Abacus Made Easy Si 00 Mae e

iy Dr. Mae,

avidow

THE ABACUS MADE EASY A

Simplified Manual for

Teaching the Cranmer Abacus

by

Mae

E. David ow, Ed.D.

Teacher of Mathematics Overbrook School for the Blind 64th Street and Malvern Avenue Philadelphia, Penna. 19151

Published By

OVERBROOK SCHOOL FOR THE BLIND

Copyright, 1966

OVERBROOK SCHOOL FOR THE BLIND All rights reserved.

No part

of this

book may be reproduced

any form without permission in writing from the author, except by a reviewer who wishes to quote brief passages in connection with a review written for inclusion in a magazine or newspaper. in

ABOUT THE AUTHOR Dr. Davidow has been a teacher for the Blind since 1935.

came

Overbrook as

to

age of 10. for

Overbrook School

at

This gifted teacher, however, first

a student, having lost

She received a B.A. from

Women, now Douglas

her sight

New Jersey

at the

College

College, part of Rutgers University.

Temple University granted her

a Master's

Degree

in

1949

and then a doctorate in I960.

Here

at

Overbrook, Dr. Davidow was instrumental

in establishing the use of the

the regular curriculum.

method

of teaching

eagerly.

with the results

In

Cranmer Abacus

Her enthusiasm

as a part of

for this pioneer

mathematics led others

to follow

her

her role as coordinating teacher, she worked

members

of the

Mathematics Department and the

Were highly successful.

Hopeful that this success

at

Overbrook might be experienced by many teachers elsewhere, she was encouraged to write the following manual.

We Dr. Davidow

are indeed appreciative of the unstinting effort of in

presenting Abacus Made Easy.

Joseph

J

.

Kerr

Assistant Principal

Digitized by the Internet Archive in

2012 with funding from

National Federation of the Blind (NFB)

http://www.archive.org/details/abacusmadeeasysiOOmaee

11

ACKNOWLEDGMENTS manual

In the writing of this

aid and suggestions

whom;

I

the

have received invaluable

many teachers and

students with

have worked on the abacus. I

of the

from

I

owe a great debt

of gratitude to the

Overbrook School for

more deeply

me

sible for

America.

I

work with

the Blind

into the study of the to attend the first

administrators

who encouraged me

abacus and who made

Abacus

Institute

the use of the

pos-

it

ever held

in

me

appreciate the opportunity that was given to

the instructors of

to delve

to

mathematics as they were teaching

abacus to their students.

In this

manner both

teacher and student learned the language and method of operation, I

the

work

feel

of

it is

appropriate to acknowledge with gratitude

Fred Gissoni whose

text,

Using the Cranmer Abacus,

helped lay the foundation and groundwork which enabled

me

to

write this manual. I

wish

to

express

my

special thanks to the

teers

who gave

me

compiling this manuscript.

in

unstintingly of their time to

work

many

volun-

directly with

Ill

CONTENTS Acknowledgements

ii

Foreword

v

Introduction to the Tool

1

Numbers

3

Setting

Addition Addition of

5

Two

Numbers

Digit

11

Subtraction

14

Multiplication

21

Treatment

of the

Multiplication of

Zero

27

Two

29

Digits

Division.

31

Long Division

35

Addition of Decimals

43

Subtraction of Decimals.

.............

46

Multiplication of Decimals

48

Division of Decimals

50

Fractions

.

54

IV

Addition, of Fractions

56

Subtraction of Fractions

60

Multiplication of Fractions

63

Division of Fractions

•

.

.

.

.

.

67

Per Cent

69

Square Root

71

Conclusion

76

.

FOREWORD Using the Cranmer Abacus ten by

Fred L. Gissoni

is

Bureau

of the

an excellent text writof

Rehabilitation Ser-

vices, Kentucky Department of Education, at Lexington,

Kentucky In

1964

I

attended the Abacus Institute

Kentucky, held under the direction

sity of

of

at the

Univer-

Mr. Gissoni.

This was the first such institute ever conducted in America.

We were

also privileged at that time to have with us Professor

Paske, an instructor in

of the

abacus

School for the Blind

Copenhagen, Denmark. In the

past two years,

Abacus using Fred Gissoni's of the to

at the

abacus

to both

I

have taught the Cranmer

text.

However, as an instructor

teachers and students.

have a simplified manual

period

I

at

I

found

our fingertips.

it

During

desirable this

have made, with Mr. Gissoni's permission, adapta-

tions of his text.

This has been done while working with

teachers and with children from second grade through high school-

It is

my

hope that this manual will make the teaching abacus more meaningful

and the learning

of the

and teachers.

shall attempt to explain in the simplest,

I

to both students

most

VI

concise manner how to add, subtract, multiply, divide,

handle decimals, fractions, percent and square root.

The method class

is

abacus

of instruction for

essentially the same.

is

likewise the

same

an individual or for a

The approach

to teaching the

for a second grade student, a

senior in high school, a college student, a teacher, a parent, or any interested person, either blind or sighted, who wishes to learn the abacus. all

When we

first

examine the abacus, we

We must move

learn about the tool in the same manner.

slowly; the language

understood.

must be simple; every step must be

The slow learner, as well as the gifted

learns the abacus in the same manner.

child,

However, after

the

learning has taken place, and the individual knows how to

manipulate the tool, then each one operates

it

at his individual

speed. In

patient.

One must use

The abacus learned,

move

teaching the use of the abacus, one must be very

it

is

studied one step at a time.

As one step

is

must be practiced again and again.

Then one may

One takes

it

to the next step.

thoroughly.

the language level of the individual.

his time

and learns

INTRODUCTION TO THE TOOL The

first lesson consists of handling the abacus.

must become acquainted with frame

is

oblong

shape.

in

We

the tool.

notice that the

There are thirteen rods, or columns.,

On each column there are

attached to the frame.

which travel up and down.

One

There

is a

five

beads

separation bar which

cuts across all thirteen columns, approximately two-thirds

from

bottom

of the

frame.

four of the beads or counters

from

the fifth bead.

the distance

We

the

now examine

shall

ting position.

the abacus in

its

proper opera-

The abacus will always be on our desk

position which is about to be described. rates the one bead to left.

This bar separates

from

in the

The bar which sepa-

the other four beads runs

The section containing the single beads

from

is

right

on the upper

part of the abacus, and the four beads are toward the bottom of the

frame

.

First let us run our fingers along the face of the

lower edge

column.

of the

frame.

In addition to

Here we

these dots,

find a dot at the end of each

we

cal line between every third column.

will find a short., verti-

The dots are

to help

us locate our place, and the lines, which are called "unit

marks"

or

commas

in

writing numbers, can also serve as a

.

decimal point.

These same dots and lines are found on the

separation bar between the four beads and the one bead.

Now

let

us place the index finger of our right hand

on the dot to the extreme right, on the bottom

of

our frame

This dot corresponds with the column to the extreme right. It

is

the units column.

column

is

the tens

Then we touch the hundreds

Moving from right

column and

the next is the hundreds column.

the line between the

column from

to left, the next

columns which separates

the thousands column.

Next we

have the thousands column, ten thousands, hundred thousands, separation line, millions, etc.

As our fingers move along left

hand

is

of the right

important

in all of

up to one trillion.

the frame, the index finger of our

always immediately to the hand.

,

The position

of our

left of the

hands

is

is

extremely

our operations as we manipulate the abacus.

The keyword, while learning the fundamentals Agacus,

index finger

patient repetition.

of the

Cranmer

SETTING NUMBERS When we we write

When we want we ^clear" it,

it

remove

to

it,

show the ten

to write

1

write

we

2,

3;

,

it

loses

digits

moved toward we

a

number

or erase

As we move a bead toward

it.

When we move

takes on a value.

or "clear"'

have

number on our abacus, we do

its

from zero

set one

to 9-

column

set another bead; then

the bar or

single

When

we

bead and we have

1

number

1

set the

beads on a column up 2.

we would

move

3

set 2

beads up

up to the bar. to the bar,

to the

set a third bead, and

Each one

4.

However.,

5,

,

The single

To

set 4,

we must be sure To

each number

we would move one

To

To

of the

set the

number

we would

set 3,

we move

we would move

below the bar on that column.

we

of the

of one.

to 5 --writing

beads up to the bar.

To set

Then

To

up to the bar.

it

separation bar.

to the bar.

column we can

of five.

Let us now write from

To

set"

there are no beads

beads below the separation bar has a value

bead above the bar has a value

rr

that

a zero position.

is in

bead by pushing

we say

it,

away from the bar,

it

On a

value.

the bar, the

set a fourth

separately.

not say

abacus language we use the word "set."

In

it.

put a

all 4

beads

the upper bead

down

that there are no beads

set the

number

6,

we would

move the 5

the upper bead down, and

same column.

and then set

To

set 8,

set the

5

we

2

Five plu$

equals

above the bar and

bead and the four

l's.

To

6.

on the same column.

set 5

the beads on a

1

move one lower bead up

3

Five

and

column are moved as close

2

set 9.

we see

to the

set a

equals

To

below.

In setting 9

we

set 7,

in

7.

we

that all

bar as pos-

sible.

Much time must

be devoted to setting numbers.

we write our numbers with index finger, our

is

left

to the left of the right hand.

in the tens

column

and the right index finger

to the right

and thumb push up or set two beads is

1

Let us

index finger rests on the hundreds column.

Then both hands move

practice

thumb and

always resting on the

The right index finger sets a

write 12. while the

index finger

left

column immediately

the right index finger, or

As

needed

in

column.

writing and reading numbers.

children can write how

many desks

how many sisters they have,

numbers according

in the units

etc.

to the level of

,

Much

Little

there are in the classroom; etc.

Each teacher gives

her group,

Older children can

write their phone numbers and then have a classmate read them.

When

the student can set and read

ready to begin addition.

numbers quickly, then he

is

ADDITION

We itself a

will start our addition by adding the

number

of

is

before,

The

the units column. is

immediately

We

tens column.

beads up to the bar.

left

number

This

is

the abacus.

hand, as we mentioned

to the left of the right

set the

hand on the

by moving one of the lower

1

done with the index finger done

right hand.

Next add

manner with

the index finger of the right hand.

1

.

Now add

column. easier. it

It

another is

We

directly.

the bar.

1

1

.

We now

if

we want

is

beads that we have

in

to

more

5

same

Now add

another

add

1

if

rf

one finds

it

more, we cannot add to

move up

to

ones ,r to add to the four

our units column, our right index finger

the bar and sets a 5.

more, but we added

in the

of the

have four beads in our unit

There are no more lower beads

Since there are no

moves above

This

to this 1.

permissible to use the thumb

find that

to

1

will place the index finger of

extreme right column on

the right hand on the

This

We

times.

number

more.

only wanted to add

We must now

many more beads were added than There are several ways

We

figure out

the one bead that

that this can be done.

1

how

we wanted.

Each teacher

figures out according to her grade level which method to use.

