Citation
A guide

Material Information

Title:
A guide arithmetic in Florida elementary schools
Series Title:
Its Bulletin no. 26, rev
Creator:
Florida -- State Dept. of Education
Place of Publication:
Tallahassee
Publisher:
[s.n.]
Publication Date:
Language:
English
Physical Description:
121 p. : illus. ; 23 cm.

Subjects

Subjects / Keywords:
Arithmetic -- Study and teaching (Elementary) ( lcsh )

Notes

Bibliography:
Bibliography: p. 120-121.
General Note:
First ed. publised in 1942 under title: Arithmetic in the elementary school.
Funding:
Bulletin (Florida. State Dept. of Education) ;

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
01820115 ( OCLC )
a 59009922 ( LCCN )

Downloads

This item has the following downloads:


Full Text




















SGcuide

ARITHMETIC IN FLORIDA

ELEMENTARY SCHOOLS


BULLETIN 26
1959


STATE DEPAkTMENT OF EDUCATION
Tallahassee, Florida
THOMAS D. BAILEY, Superintendent


1Loa ~


5-. 00"9S9
















UNIVERSITY
OF FLORIDA
LIBRARIES







12 3 Y 5 6 7 8 9/0.
12 12/3//6 /7 A/?/170
2122 23 32 272 29 30
3 1, 373393f2

77 7f 7780
V7 }qV .,-V


ARITHMETIC IN FLORIDA

ELEMENTARY SCHOOLS

BULLETIN 26
(Revised 1955-1958)

STATE DEPARTMENT OF EDUCATION
Tallahassee, Florida
THOMAS D. BAILEY, Superintendent


4








Foreword


IN KEEPING with the policy of the State Department of
Education to revise curriculum guides periodically, a small
committee began work in the fall of 1955 to make minor revi-
sions in Bulletin 26, Arithmetic in the Elementary School. At
I first, it was felt that the minor revisions needed could be made
\ within a few weeks. However, the members of the committee
soon realized that the revision should be delayed until some
up-to-date illustrative material could be secured. They also
found that a small change in one part of the bulletin necessitated
Changes in other parts. Consequently, the revision has required
months instead of weeks.
The committee recognized that most of the material in the
original bulletin is still in keeping with modern research and
literature in the field. The efforts of the committee, therefore,
were largely limited to securing up-to-date illustrative material
and to a condensation and re-organization of the content. For
example, the Learning Outcomes Chart contains material that
was originally contained in three separate charts. The new
arrangement of the material is designed to make the bulletin
more usable.
The services of those who participated in the development of
the original bulletin: co-directors, Mrs. Margaret McCurdie Mc-
Gill and Thelma Tew Bentley; consultant, Dr. W. T. Edwards;
participants, Katherine Adams, Frances Belcher, Emily Brack-
Sman, Doris Brownell, Mary Delamater, Harriet Dyer, Hazel El-
Sliot, Frances Ellis, Thelma Johnson, L. E. Jones, Winona Jordan,
Pauline Messer, Jessie Moon, Dorothy Oliver, Rosalie Powell,
Flossie Sharp, Frances Summers, Nellie Swinney, Rose Varlin
Moeller; and the resource persons, Dr. Robert C. Moon and
Mrs. Dora Skipper, are gratefully acknowledged by members
of the revision committee.








The members of the State Committee which revised the
original bulletin are Dr. Edna Parker, Associate Professor of
Education, Florida State University Chairman; Dewey de
Laire, Elementary Supervisor, Brevard County; Dr. Kenneth
P. Kidd, Associate Professor of Education, University of Florida;
Jean Matheson, Elementary Supervisor, Sarasota County; Mrs.
Margaret M. McGill, Elementary Supervisor, Duval County;
and Charlotte Stienhans, Consultant in Elementary Education,
State Department of Education.
Grateful appreciation is also extended to the following teach-
ers, principals, and supervisors who submitted illustrative ma-
terial: lantha Byrd from Ocala; Alice Clark, James Dunaway,
Jane Evans, Virgie Foster, Betty Tackett, Basil White, and Dor-
othy Wread from Sarasota; Lillian Depp, Julian E. Markham,
and Kathryn Smith from Sebring; Ruby Fratella; Z. A. Gerry
from Fruitville; Omaleah Graves from Leesburg; Cola Lewis
and Ella Rails from Arcadia; Anne McCall and Imogene Neal
from Bartow; Lillian K. McClellan and Lucille Strawn from
Waldo; Kathleen Plumb, Vera Robinson, and Jessie White from
St. Petersburg; Alice Nichols; Peter Van Deusen from Lake
Worth.
Other members of the State Department of Education giving
assistance were J. K. Chapman, Howard Jay Friedman, John
McIntyre, Sam H. Moorer, W. H. Pierce, and T. George Walker.



THOMAS D. BAILEY
State Superintendent of Public Instruction
























Table of Contents

Foreword ........................................... i

Purposes And Principles ............................... 1

Arithmetic Experiences And Resources .................. 17

Whole Number Understandings And Skills ................ 28

Fraction Understandings And Skills ..................... 60

Learning Outcomes Chart .............................. 83

Evaluation Of Understandings And Skills ................. 105

Bibliography .......................................... 120












CHAPTER 1


Purposes and Principles

T HIS GUIDE has been developed to help teachers under-
stand more clearly the purposes and principles on which
arithmetic instruction is based. There is general agreement that
the purposes of teaching arithmetic are to develop within indi-
viduals (1) competency in the fundamental mathematical skills
and (2) the ability to use quantitative thinking in solving both
personal and social problems.

These two aims-mathematical and social-are closely re-
lated. It is possible that a person may be able to sense the
functional uses of number but lack the mathematical compe-
tencies necessary to deal with particular situations. It is like-
wise true that a person with the necessary mathematical skills
may be unable to apply those skills in solving real life problems.
Hence, plans for instruction must take into account both the
mathematical and the social purposes of teaching arithmetic.

As teachers plan arithmetic instruction in keeping with both
the mathematical and the social purposes, they identify many
problems. Some of the most commonly identified problems or
questions are:

1. How does the term readiness apply to instruction in arith-
metic?
2. What is the nature and place of drill in arithmetic?
3. Should children be allowed to count in getting answers?
4. Should children be permitted to use crutches?
5. How should children learn the basic multiplication facts?
6. How can problem-solving ability be improved?








7. How can provision be made for individual differences?
8. Should children be taught to add up or down?
Before these questions are answered, it is necessary to state
the underlying principles upon which suggestions in this guide
are made. The point of view developed in this guide will be
reflected in answers to questions and will be more clearly un-
derstood if fundamental principles are first stated.


Some Basic Principles Of Instruction
In developing a point of view if effective instruction is to
result consideration must be given to the child, the society in
which he lives, and the nature of the learning process. Outlined
below with these three considerations in mind are basic princi-
ples that have some bearing on mathematical meaning, the child
and his environment, and individual differences.

Developing Mathematical Meaning

For a long time, arithmetic was taught by a plan whereby
parts were divided into smaller parts. These parts constituted
the body of the arithmetic program. The teacher's job was to
provide drill and repetition upon these parts until mastery was
achieved. This method of teaching was based on the psychology
of the time and became deeply rooted in common practice. Chil-
dren who were taught by this method did not always understand
what they were doing and frequently were not able to apply the
skills that they learned in solving simple problems.
The weaknesses of drill as the only method of teaching
arithmetic gave rise to the development of a functional method
which relied heavily on the utilization of children's activities.
According to this method, children were taught the skills of
arithmetic as they were needed in carrying out activities. It
was felt that the need for learning certain skills would provide
motivation and that use of these skills in meaningful situations
would provide the necessary practice. Subsequent use of this
method justified the faith of its proponents with respect to these
two values. However, the sequential nature of certain aspects of
arithmetical learning as well as the need for systematic instruc-
tion in developing mathematical meaning revealed the inade-








quacy of a program based wholly on the use of children's ex-
periences.
In more recent years, research findings and classroom ex-
periences of many teachers have established that drill which
follows understanding and the use of children's experiences as
a means of enriching learning do have value. These principles
take into account the importance of arithmetical relationships,
of discovery rather than memorization, and of the subsequent
development of generalizations. These principles are described
briefly in the following paragraphs.

1. Understanding The Decimal Number System

Instruction in the meaning of the number system is consid-
ered essential not only because it is basic to an interpretation
of numbers but also because these meanings are fundamental
to an understanding of procedures used in computation. The
base is simply the first collection in a number series. In the
decimal system, the first collection of ones becomes one ten, the
second collection of ten tens becomes one hundred, and so on.
These collections are used in combinations with smaller num-
bers to form the next number in a series. For example, 11 which
is more than the base of 10 means 10 and 1, 12 means 10 and 2,
20 means 2 tens, or 2 of the base. When 10 tens are reached,
the name 1 hundred is used.
This collection approach to understanding the number sys-
tem is the one most commonly used in the elementary school.
However, to supplement the collection approach, teachers may
picture the number system as a gigantic scale stretching to in-
finity on either side of a reference point zero. Number values
are represented by points on the scale. Thus, 10 becomes the
point between 9 and 11--not a group or collection of 10 ones.

