
mentioned
under
227.
bonds.
in
the
study
a
company
of
stocks
and
overcome?
indebtedness
the
liabilities
in
used
terms
of
desirable
encountered
its
can
whose
pay
$83,565?
idea
general
the
for
20
be
and
$56,340
as
bonds
difficulties
cent
per
and
they
can
bonds
regarding
19
the
are
are
with
items
the
How
28.
of
stocks
of
and
bankruptcy
the
give
meaning
connection.
BIBLIOGRAPHY
1.
Albert
Atwood,
XLV:
2.
Post,
3.
and
Atwood,
25,
Aug.
Albert
G.
1917:
31.
Journal
Van
of
Vlissingen,
Yowr
Renting
Theodore.
Lindquist,
XXXI:
Stocks
Buying
for
McClure
Profit,
's
zine,
Maga-
58.
Bonds.
4.
G.
Hints
Some
Jr.
Arthur,
The
Teaching
the
on
LXVII:
Education,
Saturday
Money.
544
StocJcg
of
580.
and
One-Man
Evening
Bttsiness.
System,
438.
Bonds.
5.
"Baby"
6.
The
Bapid
7.
Owr
Government
Increase
Literary
in
Bonds.
Digest,
LI:
Shareholders.
Outlook,
1300.
Digest,
Literary
Feb.
28,
1917:
379.
LI:
904.
CHAPTER
BILLS
Bills.
230.
frequently
with
by
their
of
the
average
real
practical
bills
may
This
be
least
at
will
store
At
in
is
United
changing
Besides
on.
few
special
ones
electric
sheets
work
in
instances.
forms
as
are
well
as
and
Much
in
practice
making
the
versa;
lumber,
accounts
grain,
ing
follow-
the
Coal,
:
gas,
statement.
be
may
filling
parts;
vice
study
tice
prac-
tion
multiplica-
monthly
the
in
;
bill
for
give
to
aliquot
of
recommended
of
The
and
amount
equation
required.
integers
money,
and'
items
as
merchandise
general
light, freight,
Weekly
some
the
the
naturally.
or
including
foreign
to
finding
;
of
playing
three
two
stages
addition
In
arise
work
easy
fractions,
States
discount
computing
by
and
of
addition
and
to
number
only
exercise.
grade.
bills
confined
be
increased
so
fourth
for
write
can
arithmetical
as
the
as
field
children
the
enterprises
the
multiplication
in
water,
early
conditions
fertile
most
a
as
introduced
better
is
soon
together
number
many
very
introduction
convenient
bills
As
This
woman.
a
make
business
would
work
so
the
as
addition
number
while
be
other
they
which
and
at
or
first
them
with
met
are
and
makes
problems.
plainly,
forms
man
them
under
arise
various
Bills
forms,
which
ACCOUNTS
AND
simplicity
business
to
XXI
the
out
computations
added
various
involved
is
advised.
231.
Accounts.
necessity
The
for
every
the
farmer,
housewife,
It
the
is
one
small
mechanic,
daily
to
be
becoming
able
contractor,
man
every
222
to
and
more
keep
the
and
simple
produce
woman
accounts.
man,
in
a
more
whatever
the
224
MODEEN
ABITHMETIC
METHODS
AND
PEOBLEMS
"
it with
credits
estimate
an
total
amount
done
much
of
the
Not
it
is
be
from
the
with
pupils
In
to
addition
for
store
be
is
kept
like
which
accounts
last
three
gardens,
making
accounts
and
so
be
where
the
in
nection
conan
other
teacher
the
mercial
com-
encourage
incomes
own
and
chickens, make
of
time
to
in
pupils play
some
raise
time
economic
kept by
Statements
on.
also
are
the
arithmetic
of
may
their
who
from
out
corporations
as
of
accounts
given
time
all, let the
Above
sell papers,
out
topics
week,
especially those
liabilities
various
of
little
a
of accounts
importance
being engaged
at
or
keep
to
expenditures,
and
can
produce.
pupils of the
business
once
enterprise.
pupils
to
some
be balanced
keeping
the
a
the
anything
phase, the study
the
to
various
the
running
account
at
and
or
in
work
by the
computation
the
involved.
pupils
field
each
simple
understood
Farming
bookkeeping
entry
but
advocated
awaken
issue
double
of
time
any
strict account
a
have
renting
or
at
invested.
has
income
the
tell
thus
will
owning
to
intelligently when
and
He
grades.
Aside
will
of
able
be
he
money
in
here
four
will
more
readily
or
and
invested, and
money
bring.
it would
rent
advisability
of
course
the
on
the
on
outlay
a
can
the
house
dwelling
interest
and
repairs,
tax,
by
for
su^ested
resources
banks
and
study
and
application.
EXERCISES
XXI.
1. When
can
2. What
importance
grades
3.
and
f
Give
state
4.
bills
be
introduced
should
into
be
the
given
to
arithmetic
the
workf
of
use
bills
in
the
Whyf
a
list of
what
Suggest
jou
data
the
topics
expect
for
a
to
in
which
bills
accomplish
bill
of
six
can
through
items
in
be
used
their
which
use
use
to
advantage
in each
is
made
case.
of
fiUquot parts.
5.
Explain
the
use
of
bills
in
one
topic for
giving reviews
in
BILLS
studied
preTiously
in
review
Thomas
cost
7. Write
8.
Make
9.
Select
from
and
11.
What
of
per
and
and
cost
two
teams
equation
and
"credit"!
Explain
the
What
report
in
taught
af
resources
of
accounts.
grades.
the
of
each
and
the
should
following^
the
the
Compare
cents
cost
for
in the
.
"
a
three
in
man
for
seed
acre,
per
13
problems
for
the
on
hand
$256.40
The
and
53
*
work
rate
acres
31
was
for
of
$55
section.
work
by
of
the
bushels
two
return
per
teams.
upon
the
14.
Sales
96.34
records
of
a
Disbursements
$
day's
business:
Balance
21.40
Tuesday
Wednesday
125.60
Thursday
104.36
109.86
95.24
76.34
92.00
178.35
68.40
Saturday
days
20
331.34
Friday
1,
bushels
at
quarter
a
upon
yield
and
following
$
on
days'
15
bushel
and
53
valued
was
$65,
planted
fertilized.
Nov.
team.
was
com
year:
April 10,
tax;
produced
$82
fertilizer
and
the
for
investment?
farm
of
tax
$24,
not
blanks
The
during
repairing roof;
which
corn,
leased
was
1, $76.40
profitable
a
yearly
seed
each
Cash
Monday
following
Feb.
bushel.
per
was
$8
per
in
acres
paid
was
cost
invested
Fill
60
experiment,
which
the
this
which
$3500,
Sept. 15, $12
Was
owner
$2.50
for
Paid
painting;
42
an
and
15.
and
years' insurance;
production
at
As
capital
a
same.
sheet
1, 1915,
month.
furnace.
the
of
farm
acre
the
a
for
Dee.
planted
worth
acre
The
be
aeeaunt
per
farmer
acre
explain
business
"debit"
should
three
for
papering
A
work
by
house
$28
repairing
same
(d)
(c) water,
same.
weekly
meant
a
for
10, $38
14.
electric
gas,
1
night
over
iron!
:
year
$18.75
the
simple
a
13.
577
does
much
How
burning
W
light, (b)
newspaper
which
Bought
$4.50
furnishing
object f
statements
Jan.
the
account
Make
one
some
is
main
12.
lamps
a
and
statement
explain
Explain
the
p. 223.
on
W
25
electric
(a)
monthly
a
10.
be
given
using
of
bill
coal.
of
form
for
lumber
a
illustrations.
bill
two
hour
per
example,
additional
light
225
ACCOUNTS
for
leaves
he
bill
a
liabilities
per
if
is the
load
Give
electric
the
waste
What
topioe;
fractions.
'6. Explain
AND
86.30
$125.60
115.34
104.36
95.24
189.75
226
Is
produce.
17.
Suggest
18.
Give
19.
Explain
this
data
for
child
the
activities
value
of
economic
and
of
its
ftelling
and
f
gardening
school
PROBLEMS
gardening
on
raising
the
on
AND
problem
a
suitable
other
arithmetical
for
data
Suggest
16.
METHODS
ABITHMETIC
MODEBN
chickens.
suitable
such
keeping
keeping
for
accounts.
from
accounts
the
both
standpoints.
BIBLIOGRAPHY
1.
Bexell,
N.
2.
Buffum,
J.
E.
Statistics
A.
Farm
A.,
19JL5:
D.
3.
XXX:
Farm
Accounts
XXXI
LVII:
Van
:
5.
E.
One
Cure
School
Boolckeeping.
Simple
4.
H.
for
National
Kept.
Country
Jr.
899.
One-Man
The
in
Instruction
Waste;
IV:
Society,
and
Arthur,
Vlissingen,
Be
Should
They
as
[
84.
Purcell,
ceedings
Pro-
925.
*
Life,
Accounts,
Standardized
and
Business.
System,
438.
Zenders,
26.
F.
A.
The
Cost
System
on
a
Missouri
Farm,
Colliers,
CHAPTER
XZn
ARITHMETIC
LITERAL
The
232.
of
kind
Literal
that
of
sum
plus
All
formulas,
statements
with
radius
the
be
can
much
illustrations
various
of
topics
will
Reason
important
that
in
to
be
their
which
able
A
may
be
Duty
part
is not
that
rather
line
as
of
the
of
these
arithmetical
of the
to
also
language.
chapter,
magazine
Literal
in
the
cussing
dis-
articles
it
Equation.
arithmetical
greatly
to
technical
A
work.
few
An
is
equation
schools
elementary
find
will
eflFective
made
recognize
this
and
for
their
tions
occupa-
advantage
articles
regarding
illustrations
be
will
242.
under
234.
the
the
intelligently
read
particular
found
teaching
leave
they
to
in
the
Teaching
for
pupils
arithmetical
shorthand
in
and
briefly:
relations.
for
reason
many
found
be
arithmetic,
quantitative
233.
this
the
height,
brevity
greater
in
clearness
slant
of
a
equals
cone
written
majority
with
a
and
be
may
the
expressed
greater
Further
squared,
indeed,
and,
of
pi, (tt), radius,
of
Thus,
surface
the
is
equation
statement.
of
area
product
times
pi
total
the
the
literal
The
mathematical
shorthand
statement
the
Equation.
serve
a
as
a
substitute
Teacher.
it
is
In
necessary
so-called
not
of
forerunner
algebra
227
high
for
that
for
shorthand
Umguage;
for
order
this
the
work
teacher
equations
are
algebra.
This
school
algebra,
the
pupils
to
who
a
work
but
can
228
MODERN
later
not
AMTHMETIC
enjoy
that
about
the
cautioned
*'
**
transpose/'
likely
terms
to
235.
prove
stimulus.
a
ambition
A
subtraction
minus
the
for
words
words
have
MvMiend
'been
itself
has
but
by
an
grown-ups.
the
law
for
This
delayed
may
the
to
has
or
once
the
the
equals
the
be
as
sixth
been
time
early
or
be
be
in the
as
seventh
sion
discus-
a
below
the
:
remainder,
made
as
soon
fourth
grade,
When
in
consumed
or
the
the
may
literal
he used
it must
the
as
like
sentences
grade.
begun, however,
already
bols
sym-
JB
write
and
proper
immediately
would
the
only
after
upon
=
problems
and
the
above
properly
may
shorthand
a
signs
When
written
are
is
there
agreed
8
read
can
tinuously
be
will in
board
which
used.
are
"
introduction
equation
arithmetical
child
a
the
that
subtrahend
nwMW
Jlf
above.
of
the
equals the remainder.
Thus
they represent
children
ing
obscur-
Introduce
is used
upon
explain
now
by the pupils, they
be
This
write
to
of the
the
This
thus
shorthand
scarcely
subtrahend
teacher
first letters
word
is
to
writing this expression in
of
way
algebraic
:
Minuend
Let
student
expression
shorthand.
is first asked
child
is
coefficient/'
other
Equation.
reference
There
know
to
Literal
shorthand
very
**
as
and
the
to
teacher
the
reason
subject.
the
a
PROBLEMS
expressions
confusing
as
The
this
fractions/'
simple
equation
statements.
of
prove
Introducing
literal
of such
use
AND
For
study.
clear
otherwise
an
METHODS
con*
learning it will
wasted.
Problems
236.
excellent
furnish
Material
as
for
material
for
Translating.
this work
Problems
of translation.
No
.
attempt
need
solution
of
upon
be
made
problems.
recasting the
at
The
first in
using the equation for the
attention
statement
into
should
this
new
be
placed wholly
language-form.
LITEEAL
Extreme
far this
early
making
this
of
translation
of
solution
shorthand
ideas
one
them
to
apply
to
the
higher
with
the
still be
for
the
any
in
making
this
their
pupils
the
be
with
made
for
next.
of
be
it
at
these
commenced
once.
advanced
to
solution
the
be
weeks
some
in
needed
of
any
introduction
the
able
unacquainted
them
to
main,
be
to
Teachers
introduce
be
the
they present
come
to
will
it
time
the
work
solution
the
attempt
problems
necessary
for
they
should
can
progress
in
fail
Grades.
pupils
equation
and
to
Later
literal
simple
prior
of
whose
it will
because
solution
grades
pupils,
often
expect
in
faster
Although
before
and
day
Treatment
237.
the
effective
in
accuracy
early practice in the
have
symbols
problems
foundation
equation
literal
Teachers
problems.
and
necessary
should
into
words
language
general
the
children
The
problems.
the
are
of
use
Facility
work.
translation
satisfactory
the
of
translation
of
selection
too, in the
is necessary,
care
229
AEITHMETIO
of
problems.
1238. Literal
Equations
application of
important
most
solution
of inverse
-For
examples.
interest
problems,
instance,
it is
problems
to
only
quantities in the general
Solution
in the
as
literal
the
be
may
solve
of Problenuu
of
any
the
following
three
substitute
to
necessary
is the
equation
in the
seen
A
inverse
the
given
equation
I=PX
Tx
100'
thus
reducing
the
is
equation
perfectly
clear
stereotyped,
and
to
solution
rules
discriminate
Mathematically
a
simple division.
easily remembered
statement
vague
to
they
thus
which
between
are
same
place
both
of their
because
the
the
hard
are
"
a
fact
mental
funda-
recognized.
and
takes
This
to
close
which
of
One
several
remember
similarity.
should
be
230
MODEEN
emphasized^
a
for
means
eflBcientlythan
more
239.
The
Literal
Still another
be
can
Equation
topic
which
has
for the
the
thing
every
solution
of such
equation
of percentage,
problems
of percentage
done
by
number
a
Ratio
with
plicity
of sim-
standpoint
equation
rules
handling
PROBLEMS
the
fundamental
The
problems.
furnishes
literal
innumerable
over
AND
From
up.
the
clearness
in its favor
far
covered
not
and
inverse
METHODS
ARITHMETIC
and
is very
greatly
is ratio
and
of rules.
Proportion.
simplified by
the
*
of the
use
literal equation
it is used
wherever
instruments
paving
usable
for
the
unifying
to
way
When
The
the
Solution
pupils
of the literal
for its
greatest
intelligible,and
more
equation
costing
buys
five times
of
cost
The
so
that
they
English equivalent they
man
a
It is wise
as
more
these
in
work
type
:
for
much
to
are
Equatioxu.
of the
substitute
can
for the
ready
select
such,
language
it
readily
first,as
at
of
solution
give
example:
a
as
certain
the
lot.
sum
and
Together
builds
they
a
house
it
upon
$3600.
What
equations
:
cost
each?''
following
When
For
lot
Literal
with
clear understanding
a
integral solutions.
"A
Problems
of
have
simple problems.
had
of the
one
rationalizing arithmetic, thus
and
simpler,
a
is
equation
Indeed,
arithmetic.