One teacher might say,

"I

asked you

to give

me

one cent and

me

you gave

how much change should

a nickel,

The student figures out

We

four pennies.

that

1

can think

from

give you"?

and he must return

4

5 is

I

beads that we have set

of the four

on the abacus as four pennies and when we clear the four beads, we are returning the four pennies.

understanding

of

addition as working with partners.

add

1

of the bar,

1.

motion.

4 to

1

,

the child

he can set the

We

down across the bar.

have the to this 5,

slide the 5

when you add

of 5

4.

5

If,

if

By doing

he

would set

bead down

a 5

this

in

to the bar,

we have added

1

and

1

.

its

one continuous

move our

to the 4

showing on the units column.

we continue as we did before.

to 4

and clear

of the 5

and clear the 4

1

5.

on the other hand,

the bar, and slide the four lower beads

digit 5

that

he must go to the top of the bar and set a

After the student learns the relationship 1,

to this indirect

The child learns

So the child clears the

we were adding

partner

can be

and there are no more beads on the bot-

to 4,

The child learns that the partners are 4 and

this

numbers.

One second grade teacher refers

tom

known as

This depends on the age of the student and his

explained.

to

is

There are many different ways

indirect addition.

wishes

This

In

finger

away from

and we now order

We move

to

add

one lower

1

We now

counter up to the bar. 1

have a

bead or counter below the bar.

we add Add

1

1

to 6

to 8

and we have

and we will have

1

to 7

We want

9-

above the bar and a

Five plus

Add

7.

5

1

equals

Then

6.

and we will have to

add

more

1

8.

to 9

and we find that there are no more beads on the unit column to

add directly,

Our

left

index finger that has been resting on the

column immediately

to the left of the right

We

column) now comes into play.

can refer to that index

When we can add no more

finger as the "helping finger."

beads with the right index finger, the play and assists.

The

left

column (push

it

10 since

in the tens

it is

left

hand comes into

hand will set one bead on the tens This one bead has a value of

up to the bar).

column.

the left index finger gave us 10

out

hand (on the tens

We

more.

only wanted

1

Now we must

more, but figure

how many extra we have. This can be done again

we wanted

many

1

extra

more; our

we have.

left

column and

value of 10.

We

several ways:

hand gave us

One from 10

so our right hand will clear the tens

in

9-

is 9-

10; let

We

We now

a zero in the units column.

see then that

9 plus 1

(1)

We

us see

have

9,

how

9 too

have a ten

had

many,

in the

This gives us a

equals 10.

(2)

We

can

speak

We

of

it

in

terms

We must

received a dime.

from

10

is 9-

partners of

1

money.

of

We want

Again we can refer

are

1

to

9-

One

give 9 cents change.

(3)

and

a penny to add to

The

our partners.

Here again, we have added

9-

indirectly.

As we continue unit

column with

I's,

we add them

When we add

to 10 gives us the digits 11.

another bead in the units column, we have 1Z.

and one

We

1

1

until

pointer finger we set tinue to add

1

until

5,

we have

we have

right index finger clear the until

we have

1

Then with

Then with our

49-

99-

9-

We

5,

in the tens

reaches over and gives us a 10

clear

We now find

with the right hand; neither can the giving us another

44.

We

column. in the

etc.

con-

index fin-

left

and with our

4,

We

have 50.

we cannot add left

,

the right

clear 4, and we have 45.

ger (our helping hand) we set the

1

Thus, one 10

gives us 11; one 10 and two l's give us 12, etc.

continue adding

adding

in the

mentioned

the right index finger, or thumb, as

One added

above.

add more

to

1

continue

more

hand assist us by So the

left

hand

hundreds column.

(This ten has the value of ten 10's or 100.)

But we now have

nine extra 10's,. so with our left index finger

we clear

10's and the right index finger clears the nine units.

the nine

Although

we wanted 9

add only

to

more, we added 10 more and thus had

1

extra to clear.

A good add from

is to

hands

1

addition exercise to introduce in the beginning

First add

to 45.

1

in starting position- -the right

and the

left

hand immediately

hand on the units column

to the left of

on the tens

it

With the right index finger we set

column.

Begin with the

to zero..

Then with our

1.

pincher fingers (the index finger and the thumb) we set

We

more beads. see that there

only

How many

set 5.

Since

is

are now ready to add

we have

2

1

many, we

abacus and find that we have

makes

6,

now add

4

sets of 3

We

will

so

The

it

What

is 6.

we gave did is

is

we give?

but

we wanted

left

we

is 6,

cleared with the right hand.

added directly.

Here again,

Now we add

the left

6.

our

only have

so

Or we may say

how much

add

5.

10.

too

The

6.

cannot add

hand sets another

beads,

child has learned

we clear

We now

We

3

index finger will set a

only 4 more;

Four from 10

5 is 2.

We check

So he will clear 6 with the right hand. 10,

We

5.

that the abacus is right.

make 10? The

4's partner to

But we

have learned that two

find that

so our left hand will help us. 10.

We

6.

We

Three from

will clear 2,

we are sure

more.

more beads.

So we must turn to the

bead.

extra did we set?

too

3

2

many 6

That can be 6

directly.

One says,

6

from

10

we must clear

10 is 4; so

We

4.

cannot clear 4 directly

with the right hand; so the right hand clears a

Here again figuring a

1

We wanted

.

nickel, so

4

from

to take

we owe

We now

away

1

give back

have 21 on the abacus. 5

We

cannot add

are ready to add

hand sets a 2.

We now

10.

add

8.

Eight from 10 9-

We now

1.

through

We

9.

add

a

This can

7.

We now

8 directly,

have 28.

so the

left

so the right hand clears

is 1,

Here again

so the right hand

have 45. continued by starting out by

is

on the units column and then adding

1

When we

When we

46.

is 2;

Nine from 10

This same exercise setting

2

.

give back

we took away

This cannot be done directly.

the left hand sets a 10.

clears

We

and

1

1.

more.

be added directly by setting

and sets

we must

tells us

4 pennies; but

We

cent.

1

5 is

5

start with

1

1,

2,

3,

etc.

on the abacus, our answer

start with 2 on the abacus, our

answer

is

47, etc

Following are some one -digit addition examples for practice: 2 +

2=

3 +

5=

7 + 2 =

6 +

1=

4 +

5=

5 +

1

=

2 +

5=

2 +

3=

3+4

=

4 +

1

=

is

11

ADDITION OF TWO DIGIT NUMBERS Addition on the abacus

When carry 3;

addition

is

done from

left to right,

On

Let us add 36 plus 27,

.

done from

is

we

to write 36,

hand

left

is

we do

put our right hand on the tens column.

immediately

to the left of the right

Then

on the hundreds column. right to the units

column

to write the 6.

The

tens column.

column, the

left

hand

is resting

to the

hand

left

is

The

hand resting

moves

the right hand

lows the right hand, and when the right hand 6 in the units

set a

we were going

Since

6.

not have to

column we

the tens

then on the units column we set a

left to right.

fol-

writing the

on the

3 in

the

For low vision students and sighted teachers,

helpful to keep the pointer fingers on the lower part of

it is

the

columns instead

this

manner we do

making

it

of resting the fingers on the

not cover the

numbers we have

possible to read with the eye what

We

on the tens column and add

2.

we must add

5 is 3,

right hand

hand) and

so the

left

We

are now ready to add 27.

5.

moves

Two from

to the units

we are ready

to

hand will set a

beads.

we do

Since

we have

written.

not have 2 beads, 3.

Then our

column (followed by the

We

add

7.

10.

Since

cannot add a

we

thus

place our right hand

we clear

so

set;

In

7

only wanted

left

directly, 7,

we

12

will say 7

from

in the units 5.

away

3,

We

column.

Three from

We

is 3.

1

cannot clear

Since

5 is 2.

we took away

will clear

2

extra beads.

Direct addition

is

directly, so

3

we cleared

The answer

with our right hand.

with the right hand

3

We

we clear

and had to take

5

will give

back

2

is 63.

not difficult and in the beginning,

one should give the student simple problems such as 123 plus

Here we

321.

hand hand)

(the left

.

set

in the

1

hundreds column with the right

hand being immediately

to the left of the right

Both hands move to the right and the right hand sets

Then both hands move

the 2 in the tens column.

and we write the

column and

the hundreds

We

the units column.

3 in

right hand sets the 2 in the tens column.

the right and set the

1

Our answer

is

Our

right hand.

Now

hand on the 4

444. let

in the

then go back to

the right hand sets the

Then both hands move

hundreds column.

in the units

left

to the right

3 in

to the right

We move

column with our

hand did not have

us add 789 to 444.

hundreds column.

the

We We

and the

again to right hand,

to assist the

place our right

cannot add a

7 to

the 4 directly, so the left hand will give us a 10 in the thou-

sands column. only

7

This 10

more hundreds.

is

equal to

1

hundreds, but we wanted

Therefore, we must clear

3

hundreds

13

(7

from

10 gives us

3)

.

Both hands move

add

8

We

tens.

assist.

The

left

do not have

We

ones.

We

do not have

by setting a say

9

from

Our answer

10 is

We

1.

1

bead

in the

tens too many, so

we

column and add

9

so our left hand assists again

Although we only wanted

10.

is

2

to the units

9 units,

to

tens, so our left hand will

have added

We now move

2 tens.

8

and we are ready

hand will give us 10 tens or

hundreds column. clear

to the right

then clear a

1

9,

we add

in the units

10.

We

column.

1233.

Exercises

in

two- and three-digit addition:

23 + 25 =

52 + 17 =

34 + 15 =

71 + 16 =

81 + 8 =

62 + 20 =

24 + 65 =

12 + 72 =

72 + 16 =

25 + 23 =

653 + 230 =

825 + 162 =

907 + 52 =

425 + 23 =

873 + 34 =

769 + 25 =

473 + 192 =

395 + 232 =

439 + 123 =

892 + 532 =

14

SUBTRACTION same time

Subtraction, can be introduced at the is

we have

2.

we have

left?

Now

us take

let

We

find that

We

Then we add

2

have

In setting

in

For example, when we add

being taught.

addition

2 left.

and

mind our hand

2.

1

away, or subtract

we have

have

4.

Now

to

1

1,

What do

1.

Clear the abacus.

left.

1

that

subtract

let us

We

2.

beads and clearing, we must always keep

position; the left

hand always follows the

right.

Let us set 32 and subtract 21 in the tens

column with

to the right

and set the

wish

of the 32

We

and take

in the units

set the 3 of the 32

Then we move our hands

From

in addition,

place our right hand on the 2

away, leaving a

Then both hands move 1

We

column.

2 in the units

Here again, as

to subtract 21.

left to right.

the right hand.

.

to the right

column from

the 2.

1

3 in

in the tens

this

we

we work from column

the tens

column.

and the right hand clears a

Our answer

is

1 1

This

.

is

subtracting directly.

For every addition problem, can be given. 1.

We

Let us subtract

a subtraction

start by saying let us add

1

from

1.