2. Understanding Place Value

It is important for children to understand that the value of
a numeral is dependent upon its position. In the number 324,
for example, each digit has a place value 10 times that of the
place value of the digit to its immediate right. Understandings
of place value and the decimal number system enable children
to comprehend the principles for carrying in addition, regroup-








ing (borrowing) in subtraction, and placing partial products in
multiplication and quotient figures in division. These under-
standings also help children to recognize decimal fractions as an
extension of the decimal system of whole numbers.

3. Emphasizing Relationships

From physics it is discovered that quantity can be measured
in terms of time, length, and mass.
An understanding of time develops as the child becomes
aware of the instruments for measuring time. Clocks measure
short periods of time, while calendars measure longer periods
of time. Speed--an interrelationship of time, motion, and dis-
tance can likewise be measured by instruments.
Length, in its broadest sense, means distance, or height, or
depth, or width--anything of a linear nature falling between
two points. Area is a part of length in the sense that it may be
obtained abstractly by multiplying the length of one side of a
rectangle by the length of another side. Area is a length rela-
tionship.

The broad term mass may be applied to all solids, liquids,
and gases. In analyzing the solids, liquids, and gases with which
the elementary child frequently deals, it is found that he uses
these in a very concrete sense. In the place of thinking of solids,
the young child is likely to refer to specific objects which to
him seem solid, such as people, food, lumber, or cloth. Water
and milk are to him common liquids; air represents a gas. In
using the objects or things which surround him in an effective
manner, the child has need for a knowledge of quantity and of
quantitative relationships.

The four fundamental processes are ways either of taking
quantities apart or of putting them together. Children need to
understand that the putting together processes are addition and
multiplication and that the taking apart processes are subtrac-
tion and division. The full meaning of all the processes is more
clearly understood because of the close interrelationship that
exists among them.
The basic facts or generalizations are much more easily de-
veloped when their interrelationships are understood. For ex-

















II











1. Children develop mathematical understandings involved
in the real-life activity of measuring the materials for a
cake.




ample, when the facts 5+ 4 = 9 and 4 + 5 =9 are taught, their
reverses (9 4 = 5 and 9 5 = 4) should also be taught.

The relationship found in common fractions, decimal frac-
tions, and per cents should be taught. They are not only inter-
related but actually have the same values expressed in different
forms.

4. Developing A Foundation For Abstract Ideas

Children need a great many concrete meaningful experiences
in order to understand abstract number ideas. For example,
before children can understand the generalization represented
by any number such as 7, they need to have a variety of expe-
riences with this idea. These experiences should include work
with real objects, later with pictures, still later with semi-con-
crete materials (dots, circles, lines), and finally with symbols.








5. Developing Mature Ways Of Dealing With Numbers
Just as it is important to develop an understanding of ab-
stract number ideas through a variety of experiences with con-
crete and semi-concrete materials, it is likewise important that
children achieve a mature level of operation by continually
substituting more mature levels of performance for less mature
levels. For example, when children are learning to add a single
7
column of three addends, such as 8 they frequently have diffi-
9
culty in adding the last addend to the sum of the first two
addends. At one level they may think 7 and 8 are 15 and com-
plete the total by counting. Later, they may use the more ma-
ture method of bridging the second decade. Children can see
that after combining 7 and 8 to make a total of 15 that 5 of the
last addend (9) can be used to make 20, and that the remaining
part (4) of the last addend can be added to make a total of 24.
Although bridging is on a more mature level than counting on,
children should not remain at this level. They should, after
learning the higher-decade addition facts, be able to combine
the three addends by thinking 7 and 8 are 15; 15 and 9 are 24.

6. Providing Effective Drill
On the basis of the principles described above, the place of
drill becomes clear. Repetition before understanding has been
developed may fix immature ways of dealing with mathematical
ideas and processes. Memorized learning that is not based on
understanding is usually wasteful of the time and energy in-
volved. Furthermore, less repetition is needed when under-
standing and insight are emphasized. However, drill which fol-
lows understanding serves to reinforce learning. It should be
remembered, though, that practice involves the repeated occur-
rence of the new idea in a variety of meaningful situations.

7. Utilizing Children's Experiences To Enrich Arithmetical
Learning

As children engage in varied activities of the school program,
they find it necessary to deal with the quantitative aspects of
larger problems. This need provides motivation for experiences
in using the mathematical ideas involved. On the other hand,








the activity may provide an opportunity for children to apply
skills that they have previously developed. Moreover, real-life
activities make it possible for children to develop mathematical
understandings through concrete situations and to see the sig-
nificance of mathematics in their everyday living.


8. Developing Children's Ability To Estimate

Estimating answers to problems before doing the computa-
tion not only provides a check on the computation but also re-
veals whether or not a child understands the mathematical prin-
ciples involved. When children are encouraged to answer orally
the question, "About how much?" before doing the written
computation, they will not be satisfied with absurd or distorted
answers. The ability to estimate should be developed along with
the ability to compute with paper and pencil. Appropriate situ-
ations may be found at all levels. The following list is suggestive:
1. Guessing the number of books needed for a particular
reading group or the number of books that are on a par-
ticular shelf.
2. Appraising the number of seats in an auditorium.
3. Estimating the number of paper cups needed for a mid-
morning snack.
4. Finding "about how much" change John will get from a
ten-dollar bill when he pays $4.80 for a bat and $2.25 for
a ball.
5. Computing from left to right:
a. 114 plus 20 124, 134, 144.
b. 598 plus 204- about 800 (598 is close to 600; 204 is
close to 200).
6. Estimating products by rounding off to the nearest mul-
tiple:
a. 4 times 22 is approximately 4 times 20.
b. 9 times 198 is approximately 9 times 200.
7. Estimating quotients by rounding divisors up or down
to the nearest decade:
a. 40 divided by 19 is approximately 40 divided by 20.
b. 38 divided by 11 is approximately 38 divided by 10.








8. Estimating common units of measure.
9. Estimating the products when multiplying common and
decimal fractions:
a. 3% times 41/ will not be less than 12 nor more than 16.
b. 3.5 times 4.5 will not be less than 12 nor more than 16.
Estimating decimal fractions will enable children to
see the reasonableness of the placing of the decimal
point in the product.

The Child And His Environment
Educators believe arithmetic has a significant role to play
in the life of the child. They are examining the child and his
environment in an effort to determine what understandings and
skills should be developed in school. They are asking such ques-
tions as "What are the things adults and children do each day?"
They have identified such activities as eating, adjusting to others,
learning to use different kinds of equipment, securing things they
want and need, interpreting signs and symbols, expressing them-
selves in various ways. As teachers study the problems faced
by both the child and the adult, they have also found that as
environments change new problems arise. For example, as man
gains more control over space, the interrelationships of speed,
time, and distance take on even greater significance.

Providing For Individual Differences
In the average classroom there are children who have the
potential for becoming engineers, mathematicians, or public ac-
countants. In the same classroom there are also children who
find it difficult to understand the simplest abstract number ideas.
No matter what the great range of differences is with respect
to mental ability, background of experiences, or rate of growth,
each child should have an opportunity to think creatively, to
relate numbers to things, to engage in a variety of first-hand
experiences.

Some Teachers' Questions About Teaching Arithmetic
1. How does the term "readiness" apply to instruction in arith-
metic?
Readiness in its simplest sense means being ready to learn
a particular idea, skill, or way of work. Many eight-year-olds,








for example, are ready to work with whole numbers and simple
fraction ideas; probably they are not ready to work with decimal
tractions or understand very large numbers.
Readiness is also a process of becoming more ready than one
was previously. It is not a question of being ready or unready!
The young child who is not ready to count his change becomes
more ready as the result of social experiences and instruction.
Good instruction, in most cases, provides the readiness for the
next steps in arithmetic.
Readiness to work with abstract number symbols is, to a
large extent, determined by a child's ability to understand the
value of numbers. To help develop an understanding for the
value of the number five, the teacher might need to let children:
a. Feel the quantity five, using the big muscles throughout
their bodies, by taking five steps, five jumps, pumping
the swing five times; feel the quantity five, using the
smaller muscles only, by placing each of the five pennies
in one hand as they pick them up and count them, placing
the five crayons in the box, coloring five objects.
b. See five pennies, five chairs-the real objects at first, and
later the pictures.
c. Hear five claps, five strikes on the triangle, five beats on
the drum.
The less ready pupils will need many or repeated experiences
in which they see, hear, handle, or in other ways react to num-
ber situations making use of physical activity and concrete ma-
terials, before they will be ready to understand or express
mathematical ideas by using number symbols.
Upper-grade children need similar instruction to gain an un-
derstanding of the value of large numbers. To understand the
value of five thousand or five million, children might need to:
a. Feel the quantity 5,000 by walking a mile, by counting
bundles of ten sticks to make bundles of one hundred
sticks, counting bundles of 100 sticks to make a bundle
of one thousand sticks, and finally assembling five bundles
of 1,000 sticks to secure the amount, "five thousand."
b. See an auditorium with 5,000 seats, see a section of the
ball stadium seating 5,000 people, see a pile or stack of
5,000 tickets, see 5,000 pages by assembling several books.