240.
is the
literal
the
proportion.
are
are
thoroughly
fractions
forms
suggested
may
mastered
take
up
for
the
the
pupils who
problems
of
the
have
ing
follow-
232
MODEBN
ABITHMETIC
METHODS
8
AND
PBOBLEMS
k"T,
=
r
d
=
diameter
r
=
radius
T
=
tension
=
constant
for
the
particular
formula
for
the
length
k
A
2.
common
of
of
B
r
I
horse
3f
=
radius
of
=
radius
of
=
length
between
their
rope
pulley
inches
in
of
centers
pulleys
in
feet.
formula
foUowing
the
use
is
belts
cross
inches
in
pulley
other
of
tested.
rope
r) +2X1,
(JB +
one
often
of
power
the
=
Automobilists
3.
rope
pulley
on
L
the
to
compute
the
machines:
(2)-fL)2A'
n.p.=
9.92
D
L
*
N
diameter
^
length
=
feet
board
cylinders
cylinders.
formula
foUowing
L
The
=
diameter
=
length
teacher
of
or
log
the work
interest
243.
and
dependence
is another
are
in
the
smaller
number
of
Such
on
Without
form
nature
in
end
inches
feet.
improbable
bewildering
Problems
numbers
the
4)2L
"
careful
particular formula.
make
at
in
log
of
be
must
assign impossible
any
give
to
log:
a
(d
d
inclies
in
stroke
cylinder
of
the
use
in
of
number
=
Foresters
4.
of
and
in
giving this work
values
errors
to
have
will detract
the
to
quantities in
the
from
tendency
the
to
pupils'
it.
Numbers.
Problems
of literal arithmetic.
of
not
questions
in
which
without
These
the
lems
probpupils
LITERAL
asked
are
with
certain
''How
Many
the
as
an
of the
but
of
subject
their
to
will
244.
Must
Be
general
The
such
''Have
produce
excellent
the
a
actually
must
his very
and
firm
binations
com-
is secured.
text
child
the
close
without
number
of the
but
Their
Problems
solution
I
as
performed
of
order
answers
of
grasp
these
teacher
English
to
by
be
question, *'How
at
questions
of
would
applied
once
little
you
to
eral
sev-
:
make
require
if I
correctly
this work
at
all times
Such
sentences.
the average
followed
are
will, of course,
the
practice in the formulation
in which
by Specific Questions.
dividing 11,941 by
in
to
work
division
a
17
Followed
division,'* is
in
examples
remainder
for the
to
answer
Again, in
clear
will
back
questions
questions
this
example
an
a
overlooked.
materially in securing
specific nature
prove
anything
by trying various
in the
the
is usually
allopathically at
taken
solved
answer
of
are
group
regard them
be
not
basic idea 'back of the
these
value.
and
the
Gteneral
Unless
a
teachers
lems
prob-
one
importance, which
grades
when
in
placed
most
should
of
price
lists of such
problems
the
the
of solution.
methods
of
be
answer
him
aid
hence
if
found
f"
often
are
These
and
be
exceedingly doubtful.
not
can
the
and
value
is
until
know
solving problems
contain
texts
they
all.
at
results
order
given
are
day
throughout
use
numbers
articles
book
importance
the
about
go
articles
of secondary
reached''
daily
would
several
present
addendum
minor
In
of
unfortunately
end
^*not
of
price
number
of the
but
at
the
the
they
given conditions, as,
can
and
one
how
to state
233
AEITHMETIC
pupil
a
quotient of
5"(
213?"
effective
definite
drill will be
and
is
get
it is necessary
statements
found
excellent
expression of ideas, an
sadly deficient.
in
art
234
MODEBN
245.
ABITHMETIC
niustrations.
mentioned
be
may
integers
1. If
2.
has
the
following
topics,
with
subtraction
the
of
and
them
of
one
how
known,
are
of
cost
of
that
number
lot
the
apples
of
when
house
a
the
John
lot
the
f
known
are
of
cost
has
after
found,
be
he
5?
John
returns
he
had
has
20
cents
many
the
and
the
can
sold
4.
of problems
f
find
you
together,
How
numbers
two
found
be
can
house
3.
of
sum
other
How
and
connection
kind
:
the
the
can
in
various
the
in
use
PROBLEMS
of the
illustrations
As
s:uitable for
AND
METHODS
from
town
when
he
with
left
20
If
cents.
how
home,
could
told
were
you
find
you
how
how
many
dpent?
he
5. John
John
cents
many
tell how
you
cents
has
Teachers
will
lists of problems
after
little
a
to
attention
of
practice
the
teacher
them,
to
their
of
reasoning
symbolical
1.
The
area
2.
The
area
whose
sides
3.
The
4.
The
base
5.
2L
The
of
of
are
L
volume
area
and
area
of
entire
and
the
altitude
of
entire
the
of
the
this
slant
the
above
entire
how
How
can
John
many
has?
them
and
raneously
extempoto
the
excellent
aid
to
the
only
the
to
will be
that
work
bring
neglected
something
at
risk
of inestimable
in
arithmetic, but
ing
develop-
EXERCISES
for
the
surface
of
surface
height
following
a
pyramid
statements:
parallelepiped.
rectangular
a
of
eight
of
square
base,
8,
pyramid.
surface
of
a
regular pyramid
of
square
h.
entire
of
well.
as
expressions
the
only caution
be
powers
how
dictate
to
is, that
XXII.
Give
able
only in their
not
know
teU
you
has?
lines of arithmetical
be
can
can
James
many
if you
The
class.
robbing the pupils
benefit
will
How
difficultyin suggesting similar
other
the
arithmetic
of
teaching
in the
how
has
little
find
James.
know
you
James
cents
many
than
more
if
surface
of
a
right
circular
cylinder.
LITEBAL
6.
The
area
of
7.
The
rate
of
8. Amount
numbers
by
11.
and
buying
the
time
of
cone.
selling price.
and
loaned.
mooey
dividend, divisor, quotient,
connecting
equation
all
that
numbers
even
27i-l,where
Distinguish
12.
of
terms
circular
right
a
and
f
Show
10.
in
of
principal, rate,
is the
remainder
surface
gain (loss)
from
9. What
entire
the
235
ABITHMETIO
''literal
2n
by
all odd
and
integer.
an
literal
between
how
Explain
is
n
represented
are
arithmetic
is
arithmetic
algebra.
and
of
sort
a
guage.
lan-
shorthand
' '
set
13.
Illustrate
14.
Suggest
15.
What
16.
Explain
the
17.
Explain
the
of
Ex.
12
solve
and
inverse
are
arithmetical
several
by
six
problems
of
literal
the
of
advantage
applicable
Ex.
to
13.
Illustrate.
problems?
use
laws.
in
equation
the
literal
solution.
their
equation
usual
the
over
rules.
State
18.
further
any
advantages
the
favoring
the
of
use
literal
equation.
19.
When
20.
Give
21.
Give
throughout
22.
What
23.
How
of
price
24.
25.
gain
the
loss
of
28.
^are
given.
30.
the
can
the
or
the
cost
loss
of
any
The
The
How
Given
Given
cost
lesson.
of
the
of
use
without
"Problems
loss
or
found
be
literal
arithmetic
numbers"?
the
from
Illustrate.
cost
and
selling
is
sale
the
selling price
cent
is
found
be
when
selling price
the
given?
found
be
when
the
cost
and
found
when
the
cost
the
given?
gain
the
other
loss
or
How
of
be
cent
per
a
of
and
weight
the
a
gain
the
price
be
of
or
full
of
loss
of
full
number
number
articles
if
found
be
the
given?
wagon
tank
of
number
any
is
the
can
can
of
price
number
weight
be
article
an
known?
are
the
of
the
the
can
buying
31.
gain
can
given.
29.
the
per
selling price
price
by
can
How
begiin?
study.
can
How
27.
be
article?
an
How
26.
arithmetic
development
is meant
can
gain
or
illustrative
short
a
in literal
work
first
a
How
the
and
should
wheat
bushels
of
water
gallons
per
cent
the
and
the
and
of
wheat
and
the
tank
the
empty
be
tank
empty
contains
wagon
found?
be
are
found?
selling price, how
can
found?
marked
found?
price
and
the
discount
of
an
article, how
236
32.
be
ARITHMETIC
MODEKN
used
33.
regular
34.
Give
outline
an
throughout
of
the
lUustrate
the
in
lesson
Consider
the
work
study
of
of
use
AND
in
PROBLEMS
numbers
without
problems
to
arithmetic.
in
problems
these
with
connection
any
arithmetic.
the
Ex.
METHODS
32
in
detail
greater
for
any
one
topic.
BIBLIOGRAPHY
1.
Smith,
D.
Review,
2.
Young,
E.
Mathematics
LIII:
J.
W.
in
391.
A.
pp.
243-46.
the
Junior
High
School,
tional
Educa-
OHAPTEB
XXni
GRAPHS
246.
relations
and
contrasts
of
columns
do
is
nations
the
vividly
from
year
of
This
quickly
the
to
is
of
range
be
can
the
the
which
it has
of
Teachers
that
fact
algebra
for
final subject
of
high
time
will
proportion
school
not,
in
alone
in
graphs
this
work
those
for
justify
those
the
will
who
who
later
will
study.
237
a
not
sons
compari-
be
to
later
one
which
tion
introduc-
the
of
ratio
of
bear
ever
as
a
average
difficulties
the
study
treated
go
mental
its
in
and
must
the
recommended,
place
prior
the
to
and
consideration
grades
is not
bringing
them,
saved
be
final
the
size
for
and
strongly
found
can
states^
relative
within
to
is
grades
which
on.
newspapers.
grades
upper
yet,
so
towns,
the
well
interesting
as
bank
publications.
comes
the
these
avoided
be
can
of
work
The
textbook.
of
exceedingly
though
even
of
children
made
into
which
of
magazines,
by
of
and
variations
Government
and
subject
a
of
idea
different
of
money;
sizes
of
sizes;
value
instrument
universally
Station
is also
it
clear
a
varying
the
of
than
strength
grains
explains
plan
effective
most
used
now
bags
relative
the
of
grain;
by
year
show
reader
Experiment
As
to
vividly
military
other
in
of
architect's
an
rooms.
and
stacks
or
and
soldiers
by
wheat
scale
to
countries;
or
the
in
sacks
drawn
Maps
relative
The
show
^aphs
and
clearly
more
portrayed
by
countries
schemes
much
figures.
productiveness
deposits
Picture
Taught.
Why
and
graphs.
in
mind
forerunner
high
school,
have
the
but
as
advantages
a
238
comparisons,
showing
magazines
will
the
of
use
lookout
and
that
in, the study
may
if necessary,
during
this material
promising
pictures
with
classes.
for
and
The
of the
of
graph
continuous-line
graph
exercises
given
in
found
excellent
idea
of
249.
a
a
motion
2
continuous
and
at
in
streets
by
the
the
east
right angles.
local
ready-made
the
as
reference
pupils
and
Latitude
of
example
inadvisable.
unless
more
for
The
use
tinction
dis-
clear.
will
240,
and
as
game,
a
page
give
to
be
the
Where
example
from
this
is the
two
case,
of coordinate
lines let
points
coordinate
west
of
plot of
a
interest
the
some
local
the houses
north
intersecting
streets
has
the
teacher
axes.
With
city
even
their
and
axes
located
condition
furnish
longitude
cities number
Most
and
and
schoolhouse, post office,library, and
pupils,
the
made
during
on
some
of isolated
be
must
Let
change.
locating points.
south, and
teacher
interest
create
to
Points.
Locating
the
football
a
3, listed
and
material
simple illustrative
use
of
placed in
or
comparisons
for
the
it.
brought
Graphs.
picture graphs
several
be looked
may
of
Oontinuous-Line
least
at
brought
Some
by
the
papers
news-
lesson, or
wall
study.
retained
be
Versus
Picture
between
cases,
future
may
the
upon
cussed
dis-
on
and
been
have
pictures
the
been
week,
a
for another
subject.
they be
within
these
time
pinned
be
now
place
248.
When
which
the
magazines
the
school
continued
be
convenient
later
in
to
cardboard
pictures has
that
from
cut
upon
explained by the pupil who
each
and
over
bring
each
illustration.
mounted
one
illustrations
pictures
above,
introducing
children, suggest
the
other
for
for
five
or
mounted
comparative
such
freely by
Four
and
material
good
PROBLEMS
mentioned
those
as
newspapers,
found
be
After
and
AND
Subject.
the
Introducing
247.
METHODS
ABITHMETIG
MODEBN
now
on
the
should
be
these
made
it, such
homes
make
a
as
of
this
240
2.
METHODS
ABITHMETIC
MODEBN
Answer
i. How
far
f toward
What
of
each
the
above
'the
Emporia
distance
kicked
that
toward
kicked
SynjDori'd^
punt
goal
Topeka
School?
High
the
of
ball
kicked
was
in
goal
of
the
what
was
above
cent
per
instances?
two
ToybeAO'./'**
to
Tof^ef^a's
pant
the
in each
ha^ii.
Emporia}^
aeeompaiiyiti{t giaph.
toward
of
goal
PBOBLEMS
instances?
two
distance
tiie
punted
actual
distance
total
of the
the
the
was
iii. The
ball
the
was
School
ii
tm
following quoBtions
Ae
High
AND
--"""
"
BmJ^orloj.
to
TopeKa^^hallr
iv.
How
toward
far
the
What
V.
Topeka?
by
vi. What
Emporia
was
the
ball
toward
20-yard
line
caught
the
15
ball
of
cent
toward
off
yards
the
the
football
''kicked
total
the
per
ball
a
Topeka
goal?
distance
that
the
ball
carried
was
bj
Emporia?
the
3. Draw
the
toward
goal?
Emporia
was
carried
ball
the
was
from
returned
to
distance
Emporia
goal?
that
show
the
it
left
10
Topeka
carried
Emporia
carried
goal?
scale
Topeka
the
that
total
Topeka
field
to"
the
and
and
side
yards
the
line.
ball
The
directly
following
was
caught
Topeka
toward
plays:
on
player
the
the
who
Emporia
241
GRAPHS
Ipoal
before
"down."
Second
right.
"down/'
"down,"
2
"down,"
"punt"
The
yards
Emporia
"down,"
yards
line
useful
in
ball
12
' '
the
will
it
from
showing
10
15
be
20
from
referred
to
' '
conditions
which
and
object
do
directly
8
to
down,
right
as
a
not
the
line.
1^
First
past
run
side
7
goal
line.
"picture
depend
side
forward
"down,"
end
graphs
to
Fourth
right.
Second
' '
the
Picture
proper.
left
the
left.
the
right.
it
and
First
graphs
from
10
Third
right.
to
the
to
brought
to
yards
12
bounds
forward
right.
the
down
following
and
and
and
ball
yards
' '
of
out
and
forward
yards
4
forward
forward
yards
5 to
and
the
distinguish
and
forward
yards
forward
The
40
"down,"
yards
forward
"down,"
5
with
4.
2
player catching
First
yards.
First
will
upon
graph"
be
to
found
continuous
changes.
Explain
5.
of
an
Make
average
the
a
graph
picture
man
5
its
graph
ft.
5
for
the
in. tall for
fully.
following
different
data
ages.
on
the
weight
242
MODERN
Make
6.
of
ARITHMETIC
States
United
New
divisions.
England
5.3
North
W.
North
Central
Atlantic
3.4
2.9
South
Central
17.4
W.
South
Central
13.2
Mountain
6.9
Pacific
6.0
of
picture graph
following
the
of
data
03
May
2.14
1.70
June
1.97
October
March
2.71
July
3.88
November
AprU
1.74
August..
extremity
vertical
and
of
sudn,
* *
represented by
water
at
the
vertical
the
the
at
.3.58
right angles
eight
"
inches
table
graph?
through
What
the
top
right
of
the
diagram
the
for
left
horizontal
the
upon
readings
the
in
feet.
in
Pressure
of
pressure
pounds.
6
2.60
8
3.40
10
4.33
15
6.49
8.66
20
curve
the
along
point
lower
the
Label
and
inch
.77
depths:
different
Beginning
09
near
ft."
each
giving
.