Our answer

is

1

example

to zero.

zero.

We

Now

to

set

.

15

zero add

We add 3

1

have

Let us add

directly, so

clear

between

and

6

I

and our answer

2

I

We

give back to

.

the-

set

.

We

1.

we cannot add

partner of

We now

is 6.

and

5

subtract

1.

words, the difference

In other

more.

But since

do not have

I

3

3

Then how much money must you

you a nickel.

me? The

We

1

could also have said that you had

will give you 3

will give

is 5

Subtract

3.

find that

and we recall that

set 5

is 5.

1

We

more.

3

have

and we have zero.

2

and our answer

1

cents and

cents

we

So we clear

3 is 2.

We

Subtract

2 left.

more.

2

We

Let us add two more.

.

child will return 2 cents and have 6

remaining

To continue with

from

6

and have

cannot clear

ner of

wanted

5

We

we

from

remainder

directly.

of 3

3 left

it

We

5.

We

away?

Now we

3.

now have

take

we can

but

recall that

to subtract 2

extra did

return

2 directly,

and 2?

Let us take away

remaining.

5

example, we took

the subtraction

clear

is 3,

see that when

3

we took

on the abacus.

There

is

1

left.

directly and our answer

is

We

We

too 2

zero.

is

5

1.

How

many.

from

subtract

subtract

5.

and 2.

5

We

more.

we return

so

or the difference between

,

What

5.

subtracted

took away

2

1

the part3.

We

many-

So we

we have

2 is 3.

a

We

This can be done

This can be done

16

We now more.

Then we add

wish

add

to

We

4.

from

start

the beginning and add

We must

3.

cannot do

left

10, but

we only wanted

Four from 10

The

hand sets

left

6

is 6.

It

We

we have column.

we

to subtract a 1,

Since there

is

1

sum

5

and 4 more

to subtract

clear 2 directly.

We

cannot subtract 5.

We now 5

and

We

in the unit

We

clear

3 is 2,

.

say,

left

and 4

is

3

In

10.

1.

Since

from 10

1

have

we turn

hand will clear a

We check

column.

We

We wanted have

of 6

how many

is 9-

We must

We now

7 left.

We

wish

9

to

But

10.

to see

So our right hand will give back

9-

gave

partner to

a zero in the units column,

we only wanted

we took away.

We

put our right hand on the units

Our

fore return

is 4's

turn again to subtraction; 10 minus

the left hand to assist us.

extra

10.

set a 10 with the left hand, clear 6

with the right, and find that the

Now we

We now

2.

with the right hand.

working with partners, we can also ask what

make 10?

2

more; how much too many did we give?

4

we clear

so

is 6,

Then

.

directly with the right hand,

it

hand will help us.

so our

and clear

set 5

1

there-

by setting a

to subtract 2.

wish to subtract

3.

directly, but our right index finger can to subtract 3.

4.

Seven minus

so

we gave back

3

So we clear

gave us

2 after

4.

5

and set

The partner

clearing the

5.

2.

of

We now

17

subtract

more

more.

2

4,

3,

1

6,

5,

Our

.

Now

to the 2.

We

This can be done directly.

We

subtract 4.

start again and add 7

8,

,

and

have zero

from

We now

9-

cleared

5

and set

1

4.

We

We

from

10

must return 5

and

2.

column.

We now We

have

9 in

subtract 4.

the units

This

with the right hand.

We now

with the right hand.

We

the units column,

10.

10.

How many

the partner of

1

7

too

6

will set

column and

3 in

the tens

We

clear 4

5.

This

is

because there

must assist

to subtract 6

so

Three

We

we subtract?

6 is 4;

3.

many, so we

done directly

are now ready to subtract

We wanted

and

we

7.

subtract

so the left hand

extra did

is

hand.

5 is 4;

done directly.

is

The right hand cannot subtract a

hand clears a

from

The right hand will give back

7.

hand

extra did we clear?

That means that we cleared

is 7.

the left

but the left hand can help

3,

How many

subtract a 10.

subtract

are now ready to subtract

The right hand cannot subtract a

We

1

2,

1,

subtract 2 directly by clearing 2

beads with the right hand.

us.

We now

left of the right

we say

directly, so

2

left.

by adding

to 45

column and

right hand is on the units

cannot subtract

1

have 45.

on the tens column immediately to the

We

Let us add

Six

us.

6

is a

from

30.

zero

The

in

left

and we subtracted

from

we give back

10 is 4; or

4 leaving us 24.

18

Now we

are ready to subtract

we must go

units column,

We

will subtract a 10. to subtract 7, 7

from

10

but

must give back

so

so

2

cannot subtract

too

many, so

give back a

We now

beads.

we clear

so

8,

too

3

ready to subtract

9-

have

We

9

wanted

"We

many, so we

gave back

We

17.

We

3

too many;

2

subtract

8.

have taken away

2

Eight from 10

the right hand returns 2 beads.

We now

hand

cannot give back

have 10.

left

we check by saying

We

5.

4 in the

many.

too

3

We

the unit column.

we can

We

2.

have taken away 10,

The

column.

to the tens

we subtracted

3 in

we must clear

gives us

we have only

That means we took away

is 3.

directly, but

Since

7.

We

beads on the abacus.

are

do this directly, and our result

is

zero.

For the next exercise,

To

add

this

1,

on the abacus. 1,

we

2,

4,

3,

set

5,

2,

3,

We 6,

7,

4,

6,

7,

8,

9-

We

This

is

get to 47

1

1

left

on the abacus.

We now

9-

same manner,

have

have 46

starting with

on the abacus.

1,

2,

3,

we begin subtracting

4,

1,

5,

2,

6,

3,

8,

7,

etc.

,

Then

Now

start out with 2 on the abacus.

we continue as before by adding

When we

and

8,

subtract in the

more bead and

1

5,

start by setting

and

9

etc

an excellent exercise for addition and subtraction. Let us examine a few subtraction examples.

We

shall

19

We

set 64 on the abacus.

column and

the left

hand

write

6

with the right hand in the tens

on the rod to the

is

hands move to the right to set the 4 While the right hand sets the

We

column.

To subtract

the 6 in the tens column.

subtract set

1

we clear

directly, so

it

with the thumb.

We

46.

hand rests on the tens place the right hand on

4,

we

must help

Since

This leaves a

Z in

we cannot take away

The

us.

back

4 and give

so

we give back

5,

so

is

Now

1

let

1

4,

6,

we

our

will say 6

too many; therefore,

from

us subtract 89

Since we cannot take

the

from

from

left

hand

from 10

cannot give back 4 ones directly,

9

182.

to subtract the 8

both hands to the right to subtract the

is

to subtract a 6

we clear

a

1.

8.

hand on the tens column

There

6

Both

the tens place.

But we had to give back 4 and we gave back

we have given back

Our answer

We

4 ones.

5.

we cannot

pointer finger will take away a 10

left

and since we only wanted to subtract is

find

a 5 with the index finger and

hands move to the right and we are ready the 4 ones.

Both

it.

column.

in the units

4, the left

now subtract

will

left of

from

2,

our

We

from

put our right the 8.

9 in the units

left

We move

column.

hand will help us.

a zero in the tens column, so our left hand will go to

hundreds column and clear

1

bead which

is

equal to

1:0

tens.

20

Since

we wanted

the tens column. in the units

to clear only

We now

column with

say

1

9

ten,

we

from 10

the right hand.

will return 9 tens in is

1

and return a

Our answer

Subtraction Exercises: 24

-

12 -

124

-

93 =

46

-

31 =

275

-

184 =

76

-

35 =

378

-

179 -

82

-

19 =

500

-

9 =

97

-

68 =

328

-

29 =

is

93,

1

21

MULTIPLICATION Since multiplication

rapid

is a

form

of addition, the

teacher can introduce multiplication whenever she feels fits into

her mathematical curriculum.

teacher

may wish

add

to

2

demonstrate

two times, we can say

can set first

answer

the

to

1

set of 2

is 4.

to

The second grade

her class that

twos are

2

And

4.

and then another set

The child may have learned

shows the child us examine

let

that 2 x

how

1

is 2,

a

name.

We

speak the same language.

all

is

problem has

the multiplicand.

done

is

factors.

way

2,

4,

6,

8,

The teacher then

etc.

etc.

,

Now

each part

of a multipli-

learn these names so that we

The number which

is

multiplied

the multiplying is

These two together are called the

the multiplier.

The answer

that

The number by which

is

Let us multiply the

2 is 4,

the child

this is set up on the abacus.

The student must learn cation

x

2

we want

an earlier

at

The mathematics becomes meaningful.

if

and see that

of 2

age to count by 2's, but now he sees how he gets etc.

it

to the left on the

called the product.

3x6.

The multiplier

extreme

left

column.

the right hand side of the abacus to figure out

3 is

placed

Now we go where

the

all

to

22

We

multiplicand should be placed.

are used

count up how

how many are used

in the multiplier,

cand and add one column for the abacus.

many columns

in the multipli-

For example,

the digit 3 there is one place; in the digit 6, there is

and we add one for the abacus giving us

extreme

column left

right,

we count

hand

is

immediately

the right

hand moves

the 6 is.

the 8 as

3

x

As

it.

1,

or the

hand has followed

6.

the

extreme

1

is.

Our answer

Let us now multiply plier 7 to the

move

1,

to the right

left.

7

x 43.

one place for the abacus.

We

We

We

is

The

where

and set

to the right

resting on

then pick up the right

will write our multi-

figure out

where

to

count up our digits:

2 digits in the

We

and

.

18.

We must now

write the 43 of the multiplicand. digit in the multiplier,

is

it

moved

1

with the

1

or the column

the right hand has

column where

hand and clear the

6

The

6.

With

and sets the

to the right of the 6

After we have set the

we say

the

As we say one,

one eight

6 is

hand rests on the

to set the 8, the left

1

we place our

-there

-

hand follows the right hand and as we write the

right hand, the left

the

From

places.

to the left of the right hand.

we say

place

places - -first column units, second

tens, and third hundreds

the right hand on 6,

left

in 3

3

1

in

multiplicand and add

go in 4 columns from the extreme

23

right

column and write, or

thousands place, and the

(The 4

set, 43.

We now

the hundreds place.)

3 in

the

is set in

must check and see how many empty columns we have after the

There should be 2- -one for the

the 43.

3 of

We

multiplier and one for the abacus.

left

move

2 in

the tens

We

units column.

now place

we

say:

column as we say

both hands to the right as

43 and our

3 of the

index finger resting on the 4 of the 43

two (setting the

set

the right hand on the 4 of the 43 and say

hand follows

it,

we

to the right as

resting on the 4).

3 is

then we

it),

we say one, and

x

7

in the

it

pick up the right hand and clear the

two (the right hand moves left

are now ready to multi-

With our right index finger on the

ply.

digit in the

7

We

3.

x 4

is

and the

set the 2

Then we move

to the

right and are ready to write the 8 of the two eight with the 2 in the tens

that

was

column.