Readiness, or number maturity, can be observed or deter-
mined. As the teacher listens to children express such number
ideas as the huge crowd, more than half, about three inches, he
becomes informed about their readiness for certain kinds of
arithmetic learning. As the teacher observes a pupil do sub-
traction examples, he might ask them to "think out loud" so
that he could determine if they were using immature or mature
ways of securing answers. The teacher notes the kind of arith-
metic errors a pupils makes in his written work to determine if
the child is ready for new work or if he needs reteaching.

2. What is the nature and place of drill in arithmetic?

Drill, or the more acceptable term practice, must be provided
if understandings and skills are to be developed. Systematic
instruction is needed to help establish many arithmetic learning.

Practice should follow understanding. After a child has dis-
covered that four cents and six cents are ten cents, he is likely
to forget this bit of learning if it is not used in days that follow.
He will need to practice to maintain this learning and to improve
his performance.

Practice provided too early, that is, before the child under-
stands the idea or process, interferes with his learning or is not
effective for him. Having developed the understanding, the
teacher can feel confident that a wise use of practice periods
will prove helpful. If practice follows the development of mean-
ing and understanding, the teacher will note that much less
practice is required.

Practice in which the child repeats abstract symbols from
the very beginning in a mechanical way may actually encourage
the use of immature procedures and interfere with effective
work habits. For example, a child who has not handled and
visualized nine groups of three should not practice 9 X 3 = 27.
Practice periods should be short and frequent, should include
variety in the kinds of practice, should be carefully distributed
so as to include all essential items and ideas. For most children
the amount of practice on any particular learning may be grad-
ually decreased, and the intervals between practice periods may
be gradually increased. If the practice seems worthwhile, the





























2. Children practice in teams, using flashcards to re-enforce
arithmetic understandings and skills.


child's attitude will contribute to his success. Each pupil should
be challenged by the drill or practice he is undertaking.


3. Should children be permitted to count in getting answers?
Counting is one of the less mature ways of securing an an-
swer. Early in the primary grades, children are encouraged to
count to find the answer. They count children, chairs, books,
pennies. Following real social experiences, they count pictures
of objects children, pennies, circles, squares to secure the
answer. Later on, as children continue counting experiences,
they learn to identify small groups-groups of two, three,
four-without counting.
During counting experiences, readiness to find the answer
in a more mature way usually develops. If that is the case, the
teacher will note that the child may use partial counting to de-
termine the answer more quickly. For example, the child who
can recognize a group of three chairs without counting them








says, "three," and then continues to count, "four, five," to secure
the answer five.
If-when children are having these counting experiences--
they also have many opportunities to match an arrangement of
three chairs and two chairs with the number symbols 3 + 2, the
number symbols will begin to take on the same meaning as the
group arrangement. After having had many such matching ex-
periences, children become more ready to secure the answer
without counting.
If a child continues to count to get the answer beyond the
time when it is appropriate to do so, he may need the oppor-
tunity to learn all over again, moving from the less mature to
the more mature way of securing his answer. In many cases,
this will call for individual or small group instruction, even in
the upper grades.

4. Should children be permitted to use "crutches"?
The principle that children grow from immature to more
mature levels of operation makes it possible to permit the use
of crutches during the initial stages of instruction. The small
2
figures used in regrouping (borrowing) A5 are one example of
-18
17
a crutch. The findings of research indicate that the crutch when
used in the early stages of instruction serves to reinforce mathe-
matical meaning.
Research shows that prolonged use of crutches decreases ef-
ficiency of operation. Hence, it seems desirable to use them
during the early stages of instruction and to discourage their
use after an understanding has been achieved.
In any case where a child continues to use a crutch beyond
the initial stages, the teacher should determine if the pupil needs
careful reteaching or if he should be more insistent about the
pupils' not using the crutch any longer.

5. How should children learn the basic multiplication facts?
Readiness for multiplication is developed by helping children
discover the relationships between addition and multiplication.








When presented with such a problem as "How many pennies
will it take if each of us (4 children) gets two pennies?" the
relationship between addition and multiplication can be seen if
an arrangement of pennies, real or pictured, is presented as
00 00 00 00. The answer may be secured in various ways:
a. By addition-The child sees and says, "Two and two are
four, and two more are six, and two more are eight."
b. By addition, counting by 2's-The child sees and says,
"Two, four, six, eight."
c. By multiplication-The teacher helps the child to see four
groups of two and helps him express the idea as four
groups of two. The child would probably express the idea
in a sentence, "There are four groups of two pennies, and
that makes eight pennies." With the help of such a ques-
tion from the teacher as, "How many times do you see a
group of two pennies?" the child will learn to express the
multiplication idea, using multiplication terms. After sev-
eral similar experiences, he will be ready to state, "Four
times two are eight."

Learning to multiply with meaning is based upon visualizing
groups of equal size, noting relationships, and expressing those
relationships in terms of multiplication. For example, after
noting that four groups of two are eight, many other groupings
of two would be considered. Eventually, all the groupings of
two would have been considered. Eventually, all the groupings
of two would have been discovered and expressed as multipli-
cation. Preferably, at this point, all these are expressed orally.

Learning to express multiplication ideas as written multipli-
cation facts should follow discovering the meaning of multipli-
cation. Most children will continue to use groupings, either of
real objects or pictures, as they record their first multiplication
ideas. After recording several multiplication ideas, many chil-
dren will be ready to record the facts without reference to visual
materials.

Learning to record or recite multiplication ideas in organized
form, that is, as multiplication tables, is one of the last steps in
learning multiplication facts. For most children, practice or drill
will be necessary to establish the facts for ready recall.








6. How can problem-solving ability be improved?
Helping a child solve a real personal problem for which he
must have an immediate solution provides readiness for further
work in problem solving. For example, the child who loses part
of his lunch money must borrow enough money to be able to
buy lunch. He needs to know the exact amount to borrow.
The teacher shares the genuine problem of any one child with
the class. In discovering the solution to the problem, the chil-
dren will use manipulative materials and pictures as well as
discussion. The teacher selects and records some of these gen-
uine personal problems, and they become verbal (printed)
problems for the children to solve. From classroom activities and
experiences emerge other genuine problems which are the con-
cern of all members of the class.

The teacher prepares children for solving the vicarious prob-
lems in the textbook. Adequate preparation includes:
a. Beginning with a genuine problem similar to the kind
treated in the textbook.
b. Using dramatization, diagrams, charts, manipulative ma-
terials, and discussion to visualize the problem.
c. Discovering basic relations needed to solve the problem.
(To do this the meanings of the processes and significant
words will not need to be understood. The exact meaning
of mathematics vocabulary will need to be developed, re-
fined, and extended at this point.)
(1) Addition-regrouping or building of a group.
(2) Subtraction-as breaking down a group or comparing.
(3) Multiplication-as a short cut to addition when the
groups are the same size.
(4) Division-as separating a group into smaller equal
groups, as sharing, as finding the number of items in
equal groups, as a series of equal subtractions.
d. Reading the problem aloud with good expression so that
pupils have the opportunity to understand and interpret
the meaning of the problem. This is essential for all pupils
who have difficulty with reading to help them visualize
the concrete situation described by the words and the
arithmetic symbols.








e. Rewriting the problem in words which the slow reader
can read.
f. Experimenting to secure the answer in more than one way.
g. Emphasizing sensible answers.
h. Applying previously acquired facts and relationships.
i. Understanding the arithmetic vocabulary used.
j. Estimating the answer.
k. Training in step-by-step analysis, to be helpful, needs to
be worked out by pupils under teacher guidance. (It is
not recommended that an adult logical pattern be used by
children in analyzing problems.)
The teacher prepares children to be able to solve real and vi-
carious (textbook) problems without the use of pencil and
paper. Readiness for this kind of performance depends upon the
amount and quality of problem-solving activities carried on as
described above.

7. How can provision be made for individual differences?
When children enter school, they differ greatly in their abili-
ty to use numbers and to understand their use. These differences
become greater as children mature and as they receive good
instruction.
There are times when all pupils in a class should be taught
as a total group so that each child can benefit from each other's
contributions. These times occur when a new experience, idea,
or skill is being introduced or explored, and after an idea or
skill has been taught, making it possible for children to share
what each has learned. Between the introduction of an idea and
skill performance, careful instruction to meet individual needs
must be given. This instruction can be provided for in a variety
of ways:
a. By dealing with the same problems but expecting and
accepting different levels of operation.
b. By providing carefully planned reteaching. (This enables
a child to learn more fully what he has learned only par-
tially. It also strengthens the learner who forgets a great
deal.)








c. By providing arithmetic experiences of varying degrees
of difficulty.
d. By using many more experiences with concrete objects
with the child who is lacking in background experiences
or who is a less mature child.
e. By not emphasizing speed during instructional periods.
(Some individual pupils are deliberate and learn best
when they are expected to be thoughtful about what they
are doing.)
f. By including in unit teaching a variety of arithmetic op-
portunities so that each pupil can identify himself with
some kind of problem he is ready to do.
g. By having available games, flashcards, and additional
work with which individuals may practice, following care-
ful instruction.
h. By grouping pupils for instructional periods to meet spe-
cific needs. (When a particular need has been met, groups
are reorganized in terms of another need.)
i. By providing a reasonable amount of homework that the
child understands and that helps meet the needs of an
individual pupil.
Faculties may work on the problem of meeting individual
differences in an in-service training program or in professional
faculty meeting (either grade-level or total school) using pro-
fessional literature and materials as well as resource persons
either on the county or state level or from State-adopted text-
book publishing companies.