2.32
long.
in
each
Locate
2.
following
Mark
.
December.....
axis, ''Depth
in lb.
Pressure
Depth
each
page
horizontal
the
along
intersecting
axes
of
corner
in
.2.18
September.
February
hand
rainfall
the
Kansas.
January
two
1000
per
16.0
E.
8. Draw
illiteracy
5.7
E.
Emporia,
the
Explain.
Atlantic
a
of
data
following
geographical
Central
7. Make
5
the
7.7
South
at
in
PROBLEMS
AND
States
Middle
1914
from
picture graph
a
United
the
METHODS
25
10.82
30
12.99
35
15.16
with
the
is
the
various
the
in
highest point
points.
pressure
at
What
a
diagram,
the
use
depth
be
can
of
draw
12
a
made
ft.?
of
smooth
of
this
27
ft.f
243
GBAPHS
what
At
extended
9.
so
What
Bureau
following
they
data
pressure
at
the
of
Kansas:
364,399
1880
996,096
1890
1,427,096
1900
1,470,495
approximate
table
hour
daily
the
to
prepared
was
relative
the
for
at
fourteen
and
the
States
Weather
observations
upon
the
Plot
the
1885?
in
United
years.
state
and
1875
based
humidity
axes
same
in
population
be
graph
ft.?
45
population
the
on
Could
1870
following
observations
the
Ib.f
107,206
the
same
give
13.5
1860
showing
the
at
the
Ib.f
7
pressure
it would
was
The
the
that
Graph
10.
is
depth
points
of
taken
two
of
sets
interest
which
show.
Dodge
Santa
Dodge
Santa
11.
City,
Fe,
Lbs.
!Feb.
Mar.
Apr.
May
June
Kan
72%
72%
64%
61%
64%
63%
Mex
55%
55%
43%
35%
36%
31%
Jul.
Aug.
Sep.
Oct.
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Kan
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64%
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the
each
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What
zone.
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table
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practical
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parcel post
use
be
can
using
rates
made
Zone
Zone
different
a
this?
of
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0
to
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mi.
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300-
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1000-
1400-
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1400
1800
1800
mi.
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12.
Make
a
13.
Make
a
division
point.
graph
for
graph
of
These
any
two
should
railroad
time
show
time
tables
trains
card.
of
trains
going
going
in
both
through
a
directions.
^iA
MODEBN
How
can
made
the
these
this
at
14.
ABITHMETIC
graphs
division
is
What
the
Explain
Suggest
17.
Select
data
18.
Graph
the
19.
the
show
status
of
graphs
board
for
and
for
conneetioiis
in
be
can
arithmetics?
Select
data
for
Graph
the
data
on
Diseuss
the
three
or
graphs
graph
to
as
17
of
ordinate
co-
tion.
illustra-
as
class.
a
for
in
problem
idea
the
streets
two
suitable
problems
(me
class
grade
longitude
any
which
featjires
a
lesson
three
data
to
as
introductory
an
what
f
latitude
using
16.
to
PROBLEMS
change.
a
upon
axes
important
point
AND
used
be
present
of
advisability
15.
METHODS
picture
and
graphs.
point
out
the
portrays.
suitable
problems
for
graphing
by
grade
pupils.
20.
features
important
21.
in
as
II
graph
of
or
done
was
which
is it not
Why
in
of
it
the
the
connect
and
of
data
extremities
made
picture graphs
graphing
19
l^e
ezpUun
portrays.
to
proper
in
problem
one
any
from
8
problems
of
the
in
data
rectangles
problema
^T
9f
and
BIBLIOGRAPHY
1.
and
Allyn
maiios,
2.
Matilda.
Auerbach,
Brinton,
W.
4.
Soldiers
Wounded
Healing
neering
Engi-
Order.
to
430.
Ow
Threaten
XXXIII:
Work,
the
6.
Covering
7.
The
8.
World's
9.
StatisticaL
Department
Will
to
Prosperity,
World's
Work,
State,
Power.
of
Atlas
Principle
Nationalism
of
the
in
Balkana.
565.
Production
Country
Gentleman,
Munsey,
Aug.
Gold.
of
of
the
Sept. 15,
8.
1917.
Munsey,
1917:
Aug.
States,
United
1917:
Bureau
423.
of
the
Census,
Co.,
New
York-
Commerce.
10.
The
American
11.
The
World
York.
the
and
Democracy
World's
New
Facts.
Presenting
561.
XXXIII:
5.
for
Methods
York.
R.
TJ 'Boats
the
Mathe-
Graphic
Boston.
New
XXXIII:
Work,
How
Go.,
in
Course
Elementary
Graphic
Lewis
Freeman,
World's
Bacon,
G.
Magazine
3.
An
Tear
Almanac
Book.
and
D.
Appleton
EncyclopedicL,
and
Press
Publishing
Co.,
XXIV
OHAPTEB.
RATIO
History.
262;
of
outgrowth
Oolden
and
similar
of
to
Rule,
of
and
of
the
All
and
word,
these
of
the
in
the
century
a
extension,
an
called
the
Chain
the
est
strict-
in
rules
were
followed
were
about
and
of
as
place
Three,
proportion
compound
formulated.
sense
of
the
as
far
as.
until
an
solution
prominent
very
Rule
to
the
back
America
Inverse
the
in
dates
a
and
referred
power
Three
is
proportion
and
also
great
held
Europe
our
was
its
of
ratio
Three,
of
Rule
Besides
ago.
Rule
Arabs,
arithmetics
of
theory
because
The
problems.
Hindus
Our
the
Rule,
PROPORTION
AND
arbitrarily
by
the
could
be
computer.
The
rules
these
in
mechanically
used
thinking
with
the
on
last
the
by
analysis.
ditches
digging
decades
problems
and
the
have
ago,
rules
in
given
be
of
to
way
any
machines,
were
unfamiliar.
largely
remarked
been
placed
re-
the
that
proportion,
compound
pasturing
without
wholly
have
also
they
problems
was
should
that
They
user
these
It
of
computer.
the
century
complicated
long,
the
fact
the
solution
the
of
part
in
lay
mechanism
whose
During
few
of
value
cattle,
problems
on
prevalent
so
more
a
nearly
probable.'
253.
the
in
were
Change
history
of
notation
the
of
Notation.
Ratio
used.
and
A
most
Proportion
Some
of
:
245
the
interesting
is the
more
phase
in
gradual
change
noticeable
forms
246
ABITHMETIC
MODERN
METHODS
4
marbles
|| 8
4
marbles
8
cents
4
marbles
8
cents
"
cents
AND
PROBLEMS
|| 9
marbles
(1)
9
marbles
(2)
9
marbles
(3)
"
: :
.
Because
of
of the
use
multiplication,
the
marbles
4
marbles
4
It
should
mark
this
that
or
reason,
number
or
selected
rational
so
9
still
: :
8
group
of
did
is
used
(5) has
term,
early forms.
only
from
largely
question
the
or
the
The
idea
real
equality of ratios.
served
to
texts,
our
(5)
appreciate
an
numbers
in
?
:
the
not
reference
any
to,
prevalent
cents
in
sign
a
as
(4)
fourth
the
proportion
a
and
marbles,
it, is absent
symbols
form
the
their
For
separate
to
one
and
others
were
meaning.
been
changed
cents
:
to
the
more
of
one
marbles
4
but
marbles
that
changed
form
the
the
without
last
The
: :
of arithmetic
writers
of ratio,
cents
9
:
representing
early
was
.
noticed
be
latter
into
evolved
has
point in decimals
8
.
which
of the
far the
9
':
marbles
meaningful
one
8
=
? cents
(6)
form
4 marbles
8 cents
9 marbles
C cents
(7)
has
yet
not
254.
Ratio.
affairs
most
and
universal
Ratios
their
expressions
one
arise
as
contains
number
one
may
or
cite
and
so
quantity
are
on.
in
Ratio
is that
comparisons
the
in
day
every
of
the
such
fa"
some,
including
terms
**they
in
continually
nomenclature
proportional to,"
times
recognition.
mathematical
common
miliar
are
received
ratio
asks:
of another.
commercial
of,"
How
*'they
many
As
trations,
illus-
activities.
248
MODEBN
used
in the
ABITHMETIC
ratio
text, it should
the
substitute
notation
is used
256.
of
correctly define
the
time
same
In
order
that
the
work
the
place of
notation
The
found
meaningless
the
use
of ratio
equality of
the
rational
thi"
to
text.
rule, the present
a
as
which
proportion
that
than
As
in the
of
teaching
they
the
on
required, thereafter,
in
text
two
hasr been
as
result
a
day
texts
ratios^
at
notation
3:4::7:f
in
to
.
be
proportion
that
f
the children
notation.
meaningless
a
it is suggested
,
greatly handicapped
more
even
for
the
of the
ProportioiL
discarded
be
PROBLEMS
of
up
and
it regardless
f
AND
atUside
first taken
be
notation
use
METHODS
(8)
one
7
3
w
N
Children
who
other
is the
last
involving
be
found,
the
for
idea
of
Through
proportion,
the
meaning
proper
proportion
given quantities
three
reducing
thus
the
of
notation, J, will readily
it.
with
text
fundamental
up
form
correct
in the
form
taught
its correct
ratio, together with
the
been
have
problem
and
supplant
equation is built
an
to
any
which
comparison,
and
cept
ac-
fourth
the
solution
the
to
one
of
a
literal equation.
Omit
257.
Complicated
complicated
problems
mentioned
above, have
proportion in disrepute.
often
teacher
later.
oflfered in
or
others
and
involving
done
much
Omission
Problems.
place the
of such
problems
highly complicated problems
the
text
which
are
either
within
be
as
will
which
experience of the pupils should
of the
be
of
is most
simplified by
the range
as
subject
The
must
long,
conditions,
unlikely
to
The
subject in toto,
the
not, however,
commendable;
discussed
Proportion
be
are
the
standing
under-
substituted.
EATIO
AND
249
PBOPOBTION
Strictly speaking, ratio is the comparison
two
quantities having the
best
to introduce
unit
same
proportion
by
of
the
C cents
4 marbles
8 cents
of
It is hence
measure.
use
9 marbles
by division
of the form
:
(10)
rather
than
by
9 marbles
4 marbles
(11)
=
8 cents
C cents
As
long
but
to use
as
the work
(10) makes
form
as
soon
as
notices
child
a
9
expresses
be
absolutely
an
permitted
form
any
carrying
is
relation
(12)
teacher's
greatest
in
work
so
make
to
as
by
an
not
relation
strict adherence
make
far.
which
are
being dealt
opportunities
children
Here
for
of
that
in
proportion
it is based
of
the
right.
the
and
of
one
rationalizing
feel that
is
with
lies
use
that
child
The
too
true.
the
feel
child
the
should
he
to
hobby
perfectly
Proportion
Unitary
solution
or
ytA"
number
will
general, and
the
that
the
the
metic
arith-
particular
a
upon
real
basis.
common-sense
258.
a
relations
It is number
too
above
form
is the
4
correct
A
it.
use
(10)
as
is
teacher
to
that
clearer
by the
Analysis.
of long,
use
arbitrary
Treated
of
a
As
the
long
as
proportion
problems
meaningless
notation
that
in
rational
the
meant
the
by arbitrary rules,
which
setting
teachers
Than
Bather
Equation
complicated
procedure
surprising
by
up
have
embodied
of
forms,
been
ing proportion for imitary analysis. A rational
also
it is
discard*
application
250
MODERN
of
the
subject through
is
suggested,
unitary
really
discarded.'
which
has
thus, be
can,
clear
as
unitary
in
his
for
259.
of
graphing
well
as
The
of
case,
will
important
most
from
it the
equation
affords
excellent
which
to
do
he
uses.
and
not
inversely
either
possess
in
help
The
teaching
by
of
portion.
pro-
far
the
children
the
literal
The
hasty inspection
a
unitary analysis does
which
peated
re-
state
Proportions.
opportunity for
in
by the
ready
proportionality.
direct
of
of solution
form
of the
idea
be
to
recognized
be
soon
of so-called
proportionality, by
direct
graph
be
can
using the simplest
directly
great
a
careful
By
about
at first
which
day
every
form
brought
of
tionality
propor-
solution
of
longer
are
some
and
be
an
value
of
pupils.
Inverse
relations, will be found
these
idea
proportion
which
as
the
required
and
relations
proportional,
and
Direct
Graphing
be
ratio and
every
The
the
is
child
ratio,the
in
form
problems,
every
is
to
as
may
of many
which
reason
pupils
This
analysis.
place, proportion
business
equation
the
to
solution
data
clear
shorthand
the
teaching,
made
by
to
discussed.
in
solution
of
many
found
to
herein
as
advantages
too
relation
referred
often
form
first
the
been
already
so
affairs
In
of the number
extension
notation
abbreviated
possesses
PROBLEMS
AND
meaningful
a
an
and
analysis
wholly
made
METHODS
ABITHMETIC
not
give practice.
Solving Problems
260.
the
proportion,
by
first,merely
of
solution
problem
to
of
A
261.
be
may
problems
Plea
may
for
often
approximate
the
be
and
a
to
from
equation;
advantage
to
Use
be
of
statement
his
process
This
in
the
Proportion.
A
fostered.
by opportunities
the
tention
at-
later
mechanical
itself,an
Reasonable
results
give his whole
number.
quality
shortened
the
solving problems
unknown
the
is, in
the
up
given
for
equation
concentration
of
power
or
the
solving
x/
should
pupil
setting
to
attention
undivided
In
by Proportion.
to
of the
cancel
equa-
RATIO
tion, which
do
not
This
for
which
been
the
such
of
solution
expeditious
262.
Ratios
texts
;
not
always
the
ratio
would
of the
be
hats
In
so.
be
sold
The
often
is
graph
between
263.
business
which
is found
is
on
a
state
a
retail.
to
as
in the
simple
a
football
baseball
usually
the
lower
This
is
as
of
as
equal
hats
the
200
hats
than
the
important
an
are
200
cost
of
direct, or
2
sideration
con-
clearing
up
indirect, proportionality
a
The
exist
over
in
one
and
modern
our
or
ones
others
biore
pluralities.
Aver-
of the
weight
men
of
salary paid teachers
Another
illustration
batting
average
three-place decimal, but
percentage.
ular
partic-
first the
come
average
average
and
Various
these
number
player's fielding
spoken
the
of
and
means
a
Among
one
the
or
as
It is
from
be
to
seem
numbers
of
illustrations.
expressed
more
a
in business.
of ratio.
case
which
So.
omitted
a
of majorities
case
team,
good
are
of
excess
any
quantities.
two
politicallife.
the
and
proportion,
ji-^ because
for
arises
as
first
be
not
of comparisons
and
or
and
Majorities, Pluralities, Averages.
forms
age
at
ratio
of 2 hats
purchase
again suggested
questions arising
any
in
for
and
or
in
case
any
is often
at
wholesale
at
which
which
would
costs
bought
would
the
analysis,
Necessarily
with
ratios which
that
namely,
having
irrational
explanation
Equal, Not
matter
a
ment
instru-
proportion.
Seem
to
plea
problems,
an
for
clearer
duty, in connection
teacher's
of
Unitary
suggested
a
than
that
call attention
to
is
it
to
use
a
of its
reason
complicated
notation.
affords
solution
a
the
application
solution
It is rather
arithmetical
disrepute by
tions
solu-
exclusive
the
problems.
serviceable
into
meaningless
form
"other
of
in
strikingly
so
plea for
a
needlessly long
to
arbitrary
largely
is not
most
a
fallen
has
applied
by
of
retention
the
themselves
solution
in the
proportion
251
PROPORTION
present
by analysis.
of
AND
Yet
another
is that
of
which
is
often
form
a
eously
erron-
which
is
-252
ABITHMETIC
MODERN
quite similar
the
numbers
which
given
as
so
arithmetic
the
it well
is the
Explain
2.
should
The
rate
common
to include
while
them
EXEBCrSES
Three
direct
"
indirect?
and
ratio.
by
closely
How
Illustrate.
tof
adhered
be
definitions
the
death
as
persons.
worth
is meant
what
clearly
3. Discuss
of
Bule
definition
this
10,000,
stating
of
work.