The

left

hand

now resting on

just written in the hundreds column.

hand cannot set an eight with the the left hand can help us.

It

wanted only

8,

from

the 2

Since the right

already in the tens column,

which

is

equal to

Because we added 10 when we

the right hand will clear the 2 that is there;

10

hand clears the

2

will give us 10

10 tens in the hundreds column.

since 8

is

is 2

2 in

and we have added the tens place.

We

2

too

many,

the right

pick up the right hand

24

and clear the

The answer or product

4.

is

There should

301.

be no difficulty in reading the answer because

it

will always be

on the extreme right side of the abacus.

Let us now multiply 43 x the multiplier to the

extreme

to set the multiplicand,

left

We

7.

7.

We

will set the 43 as

and we shall determine where

count

multi-

2 digits for the

plier, one for the multiplicand and one for the abacus.

means we go 7 in

We

the thousands column.

and the 7 is

columns from right

in 4

left

two, and

2 in the

and we write the

to left

put our right hand on the

it,

and we say.

we move our hands

to the right

hand

to the left of

hundreds column.

2 of the

and write the

Then we move both hands

position while 2

where

write a

we

the right 2

where

hand will give is 8,

hundreds column) say:

3 (of

hand

is

the 43) x

7 is

hand resting

of the tens

column.

column.

The

We

to the right left

hand

is

and

2,

which

it

we cannot The

2

1

left

from

10

(of the 21)

now resting on Ijand

set the

was resting.

and set the

pick up the right

we

Since

hand will help us.

the right hand will clear the 8 on

in the units

two, (and

Because we wanted only

Then we move both hands

to the

hold the hands in that

resting, with the 8).

the 8 is, the left

10.

We

.

7

4 of the 43 x

right and set the eight in the tens column, (our left

on the

That

the zero

and clear the

7.

25

We now

When we

read the product, 301.

must remember

4x7,

For example, we

might mention here that when a zero

corded on the abacus

it

For example,

will set the 5 to the

extreme

on the

7

3

and

and say:

x

and set

hand and clear the

it

7.

three (we

7 is

to the right

we say zero 3

which

touch as

is

if

already

zero

5

in the tens

we were recording

right hand and clear the 4.

x 4

is

we

find

to the

then say five,

We

move

pick up the right

With

4.

two (our right hand

hundreds place).

As

our right hand rests on the

place and the zero.

we

give the

3

a little

Then we pick up

The answer or product

Let us multiply 4 x 32.

We

x 47.

place our right hand

We

2 in the

of 20),

5

digits

Place the right hand on the

and sets the

(for the

the rod or

move our hands

in the units place.

our right hand on the 4 we say:

moves

us multiply

We

right and set the 3 in the tens place). to the right

be re-

and count 4 columns from

for the abacus.

1

5

left

let

By counting up our

right to left to set the 47.

we have

is to

and hold the hand there so that we

know our position.

that

we press

in multiplication,

column gently or touch

first

3x7.

and then we said

We

we

7,

to multiply the 7 by the digits of the multi-

plier in the order of their occurrence. said,

multiply 43 x

is

the

235.

Place the 4 to the extreme

26

left;

from

then count in 4 places

the

extreme

32, putting the 3 in the thousands place

We

place.

are now ready to multiply.

the 2 and the left hand on the 3.

we say zero, our hands move is

and

right and set the

hundreds

2 in the

Place the right hand on

"We say 4 x 2 is zero

8.

As

and our right hand

to the right

on the tens column as we record the zero by touching the

bar.

Then both hands move

to the right

sets the 8 in the units column.

we multiply one 2 digits.

If it is

digit

an

hand and clear the 4 x 3 is one two. setting the

1

(We must learn

whenever

that

by another digit, our answer will have

8,

2.

and our right hand

it

is

We

zero eight.)

We

pick up the right

put the right hand on the

As we say one, both hands move

immediately

We

to the right of the 3.

both hands to the right and set the

2 in the

pick up the right hand and clear the

3.

3

and say:

to the right

then

tens column.

The answer

Multiplication Exercises: =

17 x 14 =

25 x

5

37 x 6 =

43 x

7 =

63x5=

82 x 7 =

49x9=

95 x 3 =

56 x 8 =

71x6=

is

move

We

128.

28

tens place;

we move our hands

We

the units place.

The product

is

to the right

moves

2

zero.

and say:

2.

We

will put our right

Our

right

to the right

We

and the right hand

pick up our right hand and

then put the right hand on the

and say:

3

4 x 3 is one, (moving to the right and setting the

thousands column)

,

1

in the

and we say two as the hands move to the

right and set the 2 in the hundreds column. right hand and clear the 3.

The product

is

We

pick up the

1208.

Multiplication Exercises Using Zeros: 5

x 302 =

hand

and touches the rod on the tens column as

Then we move

We

3 02.

4 x 2 is zero eight.

sets the 8 in the units column.

clear the

3.

912.

to the right

we say

2 in

pick up the right hand and clear the

Let us now multiply 4 x

hand on the

and set the

703x4

=

=

902 x

3 =

204 x

809 x

5 =

207 x.14 =

7

805 x 32 =

107 x 36 =

608 x 24 =

506 x 203 =

29

MULTIPLICATION OF TWO DIGITS Let us take the example 45 x 67,

extreme

There are two

left.

two digits

We

count

5

in the ten

We

67, giving us four digits, then

for the abacus, making

rods from right to

and set 67

left

thousands place, and the

This leaves us

3

to the

digits in the multiplier, 45, and

in the multiplicand,

we add one more place

Write 45

7 in

it

5

places.

placed

(the 6 being

the thousands place)

empty columns or rods

extreme

to the

.

right.

are now ready to multiply.

With our right pointer finger on the

hand pointer finger on the

6,

we

say.

7

and the

left

We

4 x 7 is two eight.

set the 2 to the right of the 7 and then set the 8 to the right of

The

the 2.

hand on the 5

x

7 is

on the

left

8

and our

three five.

8.

We

left

We

hand

is

write the

cannot write a

3

on the 3

7.

Then we move our

the units column.

column.

The

2.

where

the 8 7,

hold our right

Now we

where our

adds a 10 and our right hand clears a is

We

hand follows the right hand.

say:

right hand is

so our left hand

is;

because

3

from

10

right hand to the right and set a 5 in

left

hand rests on the

Pick up the right hand and clear the

in the tens

1

7

from

the 67.

30

Place the right pointer finger on the

two four.

6 is

We

add 4 by setting

hand on the is

Set the 2 to the right of the

and the

7

three zero-

the

7.

Since

5

We

and clearing

1,

hand on the

left

write the

3

we cannot write

7,

because

3

Then we

2.

where

from

right

hand and clear the

1

6.

10 is

done

manner.

in the

5x6

the 7 is, our left 2 is

and our right

Then we move our

7.

with the right hand.

Our answer

say:

is

1

We record

is.

We

pick up the

3015,

Multiplication of two digits by three digits a similar

to 3,

right hand is holding

hands to the right and set the zero where the the zero by touching the

4 x

and we hold the right

hand will help us by adding 10 where the hand will clear

and say:

and add 4

6

where our

a 3

6

Multiplication by three or

more

is

done

in

digits is

same manner.

Exercises: 34 x 29 =

98 x 56 =

103 x 42 =

235 x 74 =

206 x 18 =

104 x 36 =

87 x 45 =

246 x 39 =

342 x 58 =

234 x 125 =

31

DIVISION

As we begin

division,

we must acquaint ourselves

with the names of the different parts of a division problem.

The number

that does the dividing is called the divisor.

The number

into

The answer

is

which

the quotient.

divide 6 into 48. the

dividend, and the 8 In setting

6

is

is set to the

tient will

appear

divided

For example.,

extreme

left of the

abacus.

extreme right on the frame.

is

the

The divi-

The quo-

to the left of the dividend.

the divisor into one or,

we have one

if

digit in the divisor

We

in the dividend.

then divide

necessary, more digits in the divi-

This results in a quotient figure which

the abacus.

we would

the quotient.

and as many digits as we wish

is

if

would be the divisor, the 48

In short division,

dend.

called the dividend.

is

up an example in division on the abacus, we

set the divisor to the

dend

it is

entered on

is

The figure which has been entered

in the quotient

then multiplied by the divisor and the product resulting

from

from

this multiplication is subtracted

Let us now

work an example

will be 792 divided by 4.

Our divisor

.

Our is 4;

the dividend. first

example

our dividend

is

32

We

792.

set our dividend,

right

792, to the

hand on the

the

7 of

extreme

We examine

digit 1.

digit in our dividend.

If

We

will

put our

4 will go into 7

Four

4's in 7?

The question arises where

one time.

We

left.

right.

92 and ask:

7

how many times, or how many

tient,

extreme

will set our divisor to the

will go into

to place the first

quo-

the divisor, 4, with the first

the divisor is equal to, or smaller

than, the first digit in the dividend,

we skip one column

the left of the dividend and set the

in

went into

7

hand

to the left of

we say zero, the

it,

we

the right hand

column with

1

hand on the

right

say:

x 4

1

moves

7

ing 5 and setting

with the right hand.

to place the 9?

Since

immediately right

it

say:

larger,

We

we

is

subtract 4 by clear-

We

are then ready to

larger than the

9,

we

3 in

Where number the divi-

will write the quotient digit,

to the left of the 3 in the dividend.

hand on the

As

right

the divisor with the first

Our divisor 4 is

4.

with

and rests on

4 into 39 will go 9 times.

We examine

in the dividend.

dend.

we

and subtracts

1

and the

zero four.

to the right

hand moves to the

divide again, and

is

1

to

Four

set the

Then as we say four, our

a zero.

1

our quotient.

column and

skip one

Then with our

the right hand. left

We

one time.

7

will say:

9

x 4

is

9,

Now, with

three six or 36.

the

As

33

we say three,

the right

column where there

hand leaves the

is a 3,

and subtracts

The right hand

the right and say 6.

and we subtract

in the dividend,

examine the dividend and see 4 into 32 goes 8 times. the

3 in

the dividend,

Now, with

of the 3.

immediately 32.

we have

the right

to the right

it,

it

we

say:

right hand to the 2,

We

is

x 4

8

of the

falls

tient.

We

hand

on the

3

the

We

one column for

off

Our quotient

go to

Everything

is 198,

We

can

the division by multiplying the divisor by the quo-

We

say:

left

it.

the digit in the divisor, and one for the abacus.

now check

to the left

Then we move

it.

abacus and count

to the left of that is quotient.

larger than

three two or

is

are now ready to read the answer.

extreme right

9 is

say:

and the

8

to

and we

3

We

3Z.

and the right hand

say two and clear

the

immediately

hand on the

As we say three, we clear

of the 32.

the

This leaves

will write

to the left of

We move

We move

it.

Since 4, the divisor,

we

to the next

now where

is

6.

that

moves

9,

place the right hand on the

3 in

the tens place.

and write

2 in the

clear the

8,

We

ones place.