8. Should children be taught to add up or down?
Research does not indicate that either way is more effective.
It seems logical to add down since you are ready to put your
answer down when you reach the bottom. Children should form
the habit of adding one way so that they do not spend time
making up their minds.












CHAPTER 2


Arithmetic Experiences And Resources

AN UNDERSTANDING of the meaning and significance of
arithmetic in relation to child growth and development
has been presented in the first chapter. Research findings reveal
that the number abilities which children learn and use are re-
lated to children's growth and personal needs. Since arithmetic
requires abstract thinking, teachers need to provide many con-
crete experienecs upon which this thinking can be solidly based.

Teachers need to understand how children learn, to recognize
relationships between arithmetical processes and facts, and to
feel at home with the number system. They need to plan with
other teachers in their school to insure continuity in the total
school program. Teachers need to plan with their pupils for
participation in meaningful learning situations and to provide the
necessary instructional resources with assistance from super-
visory, administrative, and materials personnel.

Selecting Meaningful Learning Situations
Children today learn and use in arithmetic many skills and
abilities that children of previous generations did not need. If
their school experiences do not develop these arithmetical skills
and abilities, children will find themselves handicapped in many
activities. Formerly, schools placed emphasis on computational
skill only; now, however, computation takes its place along with
many other skills. The ability to select and evaluate information
to solve problems that are not in textbooks, the ability to inter-
pret maps, globes, graphs, and charts of many kinds, the ability
to recognize value in consumer goods, and the skills to create
products for this technological age are a few of the many im-
portant abilities required of today's children.








Teachers in their planning make use of suggestions in the
arithmetic textbook and accompanying manual; they are alert
to recognize in everyday classroom life those situations that
provide meaningful experiences for arithmetic instruction. In
addition, they plan situations to provide instruction that will
enable children to understand and use the number system.
Suggested situations given here illustrate possible ways of
providing these experiences. Teachers will make their own lists
as they develop skill in selecting those of value for their groups
and communities. They will observe and experiment to deter-
mine special arithmetic needs of the children in their classes.
At all times, they will remember that children live in an ever
expanding environment where an understanding of the meaning
of arithmetic will be of prime importance. The teaching of some
skills and provision for some experiences have been designated
for grade placement. However, teachers will survey the abilities
their children possess and the experiences they have had and
consider both their backgrounds and needs when planning the
program. These experiences will help them recognize situations
that require a knowledge of or can be solved by the use of
number.

Activities And Desired Competencies For Primary Grades
-Building a play house, play room, or play kitchen (counting,
estimating, comparing)
-Playing games (keeping scores for games such as tenpins,
ring toss, shuffleboard; using numbers in games such as
May I?, hide-and-go-seek, and bean bag)
-Keeping class attendance for several days (counting, adding,
subtracting, developing readiness for percentage-oral)
-Preparing the morning snack (counting, measuring, estimat-
ing, adding, dividing, multiplying)
-Making model of farm, bus station, airport, or post office
(measuring, estimating, counting, grouping)
-Telling time (knowing numbers on the clock face, under-
standing how to tell time, recording time, budgeting time,
becoming familiar with other instruments for measuring
time)








-Operating postal service for school building (selling and buy-
ing stamps, preparing letters and packages for mailing, mail-
ing letters and packages, adding, subtracting, multiplying,
making change, measuring, weighing, writing numbers)
-Dramatizing activities at home and in occupations, for exam-
ple, farm, filling station, bus, hospital, grocery store (count-
ing, using money, working hours, measuring)
-Making booklets (measuring, numbering pages, balancing
materials on page, checking for legibility of figures)
-Making lunch or milk order (counting, adding, subtracting,
writing numbers)
-Going to the post office or other places of business (counting,
adding, subtracting, buying stamps, making change)
-Taking a field trip (counting money, things seen, or speci-
mens; adding, subtracting, understanding of number uses
involved, measuring distance and time)
-Making a chart or class booklet of number facts learned
(measuring, spacing of material, evaluating contents)
-Planning, preparing, and having party with refreshments-
Christmas, Halloween, birthday, or for mothers (counting,
estimating, grouping, adding, subtracting, dividing, multiply-
ing, comparing prices, buying materials, using liquid and dry
units of measurement)


Activities, Desired Competencies For Intermediate Grades

-Making a vegetable or flower garden (measuring, making
plot plans, computing cost of seeds and fertilizer, purchasing
supplies, keeping records, planning ahead, scheduling time
for talks by garden specialists, selling produce)
-Establishing and maintaining a savings account (learning to
use banking forms, forming desirable habits and attitudes
toward saving)
-Playing games (keeping score for such games as softball, vol-
leyball, three-deep; measuring off areas for courts, using
number skills in more complex games such as dominoes and
monopoly)








-Landscaping out-of-door classroom area (measuring, making
plans, computing costs of materials)
-Managing school post office (selling stamps, making change,
making out money orders, mailing packages, sending special
delivery and registered or certified mail, opening a savings
account)
-Earning money (computing earnings, budgeting, depositing
in savings account)
Operating a store for school supplies (figuring costs, making
change, estimating and ordering supplies, computing profits,
keeping records)
-Making and interpreting simple graphs (measuring, counting,
dividing whole numbers and fractions)
-Planning budgets (use of time, use of money, using forms
and records, learning to plan effectively)
-Arranging school parties (earning and saving for a party
fund, budgeting, planning, buying materials)
-Cooking simple dishes (measuring, computing and comparing
costs, reading thermometers)
-Publishing a class or school newspaper (measuring, comput-
ing costs and profits)
-Conducting cooky or candy sale (buying materials, under-
standing recipe quantities, making change, computing profits,
planning use of money)
-Selling lunch tickets (counting, adding, subtracting, making
change)
-Studying industries, for example, cattle, citrus (quantity,
counting, adding, dividing, subtracting, multiplying)
-Finding cost of new school buildings and equipment (read-
ing numbers, computing costs, measuring areas)
-Taking field trip, for example, weather bureau ( reading in-
struments, interpreting scales, quantity concepts, reading
maps and graphs)

Selecting Instructional Resources
One of the main objectives in producing this curriculum
guide is to help Florida teachers become more sensitive to the








quantitative problems of daily living. This chapter contains
many examples of situations which should suggest others to
teachers interested in helping children develop understandings
and skills relative to the number system and its use. Another
objective is to help teachers discover ways of organizing mate-
rials for instruction in arithmetic in the various grades.
In selecting materials, teachers need to keep in mind not only
the function of arithmetic but also the needs, interests, and
levels of ability of the children they teach. Resources for much
of the arithmetic program of the primary grades are to be found
within the classroom (the clock, calendar, small sums of money,
thermometers, scales, yardsticks, and mid-morning snacks).
Resources for the intermediate grades include people in the
community, reference books, magazines, newspapers, television,
radio, and government bulletins as well as textbooks-arithme-
tic, social studies, health, music, and science. From these sources
children will secure information that will help them solve
problems.
No single type of instructional material is suitable for all
situations. Any list would include pictures, books, activities,
and various teaching devices. An arithmetic shelf, table, or
corner for keeping these materials has become an accepted part
of many elementary classrooms. As the children participate in
number experiences and as the teacher's awareness of their
needs develops, many materials will be added to this center.
What is included will depend upon availability, budget limita-
tions, and the ingenuity and creativeness of the teacher and
children. Some of these materials might be used by the entire
class at one time, but many of them would be used individually
or in group work.
Some schools make instructional materials for arithmetic a
part of the school materials center. When selecting and using
materials, teachers need to keep in mind the function of arith-
metic and needs, interests, and levels of ability of the children
they teach. Many pieces of equipment can be constructed by
children or by the teacher from scrap or inexpensive materials;
if teachers wish to purchase materials and funds are sufficient,
many instructional aids can be purchased.
The most complete single source of materials and devices for
teaching mathematics is the Mathematical Teaching Aids, a sup-









plement to the Chicago Schools Journal, available from Chicago
Teachers College, 6800 Stewart Avenue, Chicago 21, Illinois.
The teacher will find in this supplement a list of free materials,
films, and manipulative teaching aids which he and the students
can make.


Manipulative Materials

These lists are suggestive; an imaginative and observing
teacher will find many more in his classroom and community.