XXIV.
1. What
th^t
is
op
1000
per
makes
tei'ms
1000
100,
many
PROBLEMS
AND
mentioned
one
of each
out
of these
use
in
is
last
the
to
METHOBS
and
notation
found
in the
of
ratio
4
in. to
a
yd.f
of
a
circle
arerage
texts.
What
4.
3
yd.
is the
2 m.f
to
What
5.
f of
its
ratio:
2, 3;
1.4, 2.1;
to
pairs
5, 7;
3.5, 4.9;
7. What
of
1
to
ratio
the
area
What
6.
pk.
2
is
ratio
bu.f
of
1, 4;
its circumference
f of
10, 15;
.3, 1.2;
is the
present
that
per
following
series
3, 12;
14, 21;
25, 35;
3.02,
3.03 f
of
$5 to
ratio
2
ft.f
1 "f
$0.25
its
eter
diam-
the
same
to
its areaf
to
the
in
6 in. to
bu.f
2
to
circumference
the
numbers
ft.f
1
pk.
6
radius
the
of
3 in. to
have
.25, 1; 1.6, 6;
1 franc
to
f $0.25
to
1 mark?
his
yrs.
8.
Show
9.
A
2
people;
the
to
at
for
What
the
mile?
2
20%.
of
cost
of
the
recipes
or
slope
distance.
is the
commercially
living
5
people
of
of
is 120%
to
ratio.
a
ratio
The
present
for
is
the
to
past
be
Illustrate.
his
income
what
it
of
cost
changed
to
10
was
living!
to
recipes
people.
gradient
horijsontal
A's
ratio
least
8
used
as
is increased
is the
Suggest
By
the
BC/AC.
150'
income
what
ago;
11.
to
man's
cent
past salary is whatf
10.
for
a
slope
of
The
of
a
slope
a
is meant
road
road
in
the
which
the
ratio
accompanying
rises
100'
to
of the
rise
figure
the
is
milef
BATIO
12.
Wliat
is the
average
gradient
gradient
is
Compare
14.
Explain
15.
The
it is
3
of
the
is
in
f
weight
cubic
a
of
oak
the
.2
of
f
of
ft.
**
that
i "=g"
his
pay
turns
in
a
him.
take
is
employee
out
received
pay
each
"A
that
''A
the
given
Explain
^
i8
each.
to
inversely
are
Suggest
B
and
proportional
B
and
A
by
^f 4f
was
is
is
means
is
Bf"
tional?"
propor-
explain
and
equal
the
to
of
product
directly proportional
of
as
is to
of
2"
in
to
If
what
12"
X
X
following
a
the
of
find
of
per
29
cubic
lb.?
weight
2"
the
weighing
122.3
X
to
is 62.5
water
wood
pounds
the
2"
copper
substance
a
557
weighing
table
of
bar
of
weight
weighing
14'
were
neither?
specific gravity
copper
XXIII
Chapter
weight
the
of
.4
cu.
ft.
10"; petroleum
X
8';
sp.
costing $4.50
only
months.
6
worn?
Which
Is
is the
costing $4.00
shoes
could
be
worn
""
petroleum
could
be
the
ratio
11
worn
of
.7
the
and
months
prices equal
a
to
economical?
more
could
7i
13.596
mercury
gr.
""
8.9
be
worn
months.
while
months
10
Answer
the
same
23.,
is in
number
7?
of
10.5
$3.00
of
graphs
water.
is the
shoes
of
the
in
ratio
of
mercury;
time
pair
What
2
the
take
loan
is the
copper
pair costing
25.
of
a
the
4' X
pair
the
question
of
inversely proportional?
silver
gr.
A
24.
to
it will
tune
Illustrate
quantities
what
pair costing $2.50
a
on
plank
4' X
**
that
witii
employee
the
if"
product
foot?
use
cu.
sp.
A
8
:
afford
the
to
of
"
volume
cubic
tank
23.
average
proportion.
"Uiy
foot,
an
By
silver;
work
saying
Bf
as
equal
an
per
22.
C
: :
can
it would
ratio
the
the
sets
lb.
of
whose
Illustrate.
of
in
that
Specific gravity
foot?
mi
20
whose
Illustrate.
ratio
the
by
directly proportional
pounds
in
railroad
of
illustrate.
interest
What
per
mi.
16
expressions.
that
Simple
21.
of
time
varies
extremes
20.
of
^f
4
:
employer
and
meant
are
Prove
elements
is
3
fully.
an
be
in
similar
19.
and
f,
amount
the
''A
18.
that
should
B
further
with
fully
What
and
distance
inversely proportional
ratio
17.
"A
feet
of
terms
that
4
:
io the
What
if the
in
proportion
statement
16.
for
amount
proportional
each
riae
263
PfiOPOETION
^f
13.
time;
AND
to
17
that
5
the
is to
same
6?
ratio
to
15
that
3
is to
2?
to
16
254
MODERN
ARITHMETIC
Divide
26.
$240
proportional
27.
yard
of
in
line
What
is the
A
in
by drawing
in
that
so
their
is 40'
shadow
which
diamond
by
3'' square.
a
mechanical
miles
be
will
sums
when
of
that
a
is
represented
are
which
in
of
ratio
length
the
called
scale.
the
cases:
represented
drawing
The
distance
actual
following
300
scale f
to
the
to
the
baseball
square;
A
whose
tree
a
drawing
a
scale
map
A
C
and
PROBLEMS
5.
of
is meant
What
any
AND
is 4.5'.
measure
28.
and
height
the
A, B,
among
3, 4,
to
Find
METHODS
by
14"
9"
a
is
6 inches.
by
by
square;
^^^
a
by 3}";
represented
20*^
represented by 5^".
Kansas
29.
used
make
to
30.
the
In
best
same
a
on
map
solve
way
9"
paper
scale
the
What
miles.
200
by
by
12"!
to
be
scale
6"
be
9"!
by
for
used
should
Illinois;
for
Carolina.
North
31.
What
miles
150
the
miles
400
is about
32.
f of
Show
is
equation
33.
will
The
Reduce
60
in
each
250
miles
f
proportion
stated
the
be
miles
50
that
a
really
a
f of
of
is used
form
30%
29
problems
of
30
literal
the
of
language
the
and
by unitary analysis.
24%
6
by adding
cream
20
24
=
30
=
reducing
in
dairymen
by
to
cream
of
in
solution
shorthand
following
gallons
length
cream:
milk.
4%
4
"
24
"
20
Hence
milk
4%
34.
cream
to
added
to
35.
ratio
36.
in order
reduce
and
State
and
should
f
of
discuss
proportion
How
mechanical
it to
in
the
milk
3%
some
the
f
to
of
the
be
How
raise
arithmetic
subject
added
be
cream
added
18
the
form
Explain
cream.
must
25%
be
must
cream
24%
a
milk
6%
gallons
10
30%
have
to
much
How
of
gallons
60
to
20%
much
it to
3.6%
of
given.
gallons
12
to
gallons
of
must
cream
30%
be
milkf
principal objections
to
including
work.
taught
in
order
not
to
make
it
OHAPTBR
ZXV
MENSURATION
264.
History.
mathematical
document,
2000
and
period
least
extending
knowledge
of
at
this
the
mensuration
time
early
of
storing
was
*'
called
5
knowledge
of
triangles.
The
of
It
is
the
the
to
c,
we
elements,
Almost
from
has
circle,'' which
equals
area
mathematical
the
ratio
For
many
the
great
of
the
area
centuries
mathematicians
a
it
language
of
finding
of
this
a
circle
problem
of
the
256
to
the
few
600
putations.
com-
in
Euclid's
;
in
the
**
of
of
side
the
ing
squara
value
of
the
square
technical
more
square
occupied
day.
a
mensuration
finding
means
as
nature.
about
problem
circle
given
well
years.
the
about
heard
means
that
300
right
from
culminated
no
tians,
Egyp-
as
practical
of
about
for
of
early
Egyptians
laws
which
studies,
the
geometry
the
present
our
one
every
the
whose
collected
sides
had
sides
Arabs
and
wholly
a
developed
continued
were
of
was
surveyors,
they
the
by
Hindus,
who
owe
These
between
with
whose
though
undertaken
Romans
facts
that
relation
early
triangle
even
attention
figures, which
plane
the
Two
connection
The
that
triangle
mensuration
Greeks,
of
knew
general
later
area
a
metical
arith-
years.
special
in
survejdng.
of
Babylonians,
fragmentary
B.
the
right
a
the
Hebrews,
that
and
part
is
bams
of
record
5000
over
of
covering
a
received
volumes
rope-stretchers,"
3, 4, and
are
the
grain;
important
an
back
problems
;
is
it
previous,
years
1700
between
Egypt
information
contained
it
1000
in
written
known
oldest
the
papyrus,
was
As
c.
b.
at
classes
Ahmes
The
of
its
tt,
or
radius.
attention
of
251
MENSURATION
to
area
considered
while
1794
In
equal
w
which
has
places by
^ades,
3i^ is
nine
to
the
at
long; this
of
The
today,
as
towards
of the
the
end
space,
what
has
construction
suggest
year
to
to
year,
intervening
brought
to
in
a
but
teacher
of
so
that
between
there
this
been
work
is not
of
development
must
for
recent
further
here.
movement
education,
the
the
the
offered
last
of
before
the
objects in
to
purpose
importance
two
or
primary
three
and
repeat
primary
its continuance
be
or
without
world
of
no
year
regarding
its
not
of
veyor
sur-
established.
relations
emphasize
may
days
the
leaving school
the
necessity
360
mensuration
into
went
plane. .It
the
year
until
Children
years
to
the
primary
mensuration
previously
degrees, probably
was
the
better
school.
work
the
The
its consideration
permit
measurements,
of lines
or
inches.
histories
Until
and
eight
the
of
ideas
r
earth
with
of
phases
not
Years.
grammar
of
in the
360
historical
the
mathematical
will
space
given
was
two
in
it
since
years
the
to
kindergarten
attention
707
to
the
and
measurement
other
or
Four
First
265.
4000
interested
directed
information
of
to
35
to
carrying
By
1706
circle into
the
standard
formulas
be
in
they considered
that
over
student
again
here
fact
the
is still the
usual
value.
Jones
ber
num-
all purposes
accurately
by W.
is, a
later
and
or
3|^.
computed
circumference
computed
divided
Babylonians
to
that
;
Ceulen
the
and
3^
was
For
(V^)^,
as
by Euler.
use
due
any
be
suggested
common
It
V2.
Shank.
places,
can
was
w
The
the
decimal
Babylonians
it
gives
irrational
von
William
the
it between
accurate
suflftciently
a
equator
symbol
into
Ludolph
pldces by
decimal
out
like
ending,
reversed
circle equal in
a
and
papyrus
it to be
proved
no
Hebrews
placed
Archimedes
Legendbe
decimal
The
of
radius
find the
3, Ahmes
to
the Hindus
passing, that
to
given square.
a
3.1605
tried
and
problem
the
in
be mentioned
It may
and
to
from
years
of
the
258
ARITHMETIC
MODERN
final years.
in
the
of the
kindergarten
offer
not
can
view
In
METHODS
for
excuse
of
years
the
excellent
the
in
and
AND
PROBLEMS
construction
early primary
the difficulties to be encountered
delaying
arithmetic
Hence
work.
done
years,
as
study of mensuration
the
work
one
adequate
an
until
the
let it proceed
last
out
through-
grades.
Fundamental
266.
with
denominate
attempt
side
measuring,
the
connected
Essentials.
inventional
numbers
is the
It
geometry.
the
on
the
other, that
of
separation
a
of
geometry
the
on
Mensuration
three.
the
is
merical,
nu-
closely
so
and
hand,
one
teacher
must
not
and
Square
cubic
.
measure
finding
are
few
a
Review
1.
2.
should
class
be
simple
Add
construction
work
time
from
with
imacquainted
introduced
it
as
Do
3.
it at
to
in
was
teach
not
This
possible.
this
:
measurements
time
to
lowing
fol-
direction
and
based
side
fascinating
beginning
The
once.
primary
the
for
formulas
of
to
logical
to
mention
rule.
by
is the
It is necessary
that
upon
grades
but
of
school
progress
quite
be
must
work
be
may
as
made
real
place
the
teacher
we
arrive
conclusions.
that
Lead
In
to
not
some
proofs
exact
to
up
the
for
formulas
inductively
application
cases,
it may
found
be
may
proofs
rigorous
be
in
the
by
but
well
the
as
rationalization.
that
realize, however,
at
of
far
as
ment
developrather
come
for
the
teacher
science
exact
more
geometry.
early
Make
4.
and
compass,
problems.
use
of
the
simple drawing
instruments,
such
ruler,
as
protractor.
Introduce
5.
of
simple figures. The
some
work
new
to
rapidly.
more
of
the
constantly
them
done.
previously
A
applying
general hints for the teacher's
learned.
already
of
volumes
and
areas
without
meaningless
both
are
drawing
devices
latter
The
scale
to
and
the
picturing
previously referred
were
of
to
conditions
under
the
in
use
object teaching.
Avoid
6.
them,
"
the
especially
7.
Pay
8.
Place
special
a
continual
as
to
standard
forms
position.
attention
premium
of
use
to
upon
local
applications.
neatness
and
accuracy.
of
figures
"
^vary
259
MENSURATION
Construction
267.
carried
triangles
it is
the
up
of
declare
to
drawing
perpendicular
be
may
any
taken
up
their
268.
Inductive
be
may
1.
for the
on
Line
out.
study
of
order
of
Such
structions
con-
parallel lines,
in
needed
as
so
A
they
or
on,
or
structing
con-
may
AE
work
to
are
in detail
who
out
contain
not
following
these
ABCD
of
(construction will
or
paper
do
The
parallelogram
is constructed
heavy
Formulas.
students
which
texts
Parallelograms:
results)
of
Proof
teaching
size
teacher
application.
suggestions
mere
one
best.
or
triangles, parallelograms, and
precede
the
the
drawing
the
that
with
unquestionably
as
given size
a
urging
along
impossible
work
as
angles
work
also
are
be
pupils
the
when
triangles they
for
should
work
that
so
Except
construction
mensuration,
taking
of
areas
accurately.
the
carry
the
Construction
mensuration
with
along
studying
are
Work.
thin
cardboard
venient
con-
any
give better
which
ABE
Triangle
is drawn.
tions.
explana-
is then
is cut
from
cut
the
D
B
figure
the
and
2.
out
placed
in
the
rectangle AEPD
ABCD
base
C
E
whose
and
altitude
Triangle:
two
equal
parallelograms
position DCP.
of the
EP
base
and
of the
area
same
as
altitude
AE
parallelogram
parallelograms
are
This
pupil is directed
Each
now
F
divided
as
by
was
the
are
will result
in
parallelogram
equal
to
the
ABCD.
to construct
done
cutting,
and
above.
one
along
cut
The
the
260
MODEBN
ABITHMETIC
longer and
the
other
also the proper
from
three
to include
place
The
in
order
with
acquainted
the
Trapezoid:
for
formula
The
formula
of
RighULine
figures for
which
divisible
them
Circle
5.
heavy
ruler
cardboard,
or
paper
is rolled
Let
again.
circumference
which
will
etc., and
will
as
trapezoids.
class
value
See
of
make
rule
on
the
of
area
circle
shown
is the
to
sum
come
circumference,
advised
proved
call
to
a
The
used
ir
of the
straightedge
as
explanation
Each
is
sector
of
nearer
the
of
areas
and
will also be found
=
fact
now
this ratio
which
its decimal
and
circle
into
equal
in
of
formula
the
of
the
triangle
total
area
triangles and
B^.
that
for
whose
of the
can
i radius
formula
'r
usual
The
this
teacher
formula
be
X
is
is
the
problem
interesting. The
inscrib-
The
very
may
The
the
nearer
to. the
teacher
used,
shown
sort
a
the
oriJ2X2X'rX^
attention
has
tt
the
this time.
at
rigorously in geometry.
finding
name
own
of the
measure
he
for
value
division
The
The
his
touches
diameter
3.
of
mark
this
divide
now
by the triangle formula.
is found
area
the
use.
common
circle.
a
until
also be
along
most
ruler
from
out
circumference
the
on
cut
radius
a
mark
closer
much
Area:
"
placed
is
text
should
circle with
all close to
thereafter
3.1416
sectors
a
a
child
each
length of the
the
3^ is
will
Circle
6.