We We

and the

We move

4 x 8 is three two or 32.

and set the

8

then

left

on the

9

to the right

move

to the right

pick up the right hand and

place the right hand on the

9

and say:

4x9

34

is

right of the 6

9,

where the

and say:

moves

1

move our hand

then

x 4

is

We move

hand sets the 4 with the

had

in the

It

1.

we

good policy

over until the process

is

In this

ting, adding

it

7.

We

have 792.

the

to

hand

pick up the right

This

what we

is

it

It is

also good

through by multiplica-

the practice of dividing, subtrac-

and multiplying.

Exercises: 255

5 =

f

1

do the same problem over and

well established.

way we have

1

first started to divide 4 into 792.

practice to divide and then check tion.

hand on the

the right

on to the right and the right

making

3,

We now

dividend when is a

pick up the right

and touches the column next to the

record the zero.

hand and clear the

and set the

As we say zero,

zero four.

to the right

We

Then we place

9-

the 3 to the

to the right

the tens column.

3 is in

hand and clear the

to

As we say three, we put

three six or 36.

357

7 =

*

8199

t

9 =

2112

2418

t

6 =

896

t

6 =

3936

+

8 =

812

*

4 =

9876

-

6 =

2348

t

s

3

=

9 =

35

LONG DIVISION Long division

made up

of

is

the process by

any number of digits

up of two or more digits.

same manner

we

as

We

is

set

which a dividend

made

divided by a divisor

it

up on the abacus in the

example.

set up the short division

Let us take an example with a divisor of 45 and the dividend of

We

We

015.

3

write our divisor to the extreme

left.

write our dividend 3015 to the extreme right.

When we

divide with

for a trial divisor.

Since 45

trial divisor should be 5

from 43

49

to

We

it

is

.

I

more than one is

close to 50

we say

a good idea to use 5 as a trial divisor.

are now ready to check and see how

say that digit of

Since

5 will

it

would not go into the

go into 30 six times.

numbers

of the

We check

we

the first

our divisor with the first digit of the dividend and we

immediately

hand on the right

many times

first digit of 3,

find that the 4 in 45 is larger than the 3 in 30.

the 6

our

that

have found that when dividing

the trial divisor, 5, will go into the first two

dividend.

we look

digit,

6

we

hand moves

to the left of the 3 in 30.

say:

6

x 4

is

to the right

two four.

So we write

With our right

As we say two, our

and we subtract the

2

from

the

3.

36

Then our hands move right

to the right

and as we say four our Since we cannot

hand touches the column with a zero.

from zero, we subtract

take 4

wanted

hand cleared a

to subtract only 4 but the left

Our

the right hand will return 6.

right hand

6 of the quotient

say:

divisor, which

is 5,

right hand,

which

and setting

2.

is

on the

6 will

subtract a

records the zero by pressing the rod.

We

see that

go into 31 six times.

sor

is

5 will

We

3 of the 31 in

5,

hand on the

divisor, 4,

is

6

we

say:

two four.

slides over to the 3 and

move

to the right as

subtract 4

from

Now we wanted

1,

6

is

5

we say

now on

the

1,

are now ready to into 31

and we

Since the 4 in the divithe dividend,

the 6 immediately to the left of the 3 of the 31 right

the

by clearing

3

which

take the trial divisor,

greater than the

in the

to the right as

the right hand,

divide again.

and

it,

As we say three,

Then both hands move

As we say zero,

zero.

times the second digit

three zero.

is

so

10,

remains on

the 6 and the left hand is on the zero to the left of

we

We

a 10 with the left hand.

times the

.

we write

And with our

first digit in the

As we say two, the right hand

we subtract

we say

4;

2.

Then both hands

and since the right hand cannot

the left hand assists by clearing a 10.

only 4 and the left hand cleared 10.

That

37

means we took away

With our right hand on the

a 6.

we

zero to the

left of it,

we have

our quotient

zero. 5

many; so the right hand will return

6 too

in

say:

6

and the

"We

2.

times the

move

5 in the

column where

We

the 5 is.

from

it

to the right

hand record the subtraction

left

hand on the

by going back to the last digit

As we say three we subtract

and setting

We

7

divisor

is

the 7 by clearing

and with the right

zero by pressing the

of the

are now ready to divide again.

We

look at our divisor and we examine our dividend.

see that

we have 45

45 one time.

left.

Then we say

Since the 4 in the 45

that 45 will go into

same

is the

as the 4 in

same, we

the divisor and the second digit in each are the

must skip with the

a space

from

We now

6.

the 4 in the dividend and set the

multiply again, the

1x4

four.

We

say zero, then

As we say

four,

we subtract

hands to the right.

As

where we cleared

the 4,

zero clear

five,

we move

move

the right

we

say:

to the right

to the right

hand rests 1

x

5 is

in the

zero

left

and say

We move

the 4.

1

giving us

zero four, the right hand moving to the right and the following.

three

both

column

5.

As we say

and when we reach the

5

we

it.

We

are now ready to read the answer.

In

order to

38

read our answer we must figure out how many spaces we

must go

from

in

right to left before

count the

number

we add

column for

1

count in

3

of digits in

places from right to

We

is

it.

left.

1

is 3.

we

First

our divisor- -we have

the abacus-- 2 and

Our quotient

quotient.

we read

then

2;

So we will

Everything after that

is

67.

can now check our answer by multiplying the

quotient 67 by the divisor 45. In

examining how the example

moment, we

spaces or rods from right to

that there are 2

plicand and

up at this

find that the quotient, 67, is located in the

proper position for multiplication. 5

is set

1

left.

The

6 of the

We

67

is in

can check and see

columns for the multiplier,

multi-

2 for the

for the abacus.

Every division example should be checked by multiplication; and every example that is multiplied should

be checked by division.

These are excellent exercises and the student requires continual practice. Let us division

is

work an example with

a remainder.

done in the same manner, but

let us

remainder will be and how we should treat

it.

The

see where the

39

We

Let us take the simple example of 4 into 1234.

extreme

set the 4 to the

extreme the

1

in the

place the

Then we

move

We

right.

dividend

we

TThen

left.

there are

say:

3

set the 1234 to the

fours in

smaller than the 4

is

say:

3

x 4

is

2

Since

.

left of the 1.

As we say one, we

one two or 12.

the hand to the right and subtract

When we

1.

say two,

we move

the right hand to the right and subtract the 2.

Then we

say:

divisor

is

immediately or 32.

Then we move

the right

the right

from

4.

to

hand

hand

2,

we

the dividend,

We

say:

set the 8

x 4

8

to the right

is

three two

and subtract

3.

we say two and

to the right as

This leaves a

Since 4 will not go into

now ready

3 in

to the left of the 3.

We move

subtract 2

Since the 4 in our

4 into 34 will go 8 times.

larger than the

we

in the divisor,

our answer immediately to the

3 of

1

2 in the units

column.

the 2 is our remainder.

check our answer, or quotient.

We

We count

are off

one place for the divisor, one place for the abacus --this

makes answer

2

places from right to

is

in the last

left.

308 and a remainder of

We 2.

read our answer.

Any figure

that

Our

would be

columns would be remainders.

Now

let us

check our answer.

quotient by the divisor.

We

We

will multiply the

will put our right hand on the 8

40

of the quotient

and say:

we move our hands

8

x 4

is

to the right

and add

3

x 4

is

to the right

3

and say:

As we say one, we move our hands

When we say

1.

and set the

answer 1234.

remainder

the right hand

Then we place our hand on the

one two.

right and set the

we move our

2 to the

We pickup

that is already in the units place. 8.

then

3;

hands to the right as we say two and add

and clear the

As we say three,

three two.

This

is

two,

Clear the

2.

to the

we move our hands

We now

3.

have

what we had when we started.

in

our

We

divided 4 into 1234.

This

is

an excellent exercise for division.

out by dividing 2 into 1234.

Then try

3

into 1234,

Start

and so on.

Both young people and teachers should do the same

example over and over. traction as

it

We

is in

is

as true in addition and sub-

division and multiplication.

will notice in division that

mated correctly the dividend,

along.

This

When

the

number

we can always the divisor is

of

if

times our

we have trial

not esti-

divisor goes into

divide another time as

compared with

we go

the dividend and

found to be smaller than the first digit in the dividend, we skip a column.

At this time the correction

estimated quotient.

If

we have

not put

down

is

added

to the

the correct

.

41

answer

we go

each part

in

of the quotient,

of

use

will go 7 times.

it

than 22, we set the quotient

We

2 is 14;

We now

subtract the 14

is

We

43,

greater than the divisor, 26.

4.

We

1

rod and add the

multiply the

3

1

1

.

to the left of the 7,

times 26.

the 22; the difference

Seven times

Hence, our estimate

Since 26

Our quotient

is 8

text,

"downward correction" and how

make our corrections. it

I

6 is 42,

is

is

of 7 is

and divide

3,

it

into

smaller than 43, we

making

the quotient

8.

from

We 43,

and 17 remaining.

Using the Cranmer Abacus,

he refers to this as "upward correction."

correction

greater

is

note that the remainder, 43,

to the 7,

Mr. Gissoni's

duced early,

with the 22

just added times 26 and subtract this

1

leaving us 17. In

from

will take our trial divisor,

The answer will be

skip

greater than 22

Since 26

immediately

have 85 on the abacus.

and the difference

too low.

7

is

multiply the estimated quotient figure,

Seven times is 8.

Since 26

We compare

for a trial divisor-

3

225 and find that

22.

as

along.

Let us divide 225 by 26, let us

we correct our answer

to

find that

work if

He also discusses

the

examples and

a trial divisor is intro-

will not be necessary to

make

a

downward

42

Examples: 444

12 =

ft

9734

18 =

*.

56883

-

38 =

73983

33456

-

272 =

63345

ft

=

27615

-

4933 25025

11

-

-

25 =

9826

-

t

72 =

123 = 95 =

64 =

43

ADDITION OF DECIMALS The fundamentals are the same as those

whole numbers.

decimal point

The

in the

the

in both addition

or "unit

the

first

decimal point from right 3

us add

let

Since the smallest decimal

is

we

That

left.

3.

1

plus

2.

04 plus 2, 005, 3

decimal

Therefore, we will put our first whole number to

find our first

immediately

to the left of

We

Going from right to

left,

decimal place and set the whole number,

decimal point we will 2. 04.

to

places in the decimal,

thousandths, we need

the left of the first; decimal point.

add

decimals.

point for that particular example.

For example,

we

of

second decimal point from right to

becomes our decimal

places.

used as a

is

and subtraction

Should there be more than

must use

mark"

one, two, or three decimal places in

example, we use the

left.

addition and subtraction of

comma

When we have

and subtracting decimals

of adding

it.

set our

Immediately 1

tenth.

add the whole number,

the left of the first decimal.

We 2,

the 4 hundredths in the hundredths place of the tenths place.

This

is 2

to the right of the

are now ready to

to the 3

This gives us

3,

5.

which

which

is to

Next we place is to the right

places to the right of the

44

decimal place.