1. Useful for counting and grouping
Buttons
Small sticks
Boxes Small
Large beads
Dominoes




2. Useful for study of measurement
Weight
Kitchen scales Cale:
Postal scales Cloc
Bathroom scales Cloc
School scales Cloc
Science scales Stop
Cloc
Time
(X
be
Tim
Sun-


Bottle tops
Stickers
Paper plates
Empty spools
Shells
Clothespins
Egg Cartons
Money (real or play)


Time
ndar
k cardboard
k room
k alarm
watch
k with second hand
ers egg, stove, process
:-Ray, photography,
auty parlor)
e line strip
-dial


Volume
Measuring cups-marked into thirds, fourths, halves
Separate measuring cups holding 1/3, 1/2, and 1/4 cups each
Measuring spoons
Pint, quart, two-quart, and gallon jars
Half-pint, pint, quart, and two-quart milk bottles and cartons
Half-pint, pint, quart metal containers
Pint and quart berry boxes
Peck, half-bushel, and bushel baskets
Paper bags-many sizes
Plastic containers
Cubic Measurement
Cubes of wood, soap, starch, or paper
Models of cubes, prisms, cylinders, cones, pyramids, spheres
and hemispheres
Rectangular solids-boxes, cartons
Cylindrical solids-tin cans, salt boxes, pipe, jars, pillars, cake
pans
Spheres-balls, fruit









Temperature
Thermometers
Clinical
Weather
Cooking
Thermostats
Square
Ruled paper or cardboard
Graph paper (divided into tens or hundreds)
Objects in classroom or immediate environment which can be
used to find area
Distance and Measurement
Instruments
Rulers, meter sticks
Compasses
Protractors
Triangles and T-squares
Transit
Plumb-bob
Carpenter's level
Illustrative material
Pictures of samples of geometric designs in everyday living

3. Useful for a variety of purposes
Peg boards
Flannel boards
Toys, figures, and symbolic materials
Magnetic boards
Fraction wheels
Abacus-type devices
Place-value holders
Toy telephones
Toy cash registers
Toy adding machines
Speedometers
Various types of dials-radio, television, washing machine
Hundred boards
Number frames

Audio-Visual Materials

The use of carefully selected pictures to illustrate uses of
arithmetic in everyday life is important. Picture material in-
cludes not only flat pictures but diagrams, charts, tables, graphs,
maps, cartoons, and clippings dealing with facts and events
having quantitative aspects. From textbooks and from vertical
files teachers can make use of pictures that have meaning in the
situation being studied. Many times it will be desirable for the
teacher and pupils to create picture materials to represent arith-
metical ideas. The making of sketches and the preparation of
diagrams and charts can be valuable learning activities. In the
primary grades they help to minimize the amount of reading





























3. Charts, diagrams, and simple mechanical devices make
number relationships more meaningful.

needed and to provide meaningful relationships which might
be difficult if only reading and teacher-explanation were used.
In the intermediate grades continued emphasis on the use of
these visual materials enables the children to see the many uses
of arithmetic in daily life. Picture materials that have proved
to be helpful should be filed for future use. If placed in folders
in the vertical file, they make a valuable resource.
Teachers can use films, filmstrips, recordings, and slides as
instructional materials in arithmetic. They will find the follow-
ing questions helpful in determining the use of audio-visual ma-
terials: Is it suitable for the varying needs of the group? Will
it be useful for introducing, enriching, or reviewing an arith-
metical idea, principle, or process? Is it appropriate for the
maturity level of the group? Is it the best material to use in a
given situation? Usually the best teaching results when the use
of audio-visual materials is combined with other methods of
instruction.
Business life furnishes another source of valuable material.
Understanding and interpreting sales slips, statements, bills of








all types, money orders, deposit slips, withdrawal slips, insurance
policies, charge accounts, cash register and adding machine tapes,
and inventory forms must of necessity involve actually working
with these materials. The function of the form, the items on it,
and the ways it is used are emphasized. Forms sufficient for
class use should be obtained.
Information found in advertisements can be related directly
to classroom work. Magazine or newspaper advertisements,
handbills, and catalogs make possible comparison of quantities,
prices, and discounts.
Most textbooks and school supply catalogs list games which
can be used to supplement other classroom activities. These
should be used only after careful evaluation to determine if
they are worthwhile. In many instances teachers and pupils may
wish to create games.
Textbooks in arithmetic are State-adopted in Florida. The
beginning teacher will find that textbooks with their accompany-
ing manuals or teaching guides will help in planning a good
program. The textbook gives a step-by-step development of the
fundamental processes. Many practice exercises are given to
develop skills. Attention is given to the meaning of our number
system. A series of problems about the social uses of arithmetic
parallels the development of number processes. Suggestions for
meeting individual differences include diagnostic tests, special
problems and reports, and such activities as construction proj-
ects. The manuals give detailed suggestions for teaching and
include achievement tests, provide exercises for meeting indi-
vidual needs, and give directions for constructing teaching aids.

When other textbooks are added to the materials for instruc-
tion, they should be carefully evaluated to see that they empha-
size the meaning and significance of the number system and
meet the needs of the group or individual using them. Whether
the teacher uses some State-adopted texts from other grade
levels or supplies children with sets of other texts or supple-
mentary materials as needed depends upon school policy and
availability of funds and materials.
Workbooks provide a source of practice material. However,
their use should be determined by need. Insofar as possible their
use should grow out of needs derived from meaningful experi-








ences or be readily integrated with such experiences. There is
a danger that use of a workbook may lead to isolated and mean-
ingless drill. A careful evaluation of available workbooks should
be made before one is selected. Of course, pupils with similar
needs could use the same workbook, but it would be improb-
able that all pupils in a class would need the same exercise or
the same workbook.

Elementary school libraries contain books that provide many
number experiences, for example, The Three Bears, Billy Goats
Gruff and Millions of Cats. Using approved lists, the classroom
teacher and the materials specialist should work together to
enlarge the school collection of these books.

Easily Constructed Materials

If a teacher and his pupils wish to construct materials, they
will find many resources readily available. Following are ex-
amples of some possibilities:
-Abacus-type devices can be made from such things as wire coat
hangers and spools or beads.
-Fraction wheels can be made from bottoms of dress boxes, pieces
of plywood, discarded film, and colored paper plates.
-Stickers from community drives might be used for making charts
showing number groupings.
-Clock faces from the large advertising clocks that have been
thrown away are excellent devices to show time relationships.
-Hundred boards have been made from scrap plywood, pieces of
paint-sample cards, and blocks of acoustical tile.
-Pocket-chart holders can be made of plywood and oilcloth or of
heavy cardboard. Small strips of oak tag board, dowel sticks, or
tongue depressors could be used as single l's, in bundles of 100's.
These bundles may be placed in the pockets of the chart holder
as needed in teaching "carrying" in addition and "borrowing"
in subtraction.
-Flannel boards can be made by stapling flannel to a heavy piece
of cardboard. Paper cutouts of animals, fruits, circles, and
squares will stick to the flannel board if small pieces of sand-
paper are pasted on the back of the cutouts.
-Fraction charts or diagrams made of cardboard can show frac-
tional equivalents.

Community Resources

"Trips" by a pupil, a group, or by the entire class to places
in the community will stimulate interest, give first-hand expe-
riences, and provide authentic information. The teacher is
charged with the responsibility of planning carefully for the
trip so that pupils will relate what they see and do on the trip




























4. Pupils visit a house under construction for first-hand in-
formation on relating problems in building a house to their
arithmetic learning activities.

to the purposes of the trip. Sometimes an interview in the
classroom with a resource person is more desirable than a trip.
A resource person will often give the pupils materials which
they can use in the classroom.

This list suggests trips rich in mathematical possibilities:
-Stores: hardware, grocery, bakery, drug department.
-Business places: bank, automobile salesroom, garage, creamery,
stock exchange, dress shop, hotel.
-Government: post office, fire department, highway patrol offices,
city hall, county courthouse.
-Health: hospital, sanitation or welfare departments, clinic.
-Offices: shipping room, income tax and accounting service, pay-
master.
-Sports events: baseball, swimming, football, golf, shuffleboard.
-Transportation: depot, airport, freight yards, ticket offices.
-Cultural: library, museum, special interest parks, TV or radio
stations.
-Rural: farms, groves, dairies, forestry projects, wild-life projects.
-Construction: homes, large buildings, shopping centers, bridges,
airports, small household or industrial furniture.

Many short excursions on the local school grounds can be
planned. "Trips" should be used when they will make a valu-
able contribution to the on-going learning activity.












CHAPTER 3


Whole Number Understandings And Skills

T HE DESIRABILITY of helping children become more sen-
sitive to the quantitative problems of their environment
and to the development of the necessary readiness for partici-
pating in the solution of these problems has been pointed out
in the two preceding chapters. Children have a need for de-
scribing and measuring-two processes-describing and meas-
uring the size, weight, shape, position, and amounts of various
things in some adequate fashion. This is done by means of a
vocabulary (little, big, far, near, like), and by means of the
number system. Use of the number system makes it possible
for the description and measurement of amounts to be accurate.

In the beginning, it is possible for children to satisfy their
need for accurate description and measurement by means of
counting. Before long, however, they find themselves combining
groups of objects and with teacher guidance learn that there are
more economical ways of finding out how many other than by
counting. It is at this point that readiness for computation de-
velops. Although children may be ready for computation, they
may become lost in a maze of abstraction if the teacher fails to
help them bridge the gap between the concrete and the abstract.