Let
some
child
each
give values
that
explain
along
by
a
with
Beginning
circle
the
of
given
are
and
Circumference:
"
choosing.
the
dimensions
triangles
also be found
may
265.
page
the
triangles,
two
hexagons,
pentagons,
as
such
into
trapezoids is
of
area
Areas
Figures:
regular figures
some
thoroughly
is known.
for which
Oeneral
4.
triangles
altitude.
and
the
is
shaped triangles
trapezoid into
a
This
of
become
may
base
of
easily developed by dividing
the
of various
pupils
meaning
PBOBLEMS
construction
the
use
the
that
AND
shorter, diagonal.
the
along
given parts.
is advised
3.
METHODS
history
of
261
MENSURATION
circumscribing
ing of circles in and
vice
furnishes
versa,
Bight
7.
between
relations
Take
of
sides
the
right triangle using
the
measure
of
the
be
in
found
the
upon
that
out
that
given
For
is well
inaccuracies
in
the
sufficient
law
take
the
it up
in
the
in square
it in
the
school
same
is the
root,
with
the
If
proof is
sides 3, 4, 5,
for
or
a2
Hence
4- 62
a^
factors
now
(c +
be
")
be
any
c^
=
and
thus
As
for
"
show
rule
a
are
The
will
the
those
ing
follow-
form
a
:
(2)
62
(c
(c4-6)
(c
lated
stimu-
(1)
"
selected
of
ing
subject, by introduc-
c2
=
the
on
existence
integers which
as
sidered
con-
better
no
can
curriculum.
=
may
for
interest
dull
simple
be
this work
multiple of these.
selecting three
as
will be
integers for sides
with
some
is to
right triangle and
the
text
right triangle is about
two
to
results
this
root
cause
More
otherwise
an
with
triangles in the
into
point
to
topic, a knowledge
with
primary
necessity for its place in the
square
before
there
year
curriculum.
only
Any
erected
by geometry.
square
connection
in
than
connection
scheme
sides will
give inexact
of the
If
is necessary.
root
root
sum
inability
and
used
be
then,
The
teacher
rigorously
consideration
right triangle which
square
the
two
class.
during
to
and
on,
a
the
closely that
will
accuracy
proved
for
perpendicular
for
the
up
construct
possible.
drawings
of course,
may,
complete
square
time
be
can
so
as
quite
equal
to
It
the
the
the two
upon
relation
the
taking
lengths
accurately
as
cases
the law
to
convenient
hypotenuse.
the
necessary
of
all
with
measure
but
erected
of
pupil
each
3, 4; 5, 7; and
as
hypotenuse
squares
construction.
study
Let
figures.
any
sides,
perpendicular
the
up
and
squares,
right triangle before
a
solid
between
in
problems
excellent
Triangle:
about
them
a^
6).
"
6)
which
The
(3)
is factored
sum
of
the
262
MODERN
latter
gives
two
factors
of a^
will
2"
ARITHMETIC
be
both
Suppose
2c
selected
so
be
both
be
6
and
hence
for
then
a
PROBLEMS
difference
to
as
and
even
is selected
12
their
and
AND
26.
will
c
above
the
odd, 2c and
or
even,
the
Should
be
integers.
becomes
6) (C"")
(0 +
(72)
which
from
METHODS
2c
=
26
Hence
there
integers
(36)
or
(4),
74, 40, 30, 26,
hence
c
=
70, 32, 18, 10,
hence
6
=
=
the
are
and
(2),
37, 20, 15,
37, 35, 12
:
20, 16, 12
:
(24)
or
(6)
13.
5.
35, 16, 9,
following right triangles
12
one
(8),
(18)
or
with
all sides
15, 12, 9
:
13,
:
12, 5.
This
joint
the
of
areas
Cube
8.
two
the square
and
that
of
pipeds
finding the
arise
bear
rectangle do
finding the volumes
of
Parallelopiped:
parallelopiped
of
seldom
so
the
that
former.
not
solids
method
similar
to
parallelo-
Oblique
need
and
to
The
absolutely
is
they
cube
relation
same
latter
of the
The
plane figures.
to
the
areas
equal
areas
circles.
more
or
Rectangular
and
rectangular
as
of circles whose
permits the construction
in
come
for
consideration.
Bight
9.
to- the
analogous
curved
Cylinder:
surface
of
around
of the
cylinder,
dimensions
cylinder.
are
the
draw
from
of the
order
cylinder
a
and
paper
In
in
treated
a
other
a
a
one
the
cylinder.
cylinder in the form
the
height
and
the
the
does
the
to
entirely
an
than
more
of
of
the
altitude
the
ference.
circum-
paper
a
the
piece
exposed edge
The
of
area
rectangular
dimension
little
line along
the
find
to
wrap
cylinder, with
the
Then
surface
therefore, be
relation
same
triangle
the
as
the
manner.
Right
the
right prism bears
parallelopiped
should,
and
paper
the
The
rectangular
rectangle
10.
Prism:
and
will
wrap
un-
show
rectangle whose
circumference
of
the
264
MODEBN
of
area
4
the
sphere
must
of
the
volume
of whose
radius
base
from
made
the
Hence,
volume
usually found
the
noting
269.
to
industrial
such
of
lines.
problems,
however,
oflfer sufficient
interest
the
domestic
conditions
assigned
to
why
reason
Similarly
to the
take
may
as
Cutting
they
should
permitted
the
doU
place
of
possible; cutting
2.
sides.
Areas
of
similar
out
gain
problems
those
on
dustrial
in-
upon
the
boys.
in the
lems
prob-
to
these, also, there
to
to
do
those
solve
so.
signed
as-
sufficientlyinterested.
are
MATERIAL
's clothes
cloth)
much
will
permitted
be
since
too
work
the
solve
material
teacher
given
be
to
not
be
boys if they
out
this
that
Problem
composition
sufficientlyinterested
PROBLEM
1.
economics
**
entirely
girls and
the
to
girls should
the
the
gives if
girls to desire
the
and
presents
actual
take
community
the
become
boys
the
case
are
water
contain
to
left to
be
would
given
of
of
home
along
The
suggested
be
economy
vessel
a
found
be
motivation
that
of |.
R^.
tt
sequel entitled
The
must
variety
in
ratio
Mensuration
problems.
such
f
that
and
in the
be
inder
cyl-
a
irregular solids
in
problems
given below, will
number
is R
and
displaced.
for
opportunities
any
no
circle,
2R
radius
of
Problems.
for
is
of such
will be
them
of water
It is furthermore
is
great
a
weight
to
Volumes
by immersing
various
found
be
sphere
Solids:
space.
In
of
the
hence,
;
altitude
clay whose
-R will
Suggestions
Material,"
of
twice
that
The
R^.
ir
or
of the
amount
excellent
for
putty
Irregular
13.
and
times
cylinder whose
22 is 2
sphere of radius
a
PBOBIiEMS
hemisphere
four
be
AND
R^.
IT
The
of
METHODS
plane circle of the
the
cover
or
ABITHMETIC
from
in
patterns placed
such
boxes
in
figures
vary
a
the
way
same
as
the
to
make
way,
etc
as
square
eloth
upon
on
as
little
(paper
waste
correspoiLding
265
MENSURATION
The
3.
yaries
Irregular
4.
A
figure.
rule
Divide
that
twice
often
the
2"vltiply by
of
the
into
length
of the
This
whose
of surface
divided
by three
the
the
dividing
by
records
into
figure
of
that
equals
areas
number
of
trapezoids
the
of
the
sum
verticals, and
even
phonograph
record.
given
the
following:
even
an
of
part
separate
is the
play
to
found
often
used
area
takes
plajing
are
sum
it
that
the
areas
the
trapezoids,
of
width
the
as
time
of
leagth
the
times
that
number
of
vertical,
last
first and
times
four
mentioned,
as
odd
of the
divisions
ticals.
ver-
the
gives
area.
Excavations
veal
in
link
of
links.
is
30th
the
to
for
the
6.
of
In
acreage
7.
The
ordinary
link
the
of
line
AB,
The
links
30
the
B
the
If
at
is, then,
drawn
line
40
BC,
The
long.
in
B,
links
line
a
taut
long,
of
proof
dicular
perpen-
is
desired
BG
out
end
the
out
making
running
lays
surveyor
is held
of
purpose
and
land
surveyors.
chain
chain
for
of
acre
point
the
B.
from
borrowed
of
the
the
0;
at
an
manner
at
end
One
pinned
out
the
BO
to
be
by this rule.
usually computed
are
may
laying
after
perpendiculax
40
chain
work
the
Hnes
to
embankments
Gunter's
A
5.
and
of
from
will
this
equal
the
be
fact
80th
the
end
dicular
perpenis left
student.
rural
which
communities
a
architect
building.
man
uses
the
and
team
i inch
cost
can
in
of
do
his
farm
per
plans
labor
including
day
offers
to
denote
amount
good problems.
1
foot
in
an
266
MODERN
The
8.
ARITHMETIC
of
drawing
furnishes
excellent
METHODS
eehool
problem
work
Use
10.
local
One
rules
is meant
and
gardens
of
is considered
Volumes
of
the
ratio
in
of
such
BC
is
supposed
good
a
to
for
similar
figures
of
cylinder containing
vary
the
as
and
250
hanging.
sq.
one
man.
cubes
of
figure.
lathings
as
paper
cover
's work
day
in the
to- AD
computations
sidewalks, painting,
paint
scale
to
D
prices
cement
gallon
application
12.
roof
a
plastering, laying
11.
school
C
of
pitch
By
of
measurements.
on
A
9.
and
grounds
PROBLEMS
AND
and
ft.
its
corresponding
sides.
The
13.
surface
the
when
and
so
14.
is twice
depth
material
any
for
radius
the
problems
on
the
of
the
a
giv^i
volume
This
base.
is least
lent
excel-
provides
construction
of
cisterns, fruit
formula
in
approximating
jars,
on.
Lumbermen
number
i [smaller
is here
L
in
feet
board
of
the
use
end
in.]2 XL-i-S
in
given
feet
XXV.
1. Give
brief
a
the
log:
a
4
"
following
and
the
of
number
=
smaller
in
end
feet.
board
inches.
EXERCISES
summary
of
mensuration
the
measurements
developed
as
by
the
ancients.
Give
a
3.
Give
reports
the
cone,
4.
workf
surface
and
What
of
of
summary
the
on
of
sides
the
between
of
brief
2.
a
development
the
cone,
of
the
right triangle;
pyramid,
and
of
the
formulas
area
sphere;
of
circle.
for
the
volume
the
relation
triangle;
of
a
area
pyramid,
sphere.
use
can
be
made
of
these
historical
references
in
grade
267
MENSURATION
What
5.
in
work
should
6. With
allied f
what
other
fully
should
primary
the
completed
f
mensuration
should
topics
doselj
be
illustrate.
and
mensuration
given
be
rationalization
how
Explain
have
mensuration
about
two
Discuss
7. When
8.
know
arithmetic,
who
children,
the
in
curriculum
grade
the
pertains particularly
to
f
tion.
mensura-
Illustrate.
What
9.
10.
How
is meant
by
rigorous
need
inductive
an
formulas
of
proofs
the
Illustrate.
f
proof
for
be
classes
grade
f
Illustrate.
Explain
11.
the
12.
for
formulas
various
How
children
the
how
2-inch
many
of
areas
led
be
can
to
for
prove
themselves
plane figures.
be
can
squares
from
cut
3
rectangle
a
8
by
inches?
Compare
13.
the
the
triangle having
14.
side
How
walk
15.
an
The
of
18.
per
A
other
for
21.
Find
is
of
cent
how
house
a
latter f
the
much,
of
cent
per
36
ft. in
sq.
and
6
are
The
with
of
that
latter
20
garden
a
the
latter f
If
area.
ft.
10
lead
you
to
apart
of
cost
Draw
and
is what
yards
square
The
latter
is what
altitude
the
what
respectively,
and
is the
of
one
length
of
the
It
is to have
The
floor
is to
of
area
to
plastering
at
papering
the
develop inductively
the
formulas
circle?
a
windows
scale
a
cottage
a
class
a
a
18'
room
(
cents
)
above
13'
X
X
per
yard.
square
40
at
room
9', containing
cents
per
roll
work.
8' with
X
of
three
and
X
and
cost
and
the
14'
20'
the
material
22.
of
f
circumference
Find
for
sides
would
doors
two
floor
side?
How
the
5'
a
former?
trapezoid
the
for
required
be
rods?
60
ground
and
is what
the
parallel
will
cement
by
the
per
larger
yards
former
of
equilateral
an
f
former
is the
the
20.
is what
square
cent
19.
the
The
of
that
plans.
Which
20
50
of
area
former
of
cent
of
yards
park
a
the
architect's
17.
or
inclosing
with
square
a
perimeter.
same
square
Compare
16.
per
many
of
area
door
rest
in
on
the
f pitch
is sealed
an
2
end
X
elevations
two
the
to
only
and
a
6 's nailed
of
roof.
on
The
the
side,
to
X
4'8
X
are
with
windows
2
cottage
summer
outside
and
the
a
2
4 's.
placed
drop
on
both
2'
siding.
side^
268
MODEBN
23.
the
Compute
be
three
Siaggeet
24.
of
amount
will
which
weather
METHODS
ABITHMETIG
and
lumber
shingles laid
the
construct
to
necessary
PBOBLEMS
AND
problems bearing
on
above
the
to
cottage.
suitable
areas
4"
for
grade
a
class.
l7wo
25.
15'
centers
%**
pulleyB
How
apart.
placed
are
long
parallel journals
on
belt
a
is
needed
with
their
connect
the
two
longitude
in
your
to
pulleys f
ifl the
What
26.
latitude f
rotation
How
of
How
27.
earth
28.
many
If
circle
remaining
part
a
circular
a
How
f feet
square
many
around
5' wide
miles
many
the
in
length
second
of
yards
move
you
as
result
a
of
the
f
surface
5
walk
cement
a
f
acres
30'' automobile
in
there
are
park containing
a
of
degree
a
do
hour
per
per
will
turns
of
miles
make
wheel
in
traveling
milef
29.
lathe
A
30.
a
oi
in
is cut
circle
from
will
h, what
be
the
area
of
the
h?
the
manual
training
and
rooms
the
shaft
above
has
diameters
are
3
o
%m
each
a
4", 6",
set
8"
of
pulleys arranged
respectively.
Compare
as
in
the
the
figure
speed
of
whose
the
lathe
with
that
of
269
MENSURATION
fhe
shaft
and
when
the
belt
about
passes
pulleys
and
a
A, h and
B, and
e
C,
31.
Suggest
32.
How
would
relation
the
farther
fielder
first base.
the
children
sides
the
circles
on
of
does
suitable
right
a
the
discover
to
for
grade
the
law
classes.
governing
triangle f
catcher
have
to
throw
right
foul
line
210
throw
to
second
to
first basef
to
A
34.
lead
you
much
than
base
problems
between
How
33.
three
catches
How
ball
a
farther
much
the
on
he
must
third
ft.
than
of
back
to
second
basef
Suggest
35.
equal
not
36.
to
take
of
38.
Suggest
39.
How
40.
41.
and
Find
A
44.
14"
the
size
46.