Now we whole number

are ready to add

T hen we add

5.

thousandths place which

mal

Our answer

point.

is 3

.

2.

We

005.

i.e.

Example;

,

4.

2 to the

005 by setting the

5 in the

places to the right of the deci-

is 7.

145.

Let us now look at an example with

decimal places,

add

more than 0005 +

2.

3 digits in

006 +

1

.

more than

3

the decimal.

000002 +

3

.

05 +

10. 3 + 1. 00002.

Since there are

more than

3

digits in

some

of

our

decimals; we must use the second decimal point, or unit

mark, from right

to left as our

decimal point

in this parti-

cular example.

First

we

set the

whole number 4

second decimal point(from right to

left)

.

to the left of the

Since

5

ten thou-

sandths requires 4 places after the decimal point, we touch the first three

columns for the zeros, and then we

We

006 by adding 2 to the whole

then add

the whole

2.

number

We now

6.

go

in 2

number

set the 5. 4,

making

places for the zeros

after the decimal point and write the 6 for 6 thousandths.

add

2

and

making

it

2

8;

millionths by adding the whole

and the

2

millionths

is set in

number

We

2 to the 6,

the sixth place

45

We now

after the decimal point.

making

the 8,

dredths place.

it

1 1

,

and setting the

Now add

whole number and set 1

and

2

3

5

3,

05 by adding the

hundredths

10.3 to the 11, making

in the

it

tenths in the tenths, place

hundred thousandths by adding

22, and set the 2 in the 5

add

1

hun-

21 in the

Then add

.

to the 21,

3 to

making

hundred thousandths place, which

places after the decimal.

Our answer

is

it

is

22,356522.

Examples: 3.

86 +

4. 09 +

06 =

6.446 +

6.

002 =

6.29 =

7.678 +

9.

085 =

2.

3.2+9-934=

2.3367+15.00245=

67. 5932 + 84. 000945 =

7,

69-93574 + 182.37095 =

21.7765 + 19-329 =

003456 +15. 3274 =

46

SUBTRACTION OF DECIMALS We decimals

have already mentioned that the subtraction

done

is

whole numbers. the decimal.

same manner

in the

as the subtraction of

However, we must determine where

We

same procedure

follow the

of

to place

we did

as

in

addition of decimals.

Let us examine a subtraction example:

Going from right to of the first

left,

we

Then

decimal point.

right of the whole

set the

We

number.

5.

whole number

354

-

2. 13Z.

the left

5 to

set 354 thousandths to the

are now ready to subtract

2. 132.

From 2.

the whole

This leaves us

leaves us with

3.

number

From

2 tenths.

dredths and that gives us

5

1

tenth and this

hundredths subtract

From

hundredths.

3

hun-

4 thousandths

subtract 2 thousandths and we have 2 thousandths. is

number

subtract the whole

tenths subtract

3

From 2

5

Our answer

3.222.

Let us now than

3

work

a subtraction

places in the decimal:

The whole number decimal place from right

7.

7 is

to left.

decimal are written to the right

674598

example -

5.

that has

more

342165.

set to the left of the second

The next six digits of the

whole number

in the

7.

From

47

number

the whole

subtract

3

7

subtract

tenths, leaving

3

5,

leaving

tenths.

2.

From

From 7

hundredths sub-

From

tract 4 hundredths, leaving 3 hundredths.

From

1

subtract

5

one hundred thousandths.

3

5

ten

ten thousandth leaving 4 ten thousandths.

the 9 hundred thousandths subtract 6 one

sandths leaving

4 thousandths

From

subtract 2 thousandths, leaving 2 thousandths.

thousandths subtract

6 tenths

millionths, leaving

3

millionths.

hundred thou-

From

8 millionths

Our answer

is

2.332433. Indirect addition and subtraction examples are done in the

same manner with decimals as they are done with whole

numbers

.

Examples: 6.42

-

4. 01 =

9-

67.081-42.931= 89-573

49-2339

48.374

-

-

784. 03964

-

-

4.

152 =

78.403-57.132 62.7193

=

19-8745

173

=

522. 7432 =

-

=

54.0347 =

123.73092

-

54.39705 =

548. 85357

-

528 78538 =

48

MULTIPLICATION OF DECIMALS In the multiplication of

plier and multiplicand in the

we

many places

point off as

in the multiplier

our product as there were places

in

decimal places

all the

from

let

us multiply

extreme

will set our 5 to the

right to left and set the 37.

say:

right,

5

x

7 is

three five.

and then set the

and say:

and multipli-

our product by

in

5

as

tenths by 37 hundredths.

Now we 7

We Move

as

left

the right hand to the right,

followed by the

left

hand, and as we say one, set

the 3.

pick up

put our right hand on the

one five.

As we move

3,

to the

We

to the right.

3 is

right hand.

rods

hand on the

we move

x

5

in 4

are ready to multiply.

and the 3

we move 7.

5

Then we count

left.

Set the

our right hand and clear the 3

we

other words,

right to left.

With our right hand on the

we

In

in the multiplier

many places

off that

For example,

from

the process of multi-

and multiplicand combined.

cand and then point

We

the abacus as

After we have completed our multiplication, we

plication.

counting

same manner on

numbers when we learned

set whole

add up

decimals, we set the multi-

to the right

it

with the

and say five, set the

Pick up the right hand and clear the

3.

5

Our product

with is

49

185.

Since we had one decimal place in the multiplier and two

decimal places

right to left.

extreme

1

for the

digit in the multiplicand,

gives us

3

places.

and set the

4.

zero eight.

We

in

places in our product

.185.

is

2 tenths

Then

left.

we count one place

off 3

Our answer

Let us multiply to the

we add them together and

multiplicand,

Then we must point

get three.

from

in the

by 4 tenths.

and

1

count in

3

place for the abacus.

columns from

the zero,

the 4.

to the right as

we say

2

we move our hands

eight and set the

That

right to left

With our right hand on the 4 we say:

As we say

4,

multiplier one for the

right and press the bar to the right of the 4 and

hands

set the 2

order to know where to set the digit in the

1

We

x 4

is

to the

move both Then clear

8.

Since there was one decimal place in the multiplier and

one decimal place in the multiplicand, we will point off two deci-

mal places.

Thus our product

is

.

08.

Examples:

.4x9-7=

63.2 x 14.43

.6x5.32=

2. 71

7.2x9-85=

8.6 x 54. 35 =

.24 x

9-

38 =

7.

x 82.

65 x

9-

3

=

=

326 =

50

DIVISION

In

we

our practice with division of decimals on the abacus,

treat the division as

bers.

OF DECIMALS

if

Then we subtract

divisor from the

we were working, with whole num-

the

number

of

number

of

decimal places

decimal places

The difference indicates the number

in the

in the dividend.

of places to point off in

the quotient.

However,

if

we do

not have any decimal places in the

dividend and we do have one or as

many zeros

more

in the divisor,

to the dividend as there are

the divisor and divide as in

then add

decimal places

in

working with whole numbers.

Since we have changed both divisor and dividend to whole

numbers, there are no places

to point off.

Let us take the example .4 into 8.4. 4 to the extreme

left.

Four

2

into 8

goes

Our 84

times.

right

we

then say:

2,

we

to the right,

on tp the right and subtract the

We

written to the extreme right. is

smaller

will skip a rod, and write the 2.

hand on the

As we say zero we move

will set our

Since the 4 in the divisor

than the 8 in the dividend,

Then with our

is

We

say:

2 x 4 is

zero eight.

press the column, move

8.

4 will go into 4 once.

Since the 4 in the

51

divisor

is

placing the

immediately

1

We

quotient.

say:

1

when we say

four.

x 4

to the right of the 2 in the

is

zero four.

We move

we count

two columns from right

in

left of this is the

Our answer or quotient

in

our divisor, and

in the

we

many zeros

dend by

in the divisor.

two places

when there

2

in

Since we have one

our dividend as there are deci-

There

becomes

is

is

one decimal place in the

a whole

number and our

one place in the divisor.

in the divisor, then

dend by 100.

and

96

This done by multiplying both divisor and divi-

960. 1

1

divisor and no places in the dividend,

divisor, so the divisor

becomes

from

21.

is

decimal place

mal places

1

any places in our answer.

off

Let us use the example .3 into 96.

will add as

to

answer.

one decimal place in our dividend, we subtract.

do not point

the 4

one digit in the divisor and

we had one decimal place

We

our hands to

is

Everything to the

we get zero.

column

answer or

Since there

one for the abacus,

Since

will skip a

column as we say zero and subtract

the right, touch the

left.

we

equal to the 4 in the dividend,

we

If

there are

multiply both divisor and divi-

This clears the decimal in the divisor and gives

zeros in the dividend.

in the quotient.

Hence, there are no decimal places

52

We in

divide 3 into

our divisor

is

moving

and

will go 3 times.

it

smaller than the

column and place

skip a

9

3

again the

3 is

6

smaller than the

the hands in the proper

these

as

9

6 in the

we say It

we

say;

2

Here

will go twice.

So we skip a

dividend.

x

Then we

nine.

3 is

answer.

zero

manner and subtracting

six.,

the

6.

And

moving Since

one digit in the divisor and one for the abacus, we

count in two places of

2,

will

zero nine,

2 to the right of the 3 in the

with the right hand on the

is

is

3

column for the zero, moving

how many times?

column and place the

there

3x3

the 3 and say;

to the right, touching the

will go into

we

the dividend,

9 in

on to the right and subtracting the say:

Since the

2

columns

from is

right to

left,

the answer.

and everything

Our answer

is

to the left

320,

There

are no decimal places in our quotient.

Fred Gissoni,

in his text,

Using the Cranmer Abacus,

describes a completely different method for locating the deci-

mals that

in both

when

the product and the quotient,

the contents of this

interested people in the field,

method

of locating the

manual are

The writer hopes fully

we may perfect

examined by a

more

decimal point.

Exercises: .205

* .5

=

.189-* 3 =

simplified

53

.279

25.25

6.66

43.56

.9 =

r

*

2.5 =

1.8 =

t

*

.71 =

330

*

7 =

96.3

t

3

.

370. 37

27.24

*

r

.11 ^

.32 =

54

FRACTIONS A

fraction

a

is

numeral such as 1/2 or 3/4.

introduction of fractions the

we must learn

mon

fraction 1/2,

The numerator taken.

1

the

is

1

numerator and

indicates the

The denominator

to distinguish

For example,

numerator and the denominator.

number

2 tells into

In the

between

com-

in the

the denominator.

2 is

of fractional units

how many equal parts

the unit is supposed to be divided.

The whole number example, 2/2

=

=

1 / 1

1

may have many names. keep

all 6,

divide

it

1

3/ 3 =

/ 1

1

take

many forms.

=

4/4 = 1/1 =

1

;

you divide a pie

If

you will have 6/6

all 4,

all the

etc.

1,

all of it.