In the past, children have gone directly from counting to
abstract drill in simple combinations. As a result, some children
become confused and frustrated, losing sight of the real purpose
of arithmetic. Children need to follow counting of concrete
objects with grouping of concrete objects. They learn to recog-
nize the number symbol which stands for the number of objects
within the group. After many experiences of this kind, they
gradually arrive at an understanding of the meaning of such
combinations as 2 and 2 make 4, 3 and 2 make 5.








As teachers attempt to direct children in acquiring compu-
tational skills, they need to be aware of the real life problems
faced by children at home and at school. They need to recog-
nize the interests that children have in the expanding environ-
ment as they move from grade to grade. Teachers need to
understand the stages of development in the computational op-
erations. For example, they need to analyze the problems chil-
dren face when they find it necessary to bridge a decade in
addition, regroup (borrow) in subtraction, use a two-place mul-
tiplier in multiplication, or find the quotient in division when
the divisor ends in 8 or 9.
This chapter includes (1) recommendations for developing
techniques and methods, (2) an analysis of the levels of difficulty
of each of the processes involving whole numbers, and (3) an
allocation chart of abilities needed for computational skills and
the use of standard units of measure.

The recommendations for developing techniques and methods
are in keeping with research findings as to the most meaningful
and effective ways of solving problems that involve computation.
For example, the method of take-away borrowing (decomposi-
tion or regrouping) is recommended for solving subtraction
problems because it can be understood when children under-
stand the place value of numbers and because it is also the
method best suited to the subtraction of mixed numbers (whole
numbers and common fractions).

Included in the analysis of the levels of difficulty in the com-
putational processes are a description and several examples of
each skill type. These analyses should be helpful to teachers
for determining next steps of instruction for an individual child
or for the entire class.

The Learning Outcomes Chart beginning on page 83 includes
recommendations regarding the teaching of specific skills within
each of the first six grades. The chart distributes the teaching of
arithmetic skills throughout the elementary school in relation
to the increasing maturity of the learner. Analyses of the levels
of difficulty within the computational processes and study of the
problems faced by children as they move from grade to grade
indicate that an emphasis on specific skills be made in each
grade.







Teachers recognize the fact that beginnings of computational
understandings with both whole numbers and fractions are
developing before children come to school. Since these under-
standings develop gradually, teachers at all grade levels should
plan for reinforcing ideas and processes previously introduced
as well as developing new ideas. At one time, skills in adding,
subtracting, multiplying, and dividing common fractions were
treated exclusively in fifth grade. In this chart, however, the
development of meanings of fractional parts is recommended for
the primary grades. Since meanings are developed from the first
grade on, some drill on simple addition, subtraction, and multi-
plication of fractions is recommended for fourth grade. The
various skills are no longer treated exclusively in any one grade.
While many pupils will not be ready for some of the learning
allocated to a particular grade, other children will be ready to
do much more than is suggested for a grade. It is recommended
that each faculty study the recommended allocation of skills
and modify the placement of skills according to the maturity
of learners.

Working With Whole Numbers

Number symbols and their names are used to express two
principal ideas-cardinal and ordinal. The cardinal idea tells
"how many?" there are in group of objects; the ordinal idea
indicates "which one?" by reference to the relative position of
the object in the group. The same number symbols are used
to represent both ideas. For example, a child uses number sym-
bols in the cardinal sense when he says,
"How many plates should I bring to the table?"
"I want as many cookies as James has."
"How many legs does a horse have?"
"I am six years old."
"I weigh 51 pounds."
On the other hand, he uses number symbols in the ordinal
sense when he says,
"Turn the television to Channel 2."
"Watch player 37 in the football game."
"My birthday is on July 10."
"Let's begin to read on page 8."








In short, arithmetic symbols are a part of oral and written
language.
Just as number symbols and their names express cardinal
and ordinal ideas, other symbols express action ideas. The sym-
bols +, -, X and -- signify types of action taking place. Al-
though four symbols are used to indicate four types of action,
there are basically only two operations to be performed. Quan-
tities can be taken apart, and they can be put together. Obvi-
ously, then, the four symbols above are used in putting quanti-
ties together or in taking them apart. It is important, therefore,
that the meanings and the interrelationships of the processes be
understood. Children should understand that addition and mul-
tiplication are processes used when putting groups together, that
subtraction and division are used in taking groups apart.

The meanings of the processes are further extended when
children understand that the addition process is used in putting
together groups of like quantities, either equal or unequal in
size. Similarly the multiplication process is used in putting to-
gether equal-sized groups of like quantities. It is, therefore, a
quick way of adding when dealing with groups that are the
same size. The subtraction process is used, however, in sepa-
rating a group into smaller groups and is the reverse of addition.
Likewise, division is the process of separating quantities into
equal-sized groups; moreover, the division process is the reverse
of the multiplication process and is related to subtraction in that
the number of groups can be determined by repeatedly sub-
tracting the smaller group from the total. Thus, the full mean-
ing of all the processes is more clearly understood because of
the close interrelationships which exist among all four funda-
mental processes.

The numbers between 10 and 20 are perhaps the most diffi-
cult numbers to teach because they are not written as they are
spoken. The "teen" part of the number is written first but is
spoken last. For example, in the number 16, one says "six-teen"
but actually writes "ten-six." The inappropriateness of the
number names in this decade should not prevent children from
gaining an understanding of these numbers.

By using concrete materials such as bundles of sticks, chil-
dren can understand that a group of 10 ones tied together is








also 1 ten. Then, by using the bundle of 1 ten and single sticks,
they can see that 12 is 1 ten and 2 ones, and that 13 is 1 ten
and 3 ones, and so on. As children sense the meaning of the
number in this decade, they may suggest more appropriate
names, such as ten-one, ten-two, ten-three, and so on. As chil-
dren deal with larger numbers they will further extend their
understanding of the number system and the place value of
numbers. It is important that children realize that the base of
our number system is ten, that it takes 10 ones to make 1 ten,
10 tens to make 1 hundred, and that it takes 10 hundreds to
make 1 thousand.
Many classroom situations may be used to further the chil-
dren's understanding of place value of numbers. There are
twenty-eight children in the class; the teacher helps them un-
derstand that there are twenty and eight more. Then he will
write the 20 and 8 on the board and explain to them how the
20 and 8 may be put together to become one number, 28. An-
other example which may follow or precede this one of class
membership is an example in which 2 dimes and 3 pennies are
used. Children can readily understand that, while there is a
total of 23 cents, there are also 2 dimes and 3 pennies. They
begin to understand the use of zero as a place holder and that
in this connection zero means no number. Examples of this
kind should be repeated over and over until the children know
that the number in the extreme right column represents ones,
the next number represents tens, and so on. This understanding
develops gradually but is not difficult if teachers use concrete
objects for demonstration purposes. It is important that a va-
riety of concrete materials be used. Similarly, bundles of 10
tens can be grouped to demonstrate the meaning of 1 hundred.
It may be desirable to use rather small sticks or dowel rods in
working with numbers beyond hundreds. However, materials
that are large enough for children to handle with ease are more
effective in dealing with ones, tens, and perhaps hundreds.


Hundreds Tens Ones








Many teachers have found that small strips of tag board grouped
into bundles of 10's and 100's, and placed in pocket chart holders
are also effective. It should also be pointed out that number
frames, money, and an abacus may be used in developing an
understanding of the number system.
Reading, writing, and interpreting large numbers are impor-
tant skills in the upper grades where the child's interest in his
expanding environment makes it necessary to read figures rep-
resenting large sums of money, long periods of time, large areas,
population, and the like. Here an understanding of place value
of numbers as well as an ability to use commas for grouping
numbers is necessary. When the figures are mixed numbers,
the word and should be used to join the fraction to the whole
number. The use of and prevents children from becoming con-
fused as to where the whole number stops and the fraction
begins; thus, 100.026 is read one hundred and twenty-six thou-
sandths, while .126 is read one hundred twenty-six thousandths.
Likewise, it is important that children learn to read 126 as one
hundred twenty-six, not one hundred and twenty-six.

Other Number Systems
An interest in Roman numerals will develop after children
have had experiences using the Hindu-Arabic numbers and
after they begin to wonder about the meaning of symbols on
certain clock faces, in books, on sundials, or on the cornerstones
of buildings. This is the time to give them information which
will lead to an understanding that there are other number sys-
tems than the one we commonly use and an appreciation of the
way in which all number systems evolved. If an explanation of
Roman numerals comes after children have learned to multiply,
they will be more able to appreciate the fact that the Roman
system was gradually abandoned because of its cumbersome
qualities and that man has been constantly trying to find better
ways of doing things. To demonstrate this point, teachers might
add and multiply two numbers such as CXXV and V in both
Roman numerals and Hindu-Arabic numerals on the board.
Such a comparison should lead to a greater aprpeciation and
understanding of the decimal number system.
Children should have experiences with number systems
other than the Hindu-Arabic and the Roman systems in order
to appreciate their great cultural heritage. For instance, chil-








dren may be encouraged to examine a number system based
on six or twelve which might have evolved had men been born
with six fingers and six toes instead of five. Each number base
uses a definite number of symbols equal in number to the size
of the base. The total number of symbols needed always in-
cludes the zero. In our decimal system ten symbols are used-
nine figures and zero. In a duodecimal system twelve symbols
are needed-eleven figures and zero; two additional one-digit
symbols are used:-1,2,3,4,5,6,7,8,9, x (deci), E (elf) and 0
(zero). Thirty-six objects would represent 3 units (or zens)
and would be written with the symbols 3 and 0. In other words,
the x figure in the second column represents units of twelve
(or zens) instead of units of ten. In the duodecimal system the
symbols 24 represent 2 twelves and 4 ones instead of 2 tens and
4 ones.