47.
volume
cone
of
of
f
surface
bring
and
how
in
up
would
parallels, and
work
find
to
cylinder,
a
of
of
angles
a
plane figures.
on
the
cone?
a
altitude
whose
cone
surface
of the
area
and
with
pyramid
a
gallons
windows
that
of
the
floor
of
there
are
and
the
upper
tent
is
Draw
in
a
square
is
12'^
base
S"
rainfall
1"
a
of
is
cone
radius?
8'
would
the
you
whose
an
over
sector
lead
a
rectangular
Find
altitude
of
6'
entire
base
area.
radius
and
and
this
lower
a
the
scale
Will
schoolroom.
your
has
equal
the
measure
differ
angle
of
as
f
in
pupils
solid,
making
yards
many
the
square.
used.
is used
How
for
is Z"
to
should
room
altitude
pattern
circle
changed
6"
a
have
the
lighted
Investigate
base
to
well
a
space.
accurate
an
sector
in
pyramid
a
angular
an
How
the
which
5*^.
the
the
What
pupils
surface
the
of
the
with
and
When
figures.
construction
your
area
4'.
of
angle
in
would
you
perpendiculars,
for
thousand
conical
base
but
10".
of
frustum
A
integers
f
square
45.
base
many
20%
about
of
entire
the
The
lead
you
the
section
43.
its
of
How
quarter
classes
grade
work
plane
of
problems
area
altitude
an
of
all
are
used.
construction
the
solid, a pyramid,
the
Find
42.
a
six
would
radiue
and
for
be
may
construction
rectangular
a
suitable
35
sides
sizef
given
a
of
whose
5.
3, 4,
mensuration
the
up
Ex.
outline
with
of
problems
for
up
an
connection
of
three
made
Give
37.
right triangles
multiple
some
Suggest
triangles
you
three
the
to
of
a
tenting
develop
pyramid,
conical
the
the
will
tent
it
formulas
cylinder,
8'
high
require!
for
and
the
the
270
ARITHMETIC
MODERN
A
48.
volume
294
of
49.
radius
A
50.
from
much
of
forced
in
immersed
diameter
54.
Of
Apply
What
56.
cylindrical
57.
59.
the
Find
radius
a
a
its altitude
of
10".
pare
Com-
f
a
8"?
radius
the
How
surface
3"
of
X
of
a
4"
the
level
would
water
2"
is filled
raises
water
much
brick,
a
entire
and
X
with
largest cylinder
the
turn
10"
into
stone
stone
area
of
high;
4.5"
of
tomato
have
had
8",
X
been
of
can
350'
4"
of
one
reservoirs
radius
and
the
same
conclusion?
is your
and
of
in
the
in
54 f
Ex.
general.
economical
most
barrels?
material
long,
with
53
principle developed
cisterns
60
Ex.
What
the
depth
of
in
can
is
use
holding
loads
the
high.
the
be
many
9'
were
high,
and
build
to
necessary
24'
base
upper
road
rail-
a
up
wide
and
a
wide?
52'
bases
equal
of
building
Construct
58.
has
height.
cistern
base
IS''
base.
in.
has
X
dropped
f
embankment
lower
the
cu.
to
high
the
should
How
10"
if
practical
the
12"
cylinder
3.5"
it to
40
away
wood
stone
the
the
what
cut
of
4"
and
55.
of
volume
Compare
volume
be
of
the
and
Ex.
volume
place
the
Find
53.
must
A
out
of
of
areas.
block
water.
is the
what
been
in
one
cylindrical glass jar,
A
full
4";
to
altitude
an
is 770
cone
PROBLEMS
7",
the
wood
rectangular
a
52.
base
and
volumes
How
51.
the
is
and
dimension
the
circular
a
AND
base
square
Find
of
similar
cone
their
half
of
a
in.
cu.
volume
The
if the
with
pyramid
METHODS
from
heavy
and
triangular pyramid
a
paper
of
prism
altitudes.
and
Construct
a
circular
and
cone
of
cylinder
equal
bases
and
altitudes.
60.
What
61.
Outline
with
out
62.
the
a
and
be
formed
66.
for
models
these
work
in
grade
have
you
would
for
grade
work
the
surface
work?
your
class
carry
solids.
suitable
in
mentioned
formulas
the
Ex.
involving
each
of
47.
of
area
and
the
volume
developed inductively?
the
inches
and
areas
volumes
of
spheres
whose
radii
are
3, 4,
respectively.
Three
How
of
problems
two
Compare
into
of
construction
solids
sphere
65.
the
Suggest
can
10
made
measurements
How
64.
be
can
the
the
regular
63.
of
use
of
radii
single sphere.
What
clay spheres
a
much
wood
is cut
largest possible sphere?
1, 2,
off from
and
3
inches, respectively,
are
is its radius?
a
wooden
5"
cube
in
turning
out
oHftpm
zxvi
Hittorjr.
270.
rbots
ing
of
squares
of
the
which
the
1
about
4000
years
the
irrationals
of
sides
being
the
extraction
271.
Powers.
of
teaching
used
text
(15)2,
as
does
for
Several
little
][)g
g^
short
(a-|-6)2
topic
the
contain
attention
9.3
which
cuts
and
to
should
large
is written
in
involution
{a
"
is
quite
called
be
b)^,
272
3
to
(10)
the
the
7th
use
sm^l
as
or
8th
of
this
miles,
numbers
cm.
based
etc.,
the
notation,
93,000,000
very
the
If
simple.
to
as
and
eliminate
regarding
it in
mention
numbers
miles,
(10)''
be
more
general.
said
to
ing
find-
of
to
in
the
During
studied
being
exponential
the
plan
good
writing
cm,
needs
the
of
process
numbers
of
between
advisable
seems
^ell
idea
the
irrational.
is
fined
con-
as
recognition
the
were
of
is
perpendicular
the
systematic
it
roots
as
not
is written
.000003
forms
powers
Special
notation
cube
Very
\i ^jii
grade.
which
of
to
arithmetic
pupils
younger
led
no
As
of
case
which
course,
known.
by
the
doubt
roots
of
in
with
relation
The
nlins
Greeks
The
of
squares
Euclid
of
acquainted
time.
as
no
square
was
more
2,
times
early
irrationals.
of
BabyioniaH
b6ok
One
table
a
table
a
in
fonnd
old.
this
triangle,
and
1
of
roots
and
right
a
32
are
and
60
JEundEmebtal
sixth
and
ettriict-
aiid
powerd
records
to
doubt
no
before
long
sides
existence
were
1
to
of
treatment
Hindus
these
from
from
are
fifth
existing
integers
to
as
earliest
integers
the
the
isi seiise
The
operations.
to
humbers
Bailing
in
are
ftOOtS
AM)
POWERS
upon
are
both
the
algebraic
interesting
HW
IPDWBKB
ahd
they
Tisdftil.
which
(68)"
number
the
5
5)2
(10a +
iOOa?
_
is 100
3 X
X
notice
to
is hence
and
4
annexed
to
(352)2
(a+6+c)2
gives (352)2"
-fllOOa
1
+
"
kk
(52)^
ail9
2)2.
p^
khy
^5
+
l) +25.
4, hence
"
rapid
the
the
place of the
of
product
lOOa
it is
work
(300+50+2)2
_
"
a
For
takes
25
that
number
The
3,
"
1200.
^
cftses
(60
iOOa(a +
(35) 2, a
to
and
273
first becomes
"
Applying
2)2
(50 +
in
stlch
ih
u^d
Me
become
ending
JKDDTS
only
last
(a +
a
(a +
1)
sary
neces-
two
zeros
1).
from
which,
a2+62+c2+2a6+2a*+26c,
(300)2+(50)2+(2)2+2X300x50+
2X300X2+2X50X2
3522
092504
3020
12
123904
This
work
form
is
explained
from
of not
272.
the
Boots.
by
with
the
a
number
number
must
extraction
must
be
has
advantage
the
cubed
be
squared
to
get
roots
the
8?
as
pupils
to powers,
be made
may
of
After
power.
raising numbers
of root-extraction
What
It
expansion.
tained
ob-
those
carrying.
any
to
figures with
its
This
form.
accompanying
comparing
Teach
raising
acquainted
given in the
above
requiring
of
verse
be
can
a
the
are
iivwell
tion
good introduc-
by asking incidentally :
to
get
27?
36?
and
25?
so
on.
or
what
Boots
274
MODERN
of
METHODS
ARITHMETIC
numbers
simple
are
time, simultaneously
object of this work
found
now
raising
with
teach
is to
AND
PROBLEMS
to
As
powers.
relation
the
for
mentally,
little
a
the
between
main
numbers
0
the
should
examples
and
number
the
number,
273.
Square
because
of its
of
Boot.
form
of
it is used
times
is composed
32
as
the
take
of 2 used
It is necessary
repeated
produce
to
a
factor
to
teach
as
in mensuration.
use
factor
finding the
given
a
5 times.
root
square
First
of all the
a
pupils should
form,
do
The
the
274.
**
of this
to
Carpenters'
Rule"
equally good.
can
BuIq"
be used
Often
those
used
in
form
for
in
all the
mixed
the
upon
pupils
actually finding
Square
place
Root.
of
the
who
explain
to
insist
not, however,
must
the
be
may
explanation by
use
ing
accompany-
liable to get badly
drawing
Teachers
the
minutely
closely are
form
permitted
are
to follow
one
accompanying
understanding
they
other
any
below.
form
required
some
follow
not
up.
or
be
an
before
roots.
The
penters'
''Car-
above
form.
AND
POWEBS
universal.
is almost
which
is desired.
17
is
square
the
than
less
4.25
The
H
first
of
mean
is
4.125
3 and
takes
now
and
Thus,
is
4.1225,
the
old
process.
by
As
For
3^
and
is about
Forms
radicals,
of
4^
the
what
of
is
old
gives
process
in
found
in
of
root
the
both
the
usual
by
terms
denominator
the
manner
to
numerator
VH=^V"-
=
special
changed
be
to
multiplying
by
certain
A
deserve
on,
is
radical
the
VS
so
first
are
required
the
now
number
the
exact
root
V12=
under
3
X
multipjied
by
2.
to
in
extract
and
often
on,
for
several
the
=
give numbers
which
decimal
the
2
that
V3.
These
case
radical
the
taken
be
can
V4
as
divide
we
a
can
The
square
and
divided
5.
If
the
15
which
instance,
For
betaken.
by
that
and
fractions
accomplished
is
such
number
V75,
Vf,
in
ones
This
only.
proximatio
ap-
4.33
that^JDeserve Study.
as
All
consideration.
equivalent
first
work.
more
Radical
276.
the
is just the
of 3 and
mean
and
and
the
^/lS
is
.
which
much
first
the
,
root
one
and
root,
4.123, which
or
The
get 4.333
3.666
class
the
=*'12,
J77I7
is between
13
of
place
the
4.12
suflSces.
after
gives
and
of the
approximation
repeated.
found
number
is
4, which
number
last
this
divide
Second,
17.
just found,
one
of
first
a
number
process
by
number,
mean
4, gives
This
4.125.
13
given
4, whose
case
.
found,
the
number,
of
root
square
this
in
the
17, by the
given number,
the
the
that
Suppose
First, find
275
BOOTS
is
has
placed
The
of
root
give
the
required
root
arranged
radical,
has
as
been
in tables
of
the
which
radical,
is then
3
only
fractions, give smaller
under
places and
factor
outside
square
simplifications
of
a
root
one
numbers,
2, 3, 6, and
computed
on
next
so
to
page.
276
MODERN
ARIT^^^T1C
EXEBCI8E8
XXyi.
1. yfha,% are
tl^e fbc f1^^4al9$l^ta)pperatiansf
2.
irr^tioiial"wl)|9ifst lUuatxa^.
W^X
arci
siiotiad
aguMCi
3* Wxx
grade
PBOBLEMS
^ND
Ml^T^ODS.
an^
toot
^
Tao,(b^ ^^^^
cube
not
^ft
arithnieticf
'
in
Express
4.
fiiqupleindex
notation:
.000,000,56,
56,000,000,
34,000, .000,075,2.
5.
indices:
without
following
the
Express
5.4(10)6,
3.8(10)-4,
6.3(10)-", 9.3(10)7, 4.2(10)-".
6.
Giy^
orally
the
following
(^5)2, (35)^ (95)\
squares.:
(%)^
(125)8,,(54)?, (3?)2, (4a)2, (68)?, (98)2.
7. Give
VS^, VIST,VTIB; VlSl?;VR
th9 fqllowing orally:
^8^,
^"215;viBsr.
8.
inverse
9.
Explain
of
hpw
raising
How
relations f
10. Find
would
you
to
a
Suggest
power.
involution
can
introduce
of
small
the
subject
at least
numbers
six
be
of
roots
as
the
examples.
used
to
tj0aeh aumber
^llustrat^.
t9
^ecim^l plw^,;
thr^
V?P?"
V^^l^
Vl^4fc
V 67.2348.
11.
the
Explain
drawing
as
to
a
class
accompanying
the
it.
derivation
of
the
form
imder
273
from
POWERS
Find
JL2.
Find
13.
14.
y/W,
What
out'the
"
at
ten
What
cU8|gQnal
as,
by
the
VM,
is
advi^tage
the
table
VT
vT,
V2S",
the
by
using
y/1,
vf,
277
BOOTS
under
276:
^
v^lST,
V^ilT,
simplifying
of
V^
Rule":
"Carpenters'
V^
forms
the
before
ing
carry-
computations?
least
1^.
values
vis;
"i^lain
16.
Ipng
following
VT,
VT,
15.
following
the
the
AND
the
illustrative
the
ali"ove
to
a
claf3.
grade
of
9f
tl|i9 placp
'"
the
distance
around"
a
is
square!
saved
by
A
going
reQt%2|^lp
wide!
BIBLIOGRAPHY
V
Ci^joxiy
2.
Boots
F.
p.
Sv^est
examples.
percentage
in
of
tea^stu^g
51.
by^ I'napectipn*
Scientific
American^
GYIII:
354.
along
t^o^
a
9fi
CHAPTEB
GEOGRAPHICAL
276.
MEASUREMENTS
Province
latitude
of
time
and
arithmetical
treatment
in
it
where
naturally
questions
these
be
taken
children
live,
learn
its
of
because
about
candle,
they
upon
the
locate
used
which
learn
can
be
globe
at
a
all
of
meridians,
the
size
the
time
this
earth's
the
houses
having
numbered
are
lines
the
two
little
very
the
properly
elementary
early
the
ball,
huge
of
of
of
principal
similar
a
its
about
lines
as
just
have
lighted
and
pointed
or
explained
two
a
we
parallels,
apple
surface
that
apple
or
earth
equator,
the
and
globe
a
the
grades
which
upon
proximity
principal
278
of
arithmetical
very
part
means
The
and
geography
treatment
in
a
rotation
upon
in
can
In
is
close
night.
Towns
which
earth
drawn
reference
in
the
the
and
upon
town.
Subject.
the
chapter
a
geography.
the
in
that
day
of
By
arithmetical
the
consideration
flat.
be
axis
an
places
in
to
us
produces
meridians
that
immense
to
seems
rotated
axis
of
for
given
from
teachers
province
the
Introduction
277.
the
within
lies
schools
questions
successful
be
for
their
but
is
tude
longian
the
placed
a
necessarily
discussion
here,
up
being
As
belongs.
must
their
standpoint
of
instead
time
and
longitude
involved,
of
to
requiring
quantitative
Because
'
relating
between
questions
are
geography.
arithmetic
the
connection
the
to
they
;
questions
All
geographical
are
consideration
in
and
TIME
AND
Study.
the
longitude,
and
principles
XXVn
as
used
streets
streets
ready-made
to
streets
out
parallels
correspond
to
are
from
tration
illusand
to
280
ABITHMETIC
MODEBN
pupils
ahooild
directed
haw
they
specific.
so
279l
up
clearing
the
in latitude
facility
are
between
longitude
explain
print
5 P. M.
7 P.
fact
time
be
can
this
of the
the
that
the
profitably devoted
tt)
fact.
at
above
number
of
to
minutes, and
factor, 15, gives the
understood,
Hence,
the
is not
candle
and
a
this
the
in
a
good
use
stances
in-
different
little
notion
clear
of
change
study, and
of
the
can
globe, inclined
at
similar
manner
to
sun.
lem
fully understood, the prob-
are
simple
in
one
evaluation,
multiplied by
seconds
difference
seconds
multiplied by
form
of
the
longitude
in
arc.