It

and If

you

you will have 4/4 or

all

names

for

Nine-ninths, 2/2, 3/3, 100/100 are all

because we have

For

into 6 parts

your pie, or

of

and keep

into fourths

your pie.

;

may

1

parts into which the

number

is di-

vided.

Any number divided by own value.

Seven

t

1

is 7; 3

will be true no matter what

x

1

1

name

or multiplied by is 3;

is

1/2 x

Let us check this on the abacus. tor

2 to the

extreme

left,

We

keeps

is 1/2.

1

used for

1

1

its

This

.

set the

numera-

skip two rods and set the denomina-

55

tor 2.

order to reduce 2/2 to

In

first find the greatest

denominator.

The factor

denominator,

2, the

into the

common

numerator

2

factor of the numerator and the

common

is 2.

We

simplest form, we must

its

divide the

and change

it

to

1;

common

we

factor 2 into the denominator 2 and change

which equals the whole number

1

numerator, and

to 2, the

to the

denominator

8.

8?

It

is 4.

the 4 to

1

.

change the cess

is

extreme

What

is

left,

common

We

to 1.

Set the

skip two rods and set the

the largest

common

Dividing 4 into the numerator 4

factor of 4 and is

1

.

We

Dividing 4 into the denominator 8 gives us 8 to 2.

Thus, the simplest form

commonly referred

have 1/1

.

Let us find the simplest name for 4/8.

numerator 4

factor 2

divide the it

to

is 1/2.

change 2.

We

This pro-

to as the reduction of fractions.

Examples: 3/6 =

5/10 =

7/14 =

10/16 =

3/9 =

4/12

6/15 =

8/14 =

2/18 =

7/21 =

-

56

ADDITION OF FRACTIONS In the addition

and subtraction

One section

the abacus into three sections.

number, the second section third section

is

Since a

it

common denominator

will be, and then place

The whole number or the second unit

there

is

divide

for the whole

for the denominator.

The common denominator

If

is

we

,

for the numerator, and the

is

tion and subtraction of fractions,

what

of fractions

is

is

it

we

necessary

is

in the addi-

shall first determine

in the

denominator section.

placed to the extreme right.

placed to the

marker from

left of the

second comma,

right to left on the abacus.

one digit in the whole number,

it

is

placed on the

seventh column from the right.

The numerator

is

placed to the

or unit marker (from right to

mentioned before,

is

nominators should be

worked

out

whereby

and arranged

in

left).

left of the first

The denominator, as

placed to the extreme right. 3 digits

comma,

If

the de-

or more, a system must be

the three sections are

moved

to the left

such a manner that they do not run into one

another.

Since we generally need a

common denominator

in

57

we

the addition of fractions,

common denominator

shall get the

Then place

by multiplying the two denominators together. the

common denominator We

abacus. 5

of the

4 5/6

f

7/9.

One way

if

denominator section

are now ready to work an example:

We must

9,

in the

to

do this

and multiply the other

use the

first find the lowest

higher than

Two x

.

denominator

2,

6 will

which

is 3,

go into 18.

case

extreme

it

is 18;

which

right,

We

we place

so is

of the 4

common

(the 7th

to eighteenths.

to the

extreme

denominator

column) In

work

5/6 and set

the it

denominator.)

example in the

We

take the

marker or

are now ready to change the 5/6 to 18ths

common denominator

that 3 by the

.

whole number

skip a space and set the 6.

6 into the

Then we multiply

We

order to change 5/6

left,

the

the denominator 18 to the

section, which is to the left of the second unit

comma

number

the section for the denominator.

are now ready to

whole number 4

to see

(Many times we multiply

etc

is

does, we will

If it

the 9 by the next

two denominators and thereby find the In this

Then we check

9 is 18.

we would multiply

18; if nbt,

denominator.

take the largest denominator, which

is to

by 2

it

common

numerator

5

9

we

set the 5

We

divide the

18 and

we

get

and we get 15.

3,

58

We

set the

which

is to

numerator

the left of the first unit

For to

numerator section

15 in the

who

the student

helpful.

marker or comma.

learning for the first time

common

change a fraction to a

method might prove

is

abacus,

of the

how

denominator, the following

In this

example we are changing

the 5/6 to 18ths.

The student sets the numerator left;

the

extreme

then he skips a space and sets the denominator

the right hand on the the

5 to

denominator

common denominator

times.

He clears

on the numerator

and says:

that 5 and places the

1

has the whole number

5 in

5

the

4, the

he asks:

x

6

It

With

go into

6 will

how many times?

denominator

the 5

18

6,

6.

will go

3

and places his right hand

Then he clears

3 is 15.

numerator section.

numerator

15,

He now

and the denomina-

tor 18.

We

are ready to add the

4 in the whole 9-

We must

number

section.

5

We now

change the 7/9 to I8ths.

twice; Z x 7 is 14.

numerator section.

We We

We

7/9

in the

Nine into 18 will go

numerator

numerator.

examine our numerator and denominator and

numerator

is

5 to the

have the whole number

shall add 14 to our

have 29

add the

in the

We

find that the

larger than the denominator, so, we divide the

59

denominator

1

to the

whole number section.

subtract the

from

numerator.

Then we add the

once. in the

into the

1

from

whole number

We

say:

1

x 18

9

and get a 10

is 18.

We

the 2 in the 29; and subtract the 8

the 9 in the 29-

As we look

Eighteen into 29 will go

This leaves us

at the three sections

1 1

numerator.

in the

we read our answer:

10 11/18.

Examples:

1/5+2

3

1/3+4

5/6 =

4

5

2/7+4

4/7 =

12 2/9 + 11 4/9 -

6

3/8 +

7

7/8 =

5

2/3+4

5/6 =

7

3/5+4

3/4 =

9

5/7 +

5/8 =

8

2/5+3

4/9 =

11

3/5 =

3

5/6 +

7

4/9

=

60

SUBTRACTION OF FRACTIONS Subtraction examples are set up as in addition.

Again, we divide the abacus into three sections- -whole num-

bers to the

left of the first unit

the

second unit marker; numerator to the

left of the

extreme

First we find the

denominators

the 24 to the

We

8

extreme

right.

to the

To change 5/8

5

x

.

Clear the

3 is 15.

Now we

5

2/3.

24.

"We will set

number section which

5

8 of the

5/8 say:

24, set

8 into

5

and set a 15 to the extreme

numerator place which

Then

Skip a space and set the

and put the right hand on the

Clear the

5/8

24.

common denominator left.

7

is to

right to left).

common denominator

to the

by multiplying

are now ready to write

have changed our 5/8 to 15/24.

the 15 in the

comma.

8

-

3 is

marker (from

With the right hand on the times

We

extreme

the 5 of the 5/8 to the

3

Eight x

3.

will set the 7 in the whole

we change our 5/8

5/8

7

common denominator

and

the left of the second unit

8.

to

right.

Let us work the example;

the two

common denominator

marker; and the

is to

24 goes

and say: left.

Clear the 15 and set the left of the first

61

We from

will change the 2/3 to 24ths

goes

into 24

8

times.

Clear the

2 is 16.

now have 16/24,

then subtract the 2/3 from the

3

.

Set the 2 to the

Clear the

3

and set the 16

2

We

borrow

a whole one

and say to the

leave us a

number

1

1

in the

8

the whole

whole number section.

3 say:

left.

We

from 15/24.

we

number

left.

x the numerator

extreme

will subtract the 16/24

from

extreme

With the right hand on the

,

Since we cannot take 16 from 15,

will have to

2.,

which

"will

Since the whole

has 24/24, we will add that tothe 15/24 and this

gives us 39/24.

We

numerator from the 1

5

However, since we cannot subtract thirds from 24ths,

Skip a space and set the 3

We

the 7 which leaves 2.

15/24.

we

are now ready to subtract the whole number

are now able to subtract the 16 in the 39, and

we have 23/24,

Our answer

is

23/24,

Addition and subtraction must first be done with

simple examples with

common

we are demonstrating how tors.

In the

to

denominators.

At this

work with lowest common denomina-

beginning of fractions, one should

work

the

examples with direct addition and subtraction. Examples: 7

4/5-2

2/5 =

moment

9

5/7-5

3/7 =

62

15. 5)56

-4

18 11/15 9

4/9-7

17 7/8

-

-

1/6 = 9

8/15

3/4 = 11

2/3 =

23 7/9 8

-

3/5-2

12 5/9 =

4/7 = 8/9 =

14 1/2

-

7

13 3/7

-

12 3/4

=r

63

MULTIPLICATION OF FRACTIONS In the multiplication of fractions,

the abacus into sections.

We

we again divide

will place our

numerators on

the left side of the abacus and our denominators will be

placed on the right. Let us take the example 1/2 x 3/4,

numerator,

the

1, to

tor, 2, to the

extreme

extreme

left.

Then place

by 3/4, we will place the numerator,

numerator,

denominator,

two columns

On

4,

numerators

1

.

and

right

will then place the

denominator

we now have

2.

the

2.

in the multiplication of frac-

we multiply numerator by numerator, and denominator

We

by denominator. 1

two columns to the

to the left of the

The student learns that

tor

denomina-

the

and on the right section we have the

3;

denominators 4 and

tions

We

3,

the left side of the abacus 1

will set the

Since we are multiplying that

right.

right of the first

We

and say:

1x3

place our right hand on the first numera(the

second numerator)

hand moves from the numerator

the left hand is placed on the

hand clears the

1

1,

1

is 3.

to the

and as we say

and the numerator

3

As

the

numerator 3,

remains.

the left

3,

64

Then we and say:

4 x 2

tor 4 and

moves

put our right hand on the denominator, 4,

is 8.

As

to the

the right hand leaves the

denominator

2,

the left hand is placed

on the denominator 4, and as we say 4 x clears the

4,

Our answer

and the right hand sets an

is 3 in the

numerator and

denomina-

2 is 8, the left

8

where

hand

the 2 was.

denominator,

8 in the

or 3/8.

Let us now take as an example

and one -half set the

We

7 to

extreme

the

right-

extreme

8.

left.

We

Then we change

skip two columns after the first

numerator

1

1/7,

Three

changed to an improper fraction, or 7/2.

is

numerator

tor 2 to the

1/2 x

3

set.

1

numerator

Then we place the denominator

to the left of the

denominator

2.

We check

the

We

denomina-

1/7 to 8/7. 7

7

and set the

two columns

to see if

we can

cancel.

This example calls for an introduction to cancellation.

We check

to see

if

there

is a

number

that can be divided

evenly into one of the numerators and one of the denominators.

We

put our right hand on the first numerator, which is a

Then we place our checks

in the

divided by

7.

left

hand on that

7 ^yhile

7.

the right hand

denominator section for a number that can be

The right hand sees another

7.

Seven will go

65

The right hand changes that

into 7 one time.

right

hand returns to the

once; and change that

hand and we say:

left

a

7 to

7 to

1

a

The

1.

7 into 7

.

The right hand checks the second numerator. an

The

8.

moves a

1

to

left

hand

is

placed on the

and the second denominator 2

hand returns to the

and we change the

1

x 4

is 4.