Adding And Subtracting Whole Numbers
In adding whole numbers, there are many ways of finding
the total in a collection of like objects. At first, the child arrives
at the number name of a collection of 3 objects by a process of
carefully touching one object after the other as he says the
number names in serial order. Later, he should be able to rec-
ognize small groups without counting the individual objects
within the group. Some insight into the number competence and
readiness for addition of a six-year-old can be gained by ob-
serving the way in which he arrives at an answer to the ques-
tion, "How many dots on this card?"




o 0


0 0




In recognizing small groups without counting, the child deals
first with objects, then pictures, and later with semi-concrete
materials such as circles, dots, and dashes. The arrangement of







groups of circles or squares in varied patterns facilitates quick
recognition. Semi-concrete materials that are placed in straight
rows (at least groups larger than five) usually require counting
in order to determine the total. Spot patterns of circles or
squares arranged according to domino, triangular, or quadri-
lateral patterns are effective in helping children recognize small
groups without counting.
For example, the illustrations below show arrangements of
nine objects grouped according to three different patterns:

o 0 0
triangular
O0 O0 OO


quadrilateral 0
00 00 0

00 00
domino o
o0 00
Children need a variety of experiences in working with small
objects such as blocks, buttons, and checkers in order to dis-
cover number relations by putting groups together and taking
them apart. For example, a child will discover that he can group
six objects in many ways. By using small objects, he can dis-
cover that the six objects can be separated into sub-group pairs
of 5 and 1, 1 and 5, 2 and 4, 4 and 2, 3 and 3. When a child is
able to recognize small groups from 2 to 5 without counting, he
can deal with groups of 6, 7, 8, 9, and 10 by using partial count-
ing. For example, in dealing with seven objects, a child may
recognize four of the objects as a group and complete the total
by counting the remaining objects. However, a child should not
remain at this level of partial counting indefinitely. He should
proceed to the point where he can put both groups together
without counting.
As soon as children understand a few of the simple addition
combinations, they should begin to learn the corresponding sub-
traction facts. Related facts, such as 3 and 4, 4 and 3, 7 take
away 4, and 7 take away 3, should be taught together so that
children see the relationships existing between these combina-
tions. When such relationships are emphasized, children begin








to get the notion that when they put groups together, they add;
that when they take groups apart, they subtract.
When one realizes that there are so many addition and
subtraction facts to be learned, the importance of emphasizing
relationships is obvious. Research has shown that certain gen-
eralizations assist the child in learning these facts.

Relationships Among Addition Facts
From the following facts:
1 2 3 4 5 6 7 8 9
+1 +1 +1 +1 +1 +1 +1 +1 +1
2 3 4 5 6 7 8 9 10
one may generalize that adding 1 to a number gives the next
larger number. The reverse idea for each of these facts may
also be shown. (See Area A of the chart on page 37.) After
children have had experiences with doubles or even numbers
such as 6 + 6 = 12, many facts can be learned in relation to
them. For example, the sum of 6 and 7 can be arrived at by
thinking of 1 more than the double of 6 or 1 less than the double
of 7. Facts in diagonal Area B of the chart are covered by this
generalization.
When children understand that the teen numbers are ten
and so many more, they can use this knowledge in thinking
about other combinations whose sums are also in the teens.
For example, 9 and 5 may be regrouped into the familiar pattern
of 10 and 4.


0 0000 000 9
5
0 0 00 00 14

This method of thinking "ten and how many more" is one way
of helping children learn the teen combinations shown in Area C
of the chart.
It will be noted that all combinations except those in Area E
of the chart are covered by the following generalizations:
1. Adding one to a number gives the next larger number.








2. Adding zero to a number does not change the value of
the number.
3. Relating near-doubles to doubles increases or decreases
the answer by one.
4. Regrouping numbers whose sums are more than ten into
a more familiar grouping of ten and so many more will
lead to an understanding of combinations whose sums
are more than ten.
Not only the relationships among the addition combinations but
also the relationship between addition and corresponding sub-
traction facts should be emphasized.

Addition Relationships Chart

0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9 D

1 1 2 3 4 5 6 7 8 9 10 )A

2 2 3 4 5 6 7 8 9 10/ 11

3 3 4 \5 6 7 8 9 10 11 12

4 4 5 6 7 8 9 0/ 11 12 13

5 5 6 7 8\ 9 10 11 \12 13 14 C

6 6 7 8 9 18 11 12 13 14 15

7 7 8 9 10 11 12 13 14 15 16

8 8 9 10/ 11 12 13 14 15 16 17
/ B
9 9 10 11 12 13 14 15 16 17 18

D A E C

As children develop control over the basic addition and sub-
traction combinations, they should relate these facts to higher-
decade addition. An understanding that 5 and 3 are 8 will help








in adding such numbers as 15 and 3, and 25 and 3. Similarly,
an understanding of combinations such as 9 and 5 related to
other members of "family groups" such as 19 and 5, 29 and 5,
and 39 and 5 will enable children to determine sums in higher-
decade addition without carrying.
It has been pointed out that subtraction is the process of
taking groups apart. It should also be pointed out that there
are several different problem situations in which the subtraction
process is used but in which the meanings are somewhat differ-
ent. These ideas are: (1) How-many-are-left? (2) How-many-
are-gone? (3) How-many-more-needed? and (4) What-is-the-
difference? (the comparison of two groups in order to determine
how many more there are in the larger group or how many
fewer there are in the smaller group). Problems involving the
first three ideas are easier to understand and should receive
consideration before those based on the difference idea.

The language which conveys the ideas of addition and sub-
traction should be controlled by the teacher in verbal problems
so that children will be able to make intelligent decisions about
which processes to use. "How much?" "How many?" "How
much in all?" "How many altogether?" and "What is the sum?"
indicate the addition process. Similar questions indicating com-
parison or contrast-"How much taller?" "How much heavier?"
"How much more will it cost?"-indicate the subtraction process.
Children may get the answer to subtraction or addition by
"counting on." They count on a calendar to find the number of
days remaining before a holiday or on a tapeline to find differ-
ences in height. Although "counting on" enables children to
get the answer, it does not help them to understand subtraction.
Therefore, the teacher needs to help children associate "What
is left?" "loss," and "decrease in amounts" with the subtraction
process.

When subtraction is introduced in its written form, the
teacher will help the children to understand the significance of
the positions of the subtrahend and minuend through rearrang-
ing the numbers and having children observe the position of
numbers on the calendar or tapeline.

In the primary grades children have opportunities for using
numbers from one to thirty-one in connection with days in the








month, books, pencils, papers, and children in class. They will
have opportunities for using numbers greater than thirty-one
in activities involving weighing and measuring, scoring games,
and determining distances. Opportunities for using zero should
not be overlooked. By using zero to indicate no score in games,
no children absent, and the like, children come to understand
that zero in this sense means "nothing" and that zero added or
subtracted from a number does not change the value of the
number. Children learn that zero is a place-holder when they
learn the significance of zero in such numbers as 50 lbs., 20
cents. Later, they learn still another meaning of zero in connec-
tion with thermometers, sea level, and time concepts.
Because of their interests in their expanding environment,
children will need to subtract and add using two- and three-
place numbers before they leave the primary grades. At times
when borrowing is necessary, the teacher will take the initiative
in solving these problems and will not attempt to teach children
borrowing in subtraction before the third grade.
An understanding of the place value of numbers enables
children to grasp mathematical meaning with respect to the four
fundamental processes. When children begin to add two-place
numbers without carrying, they should understand relationships
of numbers in the units and tens columns. (It may be necessary
to continue using bundles of sticks to help in developing an
understanding of these place values.)
By using bundles of sticks when solving problems like the
one below, children can see that 4 ones and 3 ones are 7 ones,
and 3 tens and 2 tens are 5 tens, and that the sum is 57, or 5
tens and 7 ones. By using money in other situations children
can see the relationship of pennies to dimes.
34 3 tens and 4 ones
23 2 tens and 3 ones
57 5 tens and 7 ones

340 = 3 dimes and 4 cents
23 = 2 dimes and 3 cents
570 = 5 dimes and 7 cents
Later, when they encounter problems that involve carrying,
they will be able to understand that when the sum of the ones
column is 10 or more a regrouping is necessary and that the








groups of tens are added to the tens column. For instance, in
the next example he can see that 9 ones and 6 ones can be re-
grouped into 1 ten and 5 ones. The 5 ones are written in the
column for ones, and the 1 ten is carried or grouped with the
4 tens and the 2 tens, a total of 7 tens.
49 or 4 tens and 9 ones
26 or 2 tens and 6 ones
75 or 6 tens and 15 ones 7 tens and 5 ones
Likewise, problems involving borrowing (regrouping) in sub-
traction become meaningful. In the following example, children
can understand that 9 ones cannot be taken from 5 ones. How-
ever, the 4 tens can be decomposed into 3 tens and 10 ones.
Then, 10 ones can be added to the 5 ones making 15 ones. Now,
9 ones can be subtracted from the 15 ones and 1 ten from 3 tens.
45 or 4 tens and 5 ones or 3 tens and 15 ones
19 or 1 ten and 9 ones or 1 ten and 9 ones
26 2 tens and 6 ones
Although the take-away borrowing method (decomposition)
is recommended, it is important that teachers allow transfer
children to continue using any method they have learned if they
are successful with it. The teacher will have the responsibility
of teaching next steps to transfer pupils according to whatever
method they have previously learned to use successfully.

Skills Analysis Of Addition And Subtraction
Addition
A. Addition facts whose sums do not exceed 10
B. One-place addends, column additions, sums less than ten
2 2 2 20
5 5 0 0 00
1 1 5 50
C. Two-place upper addend
Two-place lower addend, without carrying
45 $ .45* 50 $ .50
32 .32 14 .14
*Experiences In the use of dollars and cents are provided in the selection dealing with
whole numbers because children think of amounts of money In terms of "How many
cents?" but not as a fractional part of a dollar.








D. Two-place upper addend
One-place lower addend, no carrying
32 + 4 40


E. Addition facts whose sums do not exceed 18
F. Two-place upper addend
One-place lower addend, carrying, bridging the decade
36 + 5 36(
5d
G. One-place addends, column addition, sums more than ten


4 4 4 4
2 2 2
5 50 0
9 9 5
7 70 7
H. Two-place upper addend
Two-place lower addend,
65 $ .65
16 .16


4(
2(
00
5
70
(less than one hundred)
carrying in units column


I. Two-place upper addend
Two-place lower addend, carrying
Sum more than one hundred
71 $ .71
49 .49
J. Two-place addends


(Regular)
27 $ .27
35 .35
14 .14


(Irregular)
27 27
4 4
35


K. Three-place upper addends
One-, two-, or three-place lower addend
1. Carrying from units to tens only
136 $1.36
237 2.37
2. Carrying from tens to hundreds only
275 $2.75
241 2.41







3. Carrying from units to tens and to hundreds
467 $4.67
274 2.74

4. Carrying to tens and to hundreds, two-place lower
addend
232 $2.32
85 .85
5. Carrying from units to tens and hundreds, one-place
lower addend
594 $5.94
9 .09
L. Three-place addends, column addition
125 1.25 miles 2.26 grams 189
241 2.41 miles 1.31 grams 176
250 2.50 miles 3.49 grams 9
84


Subtraction
A. Subtraction facts whose minuends do not exceed ten
B. Two-place minuend
Two-place subtrahend
Without borrowing
34 344 $ .34 34 34 $ .34
12 12_ .12 10 10 .10
C. Two-place minuend
One-place subtrahend
Without borrowing
38 $ .38
5 .05
D. Two-place minuend
One-place subtrahend
Borrowing
32 $ .32 30 30 cents
4 .04 4 4 cents
E. Subtraction facts whose minuends do not exceed 18








F. Two-place minuend
Two-place subtrahend
Borrowing
Zero to left unexpressed in difference except when deci-
mal fractions are used
42 $ .42 32 $ .32
29 .29 29 .29
3 $ 1.03
G. Three-place minuend
One-place subtrahend
Without borrowing
347 $ 3.47
5 .95
H. Three-place minuend
Two-place subtrahend
Without borrowing
436 $ 4.36 355 $ 3.55
12 .12 5 .05
I. Three-place minuend
One-place subtrahend
Borrowing from tens
345 452
7 9
J. Three-place minuend
Two- or three-place subtrahend
1. Borrowing from tens and hundreds
423 430 403 403
156 156 56 356
2. Borrowing from hundreds only
423 536
152 282
3. Borrowing, using dollars and cents
$ 4.23 $ 4.23 $ 4.23 $ 4.50
1.14 1.51 3.52 3.80
K. Three- or four-place minuend with two or three zeros
One-, two-, three-, or four-place subtrahend
1. Borrowing from tens, hundreds, or thousands
500 1,600 miles 4285
198 540 miles 1862








2. Borrowing, using dollars and cents
$ 5.00 $ 10.00
1.98 5.95

Multiplying Whole Numbers
Multiplication is an efficient means of adding like amounts.
Children come to this understanding slowly and only after many
experiences with adding amounts that are the same. As children
have repeated experiences in adding like amounts and are suc-
cessful in determining the answer, the teacher shows the rela-
tionships between the addition and the multiplication processes
and introduces the multiplication form. He then helps them learn
reverses. When they learn that 2 fours are 8, they can also
learn that 4 twos are 8.
An analysis of the teaching of multiplication combinations
will enable teachers to see that, just as in the teaching of addi-
tion and subtraction facts, development of generalizations and
emphasis on relationships are important. The multiplication
facts involving the twos and the doubles are perhaps the easiest
combinations to teach since the relationships between the addi-
tion and the multiplication processes in this instance are so
clear. By reviewing the doubles (addition) it can be pointed
out that 2 ones are 2, 2 twos are 4, 2 threes are 6, and on through
2 nines are 18. The form of the doubles can then be changed
to indicate the multiplication process:
1 2 3 4 5 6 7 8 9
X2 X2 X2 X2 X2 X2 X2 X2 X2

At this point, the expression "Two fours are eight." is more
meaningful than "Two times four are eight." Later, the mean-
ing of "times" should be made clear. The child should see that
"two times four" means "four taken two times."
The zero combinations should be delayed until problems in-
volving numbers that contain zero as part of the number, such
as 20, are encountered. Through a variety of experiences, the
generalization should be developed that when zero is multiplied
by a number or a number is multiplied by zero, the product is
zero. Likewise, the generalization should be developed that
when a number is multiplied by one or when one is multiplied
by a number, the product is the number.







Children should have many opportunities to discover the
multiplication facts. The solving of problems by using concrete
materials such as pennies, beads, and discs will help them to
discover facts and to see the relationships between facts. For
instance, in a problem involving the unknown fact that 4 eights
are 32, the children may think of several ways of finding the
product. They may suggest the following:
1. Add 8 and 8 and 8 and 8.
2. Since 2 eights are 16, 4 eights are twice as many, or 32.
3. Since 3 eights are 24, 4 eights would be eight more, or 32.
4. Since 8 fours are 32, 4 eights are 32.
5. Since 5 eights are 40, 4 eights are one eight less, or 32.

After the multiplication facts have been discovered, they may
be arranged in table form so that sequential relationships may
be noted.

Multiplication facts involving 9 may be learned easily by
pointing out the relationship of these facts to the facts involving
10. This relationship is shown as follows:
10 twos are 20; 9 twos are 1 two less than 10 twos, or 18.
10 fives are 50; 9 fives are 1 five less than 10 fives, or 45.
10 eights are 80; 9 eights are 1 eight less than 10 eights,
or 72.

After the combinations involving 9 have been developed and
have been arranged in table form, additional relationships may
be pointed out: (1) the sum of the digit in each product is 9,
(2) the right-hand number of each product increases by one
while the left-hand number decreases by one as the products
become larger.

The arrangement of multiplication facts in chart form may
help children to see relationships among the multiplication facts
as well as the relationship of division and multiplication facts.
The chart below contains all the multiplication and division
facts except those involving 0 and 1. (It has been pointed out
earlier that these facts should be taught as generalizations.)
The use of the figures at the top and at the left of the chart (as
multipliers or multiplicands) makes clearer the relationships of
all other multiplication facts. Similarly, the figures at the top and
left of the chart may be used as divisors when division facts








need to be used. If the facts involving 0 and 1 are known and
if it is understood that a rearrangement of the multiplier and
the multiplicand makes no difference in the product, there re-
main only 36 more multiplication and 36 more division facts to
be taught. These 36 combinations are shown on the chart.

Relationships Among Multiplication and Division Facts

2 3 4 5 6 7 8 9

2 4 6 8 10 12 14 16 18

3 9 12 15 18 21 24 27

4 16 20 24 28 32 36

5 25 30 35 40 45

6 36 42 48 54

7 49 56 63

8 64 72

9 81

Systematic drill with the multiplication combinations should
not be practiced before the third grade. In the third grade the
drill on facts should be limited to those having a product of no
more than 36, except for combinations of 5's and 10's. This
product (36) is selected because most problems which children
may be expected to solve independently in third grade will in-
clude combinations no larger than 4 times 9, 4 times 8, 5 times 7,
and their reverses. As children are learning the multiplication
combinations, they can also learn the corresponding division
combinations. Facts involving 