Time,
giving
15
of
a
solution
in
in
expressing the difference
accompanying
globe
Some
of the
during
the
a
and
in
explaining the
at
getting
moving
the
for
can
place
into
meridian.
by
matters
units
minutes,
time
some
candle
occur
explanation
An
about
eiarth
itself
degrees,
night
saipe
on
and
ing
interest-
as
took
brought
be
truth
and
noon,
not
reduces
hours,
again
existing
Francisco
which
The
day.
ference
dif-
by asking the
San
event
the
relation
made
in
an
same
this strange
angle, about
of the
of
can
demonstrated
proper
When
the
to
is in order
easily be
can
places with
two
the
be
can
the
important
seasons
and
lines
find
can
A startling,as well
account
M.
who
of
newspaper
a
the
places
in
Pupils
this study,
morning^
that
made
addition
township
and
study
time.
how
clear
making
the
and
previously referred
for
be
not
of
matter
longitude between
in
for
to
clfiss to
at
need
left to
be
can
selves.
them-
previously mentioned,
range
Tiixie.
and
ready
introduction
Boston
this
country,
an4
Longitude
at
references
in
for
process
more
be
should
reference.
for
used
and
more
the
even
the
out
intersections, as
In
subtraction.
be
and
do^ but
to
Street
used
be
may
find
to
older,
grow
discretion
what
told
where
and
As
their
be
never
BBOBLEMS
AND
METHODS
which
time
in
converting
expressed
in
it must
be
product in
is not
to
arc.
be
MEASXIBEMENTS
GEQGRAPHICAL
AND
12 min.
3 hrs.
32
281
TIME
sec
15
8'
48""
Tfaisi poixit
accQpted.
QKen
the pupils
uusisited upon
at aU
This
cau
of
solution
numbers
by
names,
isesults obtained, or, it may
The
inverse
reciprocal
the
process,
multiplier
form
will
inverse
be
have
we
and
found
pro^duct
a
)48. hr.
3 hr.
pupils
for
of
to the
of the
use
thing.
same
Treated
as
a
multiplication in which
one
for
the
32
sec.
34
sec.
aecqmpany^ng
solution
of the
prohlen^.
15
Often
by the
given. The
good
very
of the teacher.
simple.
as
undo
are
when
matters
interpretation of
15, and
fully
to
fee
too
by either multiplicatioii
really amounts
is
problem
auch
accomplished
be
form, which
earned
merely hobbies
aceomplished
without
accompanying
feel that
to
tixxke"are
be
he
however,
caa,
made
are
//
0
and
minutes
time
and
8 min.
"
12,min.
"
teachers
even
and
write
hr.
"
carelessly use
of
seconds
3
"
6'
14"
arc
in
in
the
viation
abbre-
the
making
a
place of
ment
state3
hr.
282
MODEEN
min.
6
14
Such
sec.
Standard
280.
United
the
States
time
solar
of Mexico
use
Greenwich
are
subject
arising
the
into
the
The
the
to
one
and
name
result
of
its
of
extent
the
which
the
four
A
of
other
east.
standard
prominent
of
this
following points
of
system
Nova
the
study
is based
zones
including Labrador,
east
Spain
deserves
of the
upon
from
and
of
use
importance.
consideration
resulting
of the
meridian
this
of
most
time
of
15th
subject
the
principle
zones
the
a
of
include
fundamental
country
2.
as
bv
Isles, France,
of
mean
over
the
most
into
integral
an
adopted
and
that
simple, but
because
must
The
1.
tolerated
advantages
Mexico
Germany
use
1883, is
been
British
the
;
time;
consideration
Its
Throughout
is used
very
in
it has
that
great
so
problems
time
be
never
time, introduced
railroads
the
by
countries
European
Standard
countries.
City
The
PROBLEMS
must
affairs of today.
are
civilized
the
AND
carelessness
Time.
of business
part
METHODS
teacher.
the
by
AEITHMETIC
division
the
United
Scotia,
States
New
and
of
time.
standard
the
:
and
Found-
land.
3.
Beason
4.
The
for
variation
5. Names
the
of
Seasons
281.
the
15
he
irregularity
in
some
for
time
of
lines
boundary
6.
the
of
International
^^
would
The
which
he
hour
per
be
at
would
back
change
in
roughly
date
usually referred
strictly speaking
no
to
and
there
international
within
and
upon
be
have
is
made
such
he
when
he
meridian.
line
in
recognition.
24
of
rate
hours
it would
change
crossed
This
date
as
the
at
and
;
did
international
no
west
noon
from
started
man
meridian, but
180th
the
a
time.
traveled
is, when
would
the
If
prime
query
as
situated
standard
of
Line.
noon
the
follows
cities
zone.
always
at
zones.
zones.
principal
Date
doiy. The
next
the
establishment
the
prime meridian
the
the
each
of
between
be
dates
a
f
line
line
is
line although
it has
so
far
ceived
re-
bIeASUBEMENTS
GEOGRAPHICAL
latitude.
Define
North
America
2.
What
3.
St.
is
its
the
reason
Louis
5.
What
6.
When
of
cities
prominent
in
America.
latitude
of
San
from
that
nearer
three
of
cities
the
Francisco
the
Ex.
fully
Explain
city.
If
Chicago
from
than
former
in
named
Give
the
difference
"add
and
cities
three
east
in
from
W.
long.
Cuba
extends
named
cities
the
two
and
of
Ex.
4f
the
finding
What
places?
this?
showing
in
in
subtract"
between
map
or
of
when
latitude
or
globe
a
longitude
in
you
longitude
of
of
longitude
London.
should
in
in
of
this.
west
made
be
can
is much
is the
difference
South
farther
longitude.
cities
three
in
three
is much
for
Define
4.
of
latitude
the
difference
the
latitude
yet
Give
and
283
TIME
EXERCISES
XXVII.
1.
AND
Illustrate
use
to
as
a
class.
is
7.
Iceland
325
mi.
84**
58'
What
factor
in
760
is
meant
Explain
class
a
difference
Will
noon
in
in
74*" T
17'
W.
to
is
What
its
Explain
of
in
and
later
24*
and
long.
discrepancy.
meaning
than
Minneapolis
long.
time?
longitude
or
W.
factor?"
the
earlier
come
and
LouiSy
to
to
the
and
longitude
W.*long.
31'
from
"reducing
by
13"*
Explain
long.
on
an
the
10.
mL
is
problems
between
St.
while
long
and
8.
9.
extends
in
New
of
existiiig
places.
two
York,
Chicago?
use.
relation
the
time
reducing
the
London,
Omaha,
How
earlier
much
or
later?
Consider
11.
with
a
12.
one
any
of
study
will
sail
one
Explain
eonnection
in
mentioned
line.
date
Why
does
it
follow
not
around
Explain
the
15.
Suggest
three
fundamental
change
three
of
problems
problems
Suggest
the
earth
to
' '
gain
' '
a
day?
to
*
"lose
fully.
14.
16.
points
time.
international
the
6
meridian?
day?
the
the
of
each
standard
Explain
13.. How.
a
fully
by
seasons
suitable
problems
for
under
mentioned
of
means
suitable
class
and
globe
a
work
under
candle.
of
each
278.
for
class
work
on
standard
time.
17.
other
Suggest
than
the
an
one
introduction
to
mentioned
under
the
subject
279.
of
longitude
and
tim"
'
^
ABITHUETIC
MOD^I"N
Al^B
MfiTHODd
PBtXidLikMS
ftlBLIOGRAt"HY
1.
diaries
OdTv^n,
Edward
Co.,
GhicagOy
Willis
2.
Johnsdn,
3.
Mill,
4.
Wilteo^,
States.
R.
Hugli
Harvard
K
Robert
E.
An
Tow
Angular
of
the
irc*id,
^dmsiS
111.
E.
92-103.
pp.
pp.
W.
Cooperative
16-19
Time
and
p.
of
Society.
25.
Suwritfe
drid
Sunset
in
the
VMlf^
ZZVni
CHAPTEB
SERIES
Arithmetical
282.
topics
the
ago
occupied
which
weekly,
summation.
series
a
arithmetical
Especially
number
of
a
and
so
The
or
of
series
yearly
;
series
geometrical
a
on
total
arithmetical
an
illustration
series
payments
the
give
of
summation
is less
ratio
the
into
in
acquire
of
than
and
metic.
arith-
infinite
an
one,
of
use
reasoning
otherwise
the
the
insight
an
mathematical
can
it show
when
terms
clubs,
savings
country.
offer
good
of
does
standing
under-
as
in
the
of
case
lH-i
It also
of
is
pupils
the
because
an
cents
or
solution.
for
processes
than
15
monthly
upon
Geometrical
advanced
operations
deposits
series
values.
iiverage
of
sum
appendix
make
place,
of
grades
the
to
5, 10,
the
years
series
unfortunate,
first
over
is the
Buying
presents
The
all
in
seem
the
few
advanced
place
a
would
In
yearly
or
the
really
deposit
deposits
monthly
regular
is
which
found
be
to
are
weekly
these
more
This
members
the
given
are
useful.
series
of
also
they
arising
are
in
A
geometrical
and
position
altogether.
conditions
for
arithmetical
of
Today
omitted
Series.
Geometrical
important
an
arithmetic.
in
and
the
shows
i +
+
law
*-h
governing
changes
such
the
as
increase
population.
Forms
283.
texts
do
to
use
Smes.
include
not
series, they
desire
of
are
offered
Since
the
forms
here
for
them.
285
of
many
or
use
the
the
present
application
by the
teacher
of
who
day
these
may
286
MODERN
ARITHMETIC
Arithmetical
I
AND
PROBLEMS
series:
H- [a +
a
METHODS
d] +
[n
a-f
=
[a H- 2"?] -f [a H- 3d] H
l]
"
Xd
[a-\-l]Xn
5
:
=
Geometrical
series:
a-\l
(I
S
=
=
last
=
number
term.
of
of
8
=
sum
d
=
difference
=
ratio
r
l
"
'first term.
"=
w
a
"
r
I
+
or^
rn-i
aX
=
+
-\- ar^
ar
terms.
first
terms.
n
between
between
consecutive
two
the
and
term
any
in
terms
an
arithmetical
preceding
next
series.
in
one
metrical
geo-
a
series.
Equal
If
a
A^ is
sum,
equaling
each
found
series:
payment
from
the
p,
the
paid
be
to
in
n
equal, yearly instalments,
of interest
rate
being
r, then
p
will
be
formula
^
[l+r]""
If the
is, less the
"
paid off
amounts
interest
on
from
unpaid
the
1
principal
portions,
are
each
a,
k, then
fc]r +
a[lH-r]
p=[c-\-d+
fc]r +
6[lH-r]
p=[d-\-
1c]r +
c[l-\-r]
P=[h-\-c-\-d
p
=
fc[l +
+
r],
h,
year,
c,
d,
that
288
MODERN
4.
Define
5.
Find
ARITHMETIC
illustrate
and
the
The
(6)
the
is
is
Is
7.
if
What
interest
4
8"
equal
the
instalments.
6
first
6
series.
terms
of
3
terms
of
5
fifth
and
term
number
total
yearly
the
is
324.
of
terms
attitude
present
equal,
of
terms
justifiable?
ten
Suggest
8
first
the
this
is
first
the
(d)
What
geometrical
a
PROBLEMS
6
-f
15
-f
12
4-
45
"
(c)
6.
AND
of:
sum
(a)
term
METHODS
H
135
H
"
geometrical
of
i
series
-j-
series
toward
i
-j-
in
^
the,
whose
first
-f*
grade
metic?
arith-
Whyf
will
payments
buy
a
$1600
tract
of
land
6%f
and
solve
two
other
problems
on
the
paying
of
debts
by
BIBLIOGRAPHY"MISCELLANEOUS
1.
HoweU,
2.
McLellan,
D.
J.
Co.,
Psychology
The
Nwnher,
of
York.
A.
Mwthods
Special
in
Macmillan
Arithmetic.
York"
John
Walsh,
4.
New
Chas.
John.
Dewey,
and
252-99.
203-251;
112-21;
pp.
Co.,
McMurry,
New
A.
and
Appleton
3.
B.
Henry
H.
Methods
D.
Arithmetic.
in
C.
"
Heath
Co.,
Boston.
5.
Words
6.
"md
for
Howell,
8.
Patrick,
of
D.
Phillips,
B.
J.
Psychology,
of Number
for
XXI
:
Forms.
487.
Edu-
41-48.
pp.
W.
E.
Number
Science
Popular
Forms.
Monthly,
Genesis
D.
F.
IV:
Howell,
12.
Scripture,
B.
E.
G.
D.
14.
Smith,
15.
Mikami,
Ball,
Journal
of
American
Journal
American
Journal
13-41.
Prodigies.
1.
The
A.
Bara
W.
New
York.
pp.
Arithmetica.
and
Court,
Introduction
Historical
Co.,
E.
W.
Prodigies,
Mathematical
Yoshio
Open
mattes.
pp.
W.
Macmillan
Literature.
American
1.
IV:
Miller,
Forms.
Mathematical
Henry
Psychology,
of Number
506.
Mitchell,
11.
16.
Value
Memory
467.
:
VIII:
Psychology,
13.
Between
of
Journal
Pedagogical
B.
Henry
G.
delation
The
504.
Psychology,
10.
V
Beview,
XLII:
of
A.
7.
L.
American
Numbers,
Hombrook.
eational
9.
Edward
Thomdike,
129-58.
Ginn
D.
Smith,
E.
Mathematical
to
and
History
Co.,
Boston.
of Japanese
Mathe*
Chicago.
B.
Mathematical
MaemiUan
Becreations.
Co.,
York.
New
17.
Science
18.
A.
Crommellin,
and
Jones,
Mathematics,
S.
I.
C.
D.
XIV:
An
Arithmetical
Curiosity.
School
451.
Mathematical
Wrinkles.
Tex.*
289
S.
I.