8

As

is

and we say:

We

the 4 remains.

1

2 into 8

.

the right

will go

will go

Then the

goes 4 times

1

we

as

hand leaves the numerator

4, the left

Then we put our

is

are now ready to multiply.

and as we say 4 the

1,

2 to a

placed on the numerator

and moves to the numerator

numerator

Two

will go into 2 and 8.

8 to a 4.

The right hand say:

What number

The right hand changes the

into 2 once.

right

Two

and 8?

denominator

first

a 2.

is

is

It

while the right hand

3

The

check the denominators.

evenly into

goes

left

hand

is

1

placed on the

hand clears the

1

and

right hand on the first de-

As

nominator and

say:

denominator

and moves to the second denominator _l the

left

1

x

hand 1

is

is 1,

1

in the

x

1

is

1

.

the right

placed on the first denominator the left hand clears the

mains on the other 1

1

1

.

Our answer

1

hand leaves the

1

.

As we

say:

and the right hand re-

is

4 in the numerator and

denominator, or the whole number

4-

66

Examples: 1/3 x 4/5 -

2/5 x 5/6 =

3/4 x 7/9 =

7/8 x 4/7 =

3/8 x 4/9 -

2

1/2 x 4 1/3 =

1

1/7x1

1/8 =

2

1/3x1

2/7 =

3

3/4 x

1/5 -

11/6x2

2/5 =

2

67

DIVISION OF FRACTIONS

Division, of fractions is done in

The only difference

multiplication.

f

he same

we

is that

sor and proceed as we did in multiplication

manner

as

invert the divi-

fractions

of

5

i.e.

,

multiply numerator by numerator and denominator by deno-

minator

.

Since division of fractions plication of fractions

3

is

we can refer

the inverse of the multi-

to the division of fractions

as multiplying the dividend by the reciprocal of the divisor.

The inverse

of a

number

For example.,

fraction.

is

a reciprocal of a

number

the reciprocal of 2/3 is 3/2.

of

any

We

rewrite each division example as a multiplication example

and then find the answer.

extreme

leftj

let

us divide 2/3 by

and the denominator

the

3 to

;

we

numerator

set the 2 of

numerator

the dividend

5

2 to

extreme

then multiply this by the reciprocal of l/5

Of the 5/l the

j

Of the dividend 2/3, we set the numerator

1/5.

We

For example

;

the

right.

which

is

5/1,

two rods removed from

The

set on the third rod to the left of the 3,

1

of the

We

denominator

then proceed as in

multiplication of fractions, by multiplying the numerators (2

x

5)

giving us

1

in the

is

numerator, and the denominators

68

(1x3)

giving us

3 in

Our answer

the denominator.

is

an improper fraction, we will divide the de-

Since 10/3

is

nominator

3 into

the

numerator

10,

and our answer

is 3 1/3,

Examples: 1/2

*

3/4 =

4/5

*

1/2 =

5/9

*

2/3 =

4/5

*

2/5 -

1

1/2

1

1/3 =

2

*

5/9 =

2

1/4*1

2/7 =

11/8*

1/4 =

5

2/3*1

2/5 =

3

*

10/3.

1/2

1/3

*

2.

1/2 .=

,

69

PER CENT means hundredths,

Since per cent

should be easy

it

for us to write per cents as fractions or decimals.

were or

.

to say 15%,yOf 125,

15 and multiply

abacus

in the

15 x 125/1.

same manner as we

set the 15 to the

from

fourth rods

.

we would change

extreme

right,

places in our answer since per cent

The answer

we would

is

of 129-

then point off two

the

same as hundredths

it

of the 1/3

and the denominator

3

to the

(of the

1

whole number 129)

the denominator 3.

extreme

1/3x129, we

to the

right.

and set the 129is

We check

to

For example:

This would set up as follows:

would be placed

rods from the numerator 1

would be quicker

Instead of multiplying .33

could multiply 1/3 x 1291

set 125 and

We

revert to the multiplication of fractions.

numerator

and

the per cent has a simple fractional equivalent,

such as 25% =1/4, or 33 1/3% =1/3,

1/3%

fifth

would

is 18, 75.

When

33

15/100

We

set up .15 x 125.

The answer would be 1875.

multiply.

to

we

This would set up on the

and on the sixth,

left

extreme

the

15%

the

If

extreme

We

the

left

skip two

The denominator

placed two rods to the

left of

for cancellation and find that

70

3

will go into 129 forty-three times (changing the 129 to 43);

3

will go into the denominator

a

1

We

.

3

one time, changing the

then multiply the numerator by the numerator and

the denominator by the denominator, and our

answer

is

43/1 or 43.

Examples;

17%

9%

of 215 = of

12.3% 6.

5

05%

0%

3 to

23%

of

368 =

33

1/3%

of 912 =

of 47 =

87

1/2%

of 168

of 103 =

25%

420 =

of 1232 =

of

=

210 =

62 1/2% of

1

84 =

71

SQUARE ROOT We know

the two equal factors of the

16=4 many at

since 4 x 4

is

much

In this

the

same way as

changes with each step.

manual we

The square root

of

Though there are

manual we

in

number by

is

done

Instead of doubling the divisor, in

working with

our dividend.

places from right to

the decimal point.

to

in long division, but the divisor

left

and set 1225.

our digits into groups oi two from right to

root as

This

division.

Let us extract the square root of 1225. 6

how

shall discover

shall halve the dividend, thereby

smaller numbers

move

one of

is

tables of squares and square roots, they are not always

our fingertips.

this

number.

or 4 squared = 16.

16,

extract the square root of a in

number

that the square root of a

We

we have groups

two groups, 12 and 25.

will have as

many

in the original

For the

We We

will

separate

left starting

from

digits in the square

number.

We

first trial divisor

square root of the largest perfect square which

is

,

here have take the

less than

the first group (12).

First we will extract the square root of the 12 and say:

what

is

the greatest square in 12?

The greatest square

72

and the square root

in 12 is 9

of 9 is 3.

We proceed

division by dividing; multiplying and subtracting.

immediately Subtract

get 5.

16 leaving us

Five x

the 35.

5 of

We now

12.

have 325-

We

5.

=25.

5

square

=15.

3

We

have 12.5.

Subtract

square the

Divide the 25 by 2 and the answer

we know

of 5,

3 = 9

divide the 325

Five x

Since the remainder on our abacus

12.5.

of the

We now

1.

We

3

divide 3 into 16 and

Set the 5 to the left of the 16.

from

is

from

9

Set the

Three x

group.

in the 12

1

and this gives us 162.

in half,

15

to the left of

as in

is

exactly 1/2

that 1225 is a perfect square and

the square root of 1225 is 35. Let. us extract the

square root

we

that 86 is not a perfect square,

decimal place.

to a

it

out to

After the 86

two more groups

decimal and dividing

from

it

into

we

groups

set the

answer

out

decimal and carry

of

two from

left to right

the decimal point. in 86 is

We

9 to the left of the 8 in the 86.

answer

in half

have

shall carry our

thus adding 4 zeros after the

t

The greatest square

the

Since we know

of 86.

.

is 5.

We now

and we get 2

(

.

2 X.9 =

2.

1

•

5000.

8)

.

have

5.

.

subtract 81

0000.

We now

Place

9(9x9

We

= 81).

from

divide

5.

Set the 86 and

0000

divide 2.5 by 9 and

we

2 in the quotient to the right

73

We

of the 9-

.

We now

7.

subtract the 1.8

from 2,5 and

We

divide 92 into

have

.

have approximately

answer

is

27

9-

7000.

Place the

7.

8 (8

x

We

When we that 6

We

We

subtract 64

from 49 leaving us it

from

we

get

have

is

is

*

is

We now

9-

and we have 498.

6(8x6

We now

1.

group, 73,

73 giving us

in half

in half (6 x 6 = 36; 36

on the abacus and that see that 7396

Our two groups are

in the first

number

divide that

divide 8 into 49,

and divide

9-2 and our

separate the

right to left into groups of 2.

8 = 64).

have 996.

7000 and we

.

The greatest square

73 and 96.

.

7 to the right of

Let us take the example 7396.

number from

the difference is

We

= 48).

We

18.

2 =

1

8)

1/2 of the square of

square the

We

.

6.

subtract

have 18

Again we

a perfect square and the square root of

it

is

86.

Let us extract the square root of 841. 841 into two groups

For

from

We

divide

right to left giving us 8 and 41

.

the first trial divisor, take the square root of the largest

perfect square which

is

less than the first group (8).

The square root

greatest perfect square in

8 is 4.

Since the trial divisor

smaller than

from

2 is

the 8 and set the 2.

We

square

8,

we

2(2x2

The

of 4 is 2.

skip a place

=4) and subtract

.

74

it

from

the 8 leaving us 4.

Although 22

Z is 11,

*

have 441

we

still

us 29us 4.

We

number

We now

have 40.

half (9 x 9 = 81; 81

(9)

We

and place

9x2(18) and

multiply

*

We

divide

must have but one

to place next to the 2 in the quotient.

possible one -digit

.

Let us divide the trial divisor

in half giving us 220. 5.

22.

We now

5.

We

2 = 40. 5)

2 into

digit

take the highest

it

next to the 2 giving

subtract

square the

it

9

from

22, leaving

and divide

Again we see that

.

it

in

it

this is a

perfect square

Here square root

s

we

right to left (l first

a five -digit

is

number- - 1 7689-

-

76

-

89).

two groups 1-76.

We

The square root

We

subtract

789-

We

halve that, giving us 394.5.

69 from 176 giving us

giving us 3 (3 x 13 = 39).

on the abacus.

We

tient, giving us 9-

We

this is half of the

of 176 is 13.

We

square the final 9»

square of the final

7,

We

subtract 39

halve the

Place

square 13 which gives us

We

1

from

extract the square root of the

169.

know

find the

divide 17689 into three groups moving

that figure to the left of 176.

4. 5

To

j3

We now

divide 13 into 39

from 39 leaving us of 133 in the

giving us 4.53 in

have

quo-

Since

our quotient, we

that 17689 is a perfect square and that 133 is the square

root of 17689-

75

Examples: 9801

4 °96

6889

15Z1

1936

864 9

6241

1849

-

7744 8649

76

CONCLUSION The foregoing manual

is

offered to both teachers and

Cranmer Abacus

students in order to advance the use of the

The writer has found that mastery been of

of inestimable help in increasing the

of this tool

speed and accuracy

mathematical computation among her students

Overbrook School for the Blind. in the

has

at the

This manual was written

hope that these step-by-step explanations will make

it

easier for each teacher to learn the procedure, as well as to facilitate the teaching of the operations to the student.

The enthusiastic teacher will find a ready response

from

the students to

whom

opened through the use

a

of the

new world

of

mathematics

Cranmer Abacus.

is

E&rUBwfiaBfl

PRINT 511.2

DAVIDOW, Mae E. The abacus made easy.

D

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