Jones,
Gunther,
INDEX
Abacus,
13y
18,
19,
97,
20,
Arabic,
113,
130
12-15,
Abstract,
concrete
Accounts
to,
Areas,
222-^4
257
50, 53-54,
irregular
motivating,
plane
42
38,
degrees,
1^7-68
figures,
problems,
52
value
Quincy,
John
Adams,
Addition
combinations,
decimals,
history,
76,
Averages,
99-102
256
Algebraic
business
102-3
work,
papyrus,
discount,
110
for
ties,
plurali-
parts,
Banking
138-39
Bankruptcy,
Applications,
193-94,
79
teaching,
Approximations,
multiplication
decimals,
multiplication
Base
249-51
69
and
division
of
and
numbers,
and
of
number-system,
Basic
subject-matter,
Bills,
222
Boethius,
Bonds,
of
222-26
20
219
Bookkeeping,
136
Borda,
52
291
interest,
173
207-15
219-20
accounts,
167-69
division
207
33
20
unitary,
207-8
208
school,
19,
results,
and
211
savings,
short
272-73
Algorithm,
usage,
functions,
formulas
mixed
majorities
Bank:
97-98
76, 104-5,
Analysis,
subtraction,
251-52
cuts,
Aliquot
of
80
133-34
of
cuts,
lessons,
method
163
fractions,
short
of
285-86
105-6
108
checks,
76
series,
Austrian
103
^carrying,
order
of, 2, 3-6,
Assignment
:
227-36
228
Arithmetical
173
265
260
literal,
introduction,
150
145,
figures,
Arithmetic,
Accuracy:
number-system,
18-19
16,
Archimedes,
24
:
biUs,
Ahmes
Hindu-,
222-24
11
28-29
292
* *
INDEX
' '
Borrowing
of
105
United
30, 32,
usage,
brokerage,
200
multiplication, 11^-16
201-2
insurance,
interest, 210-11
151-52
measurements,
partnership,
and
in
51-52
24
263
Construction
work:
93-95
early, 85,
146
problems,
173
Cone, right,
29
200
211
abstract,
to
200-1
brokerage,
interest,
Condorcet,
251
simplicity, 28,
weighing,
Commission
Concrete
203
revenue,
discount,
Computation
199
loss, 201
proportion,
Commercial
Compound
216-17
186,
percentage,
profit and
:
100-2
addition,
237
graphs,
165-67
21
Warren,
Combinations
132-33
fractions,
260-61
Colburn,
200-1
discount,
Circle,
13
Nicolas,
Circulating decimals,
217-18
corporations,
12-13
number-system,
Chuquet,
and
15-16
development,
Chinese
199
207-8
commission
224
interest, 32-33, 84-85
nimiber
146
banking,
Child:
enterprises,
200
Standards,
States,
Business
subtraction,
commission,
Brokerage,
Bureau
in
later, 259
Cajori, F.,
Converting
147
for
rule
Carpenters'
root,
square
**
in
Cases"
48-49,
out
addition
problem
Ceulen,
von,
Changes
in
Checks
of
sources
nines:
subtraction,
and
108-9
division, 122
thermometer,
183-84
and
number
schools,
Counting
Course
of
42-43
reckoning,
subject-matter, 29-30
grades,
108-
28-35
29
31-32
selection,
subtraction,
18-19
study:
arithmetic,
31
of
Criticisms
arithmetic
1-3
110
and
multiplication
division,
Cube,
Cube
122-23
problems,
problems,
curtailment,
257
and
40-43
10-12
:
addition
subjects,
Counting
multiplication
Centigrade
eolutions,
192
and
:
socialization, 36-44
103-4
addition,
in
Casting
Correlation
other
274-75
Carrying
217-19
Corporations,
138
Cancellation,
247
factor,
52,
69
262
root, elimination, 272
Curriculum,
28-35
ing,
teach-
293
INDEX
29
curtailment,
31-32
grades,
selection,
164-65
decimals,
31
fractions,
elimination
Curtailment,
122-23
checks,
of
ject-matter,
sub-
Cylinder, right,
114-15
history,
inverse
29
137-38
119
262-63
and
measurement
Date
line, international,
Date,
where
Decimal
changed,
159
and
short
in
approximations
and
division,
multiplica-*
167-69
division,
circulating,
163
Hindu-Arabic
ber-system,
num-
Degree
of
Degree
of
fractions,
Denominate
grades,
fundamental
history,
Direct
147-49
153-55
152-53
arithmetic,
Discipline, formal^
18-27
150-51
subject-matter,
28
19
Early
construction
Early
number
Early
number
work,
lessons,
85
89-95
work, 15-16, 85-87,
curtailment
and
English
in
Equation
of
29
Arithmetic, 47,
82
224
series, 286-87
payment
:
industries,
inverse
231-32
of
process,
literal,228
50, 77,
229
literal, 227-37
211
Division,
proportion,
200-1
solution
119-20
approximations
169
of
introduction
5-6
Discount:
commercial,
61
numerals,
Equal
measurements,
bank,
77
Enterprises, child,
of
62-63
good,
subject-matter,
143-47
Development
of
Elimination
operations,
reductions,
61-71
88-89
143-57
151-52
equivalents,
.
29
257
numbers,
23
165
167
accuracy,
first four
Dust
162
subject-matter,
arc,
Drills,
Duality
163-64
writing,
to
of
259
scale, 258
reviews,
159-62
and
discipline, 5-6
:
forms, 64-66;
multiplication,
Decrease
formal
essentials
158-59
reductions
of
Drawing
to
introduction,
reading
Doctrine
solving problems,
165-67
160-61
history,
124-25
cuts, 124-25
mensuration,
164-5
extension,
113-28
120-22
order,
subtraction,
partition, 120
relations,
number
158-72
addition
tion
multiplication,
282
282
point, origin,
Decimals,
tion,
multiplica-
of
(reverse)
with
decimals,
of
Estimating
Euclid, 256,
249-50
problems,
results, 52
272
229-31
294
INDEX
Geographical
257
Euler, Leonhardy
early arithmetic,
Europe,
involution,
and
Evolution
time,
18
272-77
211-12
Exchange,
278-84
early grades,
278-79
fundamental
problem,
and
longitude
81-82
''Explanations,"
123-24
use,
thermometers,
Fibonacci
(Leonardo
of
183-84
Gerbert
Pisa),
20, 130
Gobar
Sylvester II),
(Pope
numerals, 19,
20
114
31-32
Grades, work,
12-13
Figures,
Graphs,
237-44
:
essentials, 239
11
reckoning,
introduction,
cards,"
Fontana,
Nicola
Four
65
proportionality,
showing
(Tartaglia),
237
taught,
why
of
A.
Grube,
22-23
W., 20,
54-55,
problems,
250
20
discipline,doctrine, 5-6
Formulation
66,
239
problems,
Flash
Formal
238
10
signaling,
"*
256
Geometry,
Fahrenheit
Finger
280-82
time,
series, 286
Geometrical
Factors,
279-80
278
province,
272
Exponential Notation,
and
measurements
numerals, 19,
Gubar
114
76-77
Gunter,
181
operations,
fundamental
98-99
24
Habituation,
129-42
Fractions,
133-34
Hindu-
number-system,
15, 16, 18-19
History
decimals,
158-59
division, 113-15
133-34
early
varieties,187
unit,
129
what
pupils
18-20
know,
should
129-30
139
256-57
mensuration,
operations:
measures,
193
numerals,
metric
system,
moneys,
146-47
173-74
153-55
multiplication, 113-15
33, 40, 67-69
percentage,
in
arithmetic
fractions,
and
143-47
numbers,
275
three
Fundamental
subtraction, 97-9"
and
addition
165
denominate
root,
subtraction,
:
134-36
129
sexagesimal,
weights
12-
130-31
reduction, 131-32,
Boman
Arabic
144
129-30
multiplication,
Games,
England,
of
I
Henry
introduction,
square
55
home,
"Help,"
158-72
division, 137-38
history,
115
Harriot,
138-39
aliquot parts,
decimal,
101-2
combination,
addition
addition,
percentage,
16
powers
and
186
roots, 272
Europe,
296
""
INDEX
Making
change/'
106
subtraction,
Material
for
of
process
Moneys,
problems,
39-42,
54-
Mathematical
division^
Measurements
:
first four
278-84
history,
division,
153-55
operations,
143-45
Mensuration,
134-36
113-15
136
123-25
cuts, 76, 123-25
Multiple,
133-34
258
256-57
plane figures, 259-62
suggestions
115-16
113-28
relations,
short
257-58
fundamentals,
history,
decimals,
order, 117-18
256-71
early grades,
76,
numbers,
number
152-53
reduction,
history,
mixed
143-47
units,
116
163-64
fractions,
147-48
grades,
of
addition,
with
combinations,
151-52
fundamental
115-19
of
decimals,
equivalent,
42
122
checks,
:
origin
33-34, 39-40,
167-69
150-51
89-96
grades,
geographical,
Measures
173
approximations
indirect,
and
primary
Gaspard,
abbreviation
120
146-47
history,
Monge,
2S
subject-matter,
Multiplication,
series, 285-88
Measurement,
direct
of
Motivation,
264-66
55,
Modification
for
problems,
264-
Napier,
John,
Nines,
casting
addition
20
out:
and
subtraction,
108-
9
66
Mental
Mental
multiplication
61-2
arithmetic,
discipline, doctrine,
Metric
claimed
disadvantages,
in
history,
173-74
and
teaching,
174-78
games,
Number
addition
teaching, 174,
15-16
:
16,
reckoning,
174
division, 122
numeration,
early lessons,
179-82
emphasis
Notation
Number
173-85
system,
advantages,
5-6
and
84-96
67-69
10-11
relations:
and
subtraction,
179-80
76,
110
United
used
in
defining
used
in
pharmacy,
Mixed
173-74
States, 146,
multiplication
the
yard,
and
proportion,
74-75
multiplication, 136
Mode
of
of
249
Number-systems,
subtraction,
thought
and
arithmetic, 72-73
division,
123-25
174
numbers:
addition
and
146
Chinese
and
11
Japanese,
Hindu-Arabic,
language
periods of,
Boman,
16
14
12-15
12-13
297
INDEX
Number
work,
early, 15-16, 84-86,
88-89
Numbers,
introduction,
inverse
into
separating
periods,
14
language,
188-89
equation,
and
gubar,
Hindu-
19,
gobar,
or
13
Japanese,
unitary analysis,
114
Periods
16
Roman,
of
teaching,
Oral
arithmetic,
Oral
drills,good,
Pike's
23
Order
of
priority
Order
of
work:
of
value,
signs,
125-26
how
102-4
Plane
division,
120-22
Play,
subtraction,
Origin
116-19
measures,
William,
Oughtred,
Outlines
":
33
251-52
Pluralities,
off, 14
II.
(Gerbert),
the
teacher, 6-7
Pope, Sylvester
115
arithmetic
of
145
33
Pointing
143-45
79-80
67-69
store,
and
weights
of
15
figures,
bank,
106-8
161-62
lesson,
addition,
by grades,
20
272-73
Powers,
history,
31-32
20
12
taught,
Plan,
multiplication,
257, 260
decimals,
62-63
14
23
arithmetic,
Place
61-62
193-95
number-systems,
Pestalozzi, 20,
Pi, (^)
Object
188
reductions,
12
Arabic,
192
192-93
problems,
Chinese
191-92
problems,
literal
Numerals:
186-89
272
roots, 272-77
Paper
construction,
Paper
folding,
Parallelogram,
Primary
arithmetic,
259
Primary
grades, mensuration,
rectangular,
Partition, division,
Payment
aid
usage,
games,
186,
history,
186
in
125-26
262
45-60
solving,
by
formulation,
199
problems,
193
right,
changing
four
difficulties,188-90
fundamental
signs, order,
23^24,
46-47,
248-49
186-98
application, 199-206
business
88-
189-92, 201, 229, 230, 232-34,
series, 286-87
Percentage,
of
Problems,
216-21
84-96
95
Prism,
120
216
corporations,
262
Priority
211
payments,
Partnership,
of
Preparation
131
Parallelepiped,
Partial
88-95
190
steps in
history,
judgment
seasons,
34
54-57, 66, 76-77
solving,
46-53
45
in
"practical,''
solving, 4, 48,
56-57
53
298
INDEX
of
value
goody
without
232-33
the
Proportion,
Rice,
248
245-46
history,
263
261
theorem,
Pythagorean
and
drills,61,
M.,
1
J.
Right
cone,
Right
cylinder,
Right
prism,
263
262
262
of
Right
pyramid,
263
Rahn,
115
Right
triangle,
261-62
Ratio,
246-48
Roman
number
245-55
proportion,
15-16,
Reason
elimination
numbers,
writing
arithmetic,
teaching
30
3,
Reasoning:
69
Reckoning,
number,
Rectangle,
53-54
48
Rule
of
Savings
262
188-89
numbers,
fractions, 133-34,
Three,
256
158
245
208
bank,
to, 258
Scale, drawing
School
denominate
276
10-11
Reduction:
decimals, 165-67,
272
273-75
Rudolff, Christoff, 20,
parallelopiped,
Rectangular
cube,
275
Rope-stretchers,
15-16
solving problems,
33
bank,
Selection
of
subject-matter,
Separating
165-67,
into
numbers
periods,
18814
Series,
285-88
cents, 188-89
Regular
rightrline figures,
Results, interpretation,
203
Reverse
(inverse)
51, 77-78
division,
Sexagesimal
260
Shank,
53
Short
Revenue,
119
31
152-53
89
per
274-
272
history,
table,
approximations,
square,
273-76
extraction,
square,
counting,
of
fractions,
162
for
of
75
98-99
Rationalization,
and
16
rule
carpenters'
24-25
method,
Reading
notation,
Roots:
245-46
history,
Rational
system,
78-79
20
Radix
11
164
105-6
Riese, Adam,
right,
Pyramid,
Review
259
point,
191-92
subtraction,
grades,
229
decimal
percentage,
notes^ 209
in
equation,
locating
201
loss,
Promissory
Proofs
literal
45-46
numbers,
and
Profit
interest, 210
37-39
Bocialized,
processes,
49-
fractions,
William,
129
257
cuts:
addition, 104-5,
110
interest, 210
multiplication, 76,
123-25
299
INDEX
fractions,
of
multiplication
roots,
110
subtraction,
Signs,
fractions,
274-76
272-73
squaring,
97-98
methods,
105
order
of
order
36-44
correlation,
problems:
illustration, 37-39
cones,
263
cubes,
262
spheres,
Symbols
22-23
Spiral system,
53-54, 145,
Square
measure,
Square
root, 274-75
149
Table
Hindu
Tables:
158
"Squaring
the
circle,"
Squaring,
short
cut, 273
time,
Stocks,
Tartaglia
130
Subject-matter,
choice,
and
curtailment,
of,
of,
6-8
69-70
183-84
20-23
29
devoted
arithmetic,
to
socialization, 36-37
standard,
waste,
:
282
80-82
Tonstall, Cuthbert,
addition,
97-112
Austrian
method,
''borrowing,"
280-82
longitude,
29
105
105-6
Topical
system,
Trapezoid,
260
2
measurements,
278-84
6-7
modification,
Subtraction
203
preparation
geographical
31-32
knowledge,
20
11-12
Time:
28
elimination
Fontana),
116
(Nicola
value
TiUich,
31
151
151
Thermometers,
28
duality,
grades,
Tests,
28-35
149,
numbers,
(revenues),
Teacher,
33
276
100
Tally, keeping,
Taxes
Store, playing,
roots,
square
omission,
217-18
basic,
of
multiplication,
282
Stifel, Michael,
263
arithmetic, 73, 73-7^
denominate
256
20, 158-59
Stevin, Simon,
262
262
132
addition,
276
Standard
262
263-64
of
78,
carpenters' rule, 274-75
method,
series, 286
:
right pyramids,
263-64
table,
mathematical
right prisms,
23
Sphere,
106-8
work,
right parallelepipeds,
42-43
material, 39-40,
Speer,
of
'
right cylinders,
88
early,
of
Surfaces
36-37
Socialization,
Socialized
Sum
priority, 125-26
133-34
history,
52
use,
163
decimals,
138-39
square
108-10
checks,
21-2
20
300
INDEX
Travelers'
Triangle,
arithmetie,
of
Unit-fraction,
Unitary
United
75-79
129
analysis,
249-50
193-94,
right
cones,
right
cylinders,
right
prisms,
right
pyramids,
spheres,
Waste
20-21
of
bureau
263
262
262
263
263-64
partial
and
first
weights
and
metric,
180-81
151-52
four
origin
operations,
of
arithmetic,
of
multiplication,
units,
cubes,
irregular
143-45
152-53
98
John,
the
by
Conqueror,
145
31-32
grades,
69-70
tests,
122
52,
subtraction,
Widman,
Written
122-23
problems,
76
Work
108-10
division,
3-6,
William
:
addition,
Volumes,
2,
153-55
143-46
reduction,
Verification
147-49
grades,
history,
143-46
measures,
143-57
measures,
fundamental
146-47
money,
80-82
equivalents,
211
payments,
time,
of
Weights
146
standards,
Units:
Value
262
States:
arithmetie,
rule,
261-62
259-60,
Unifieatioii
parallelopipeds,
212
Checks,
Yard
69
in
defined
terms
of
meter,
146
108-10
54
Zero,
262
solids,
264
13,
dif"culties
18
in
numbers,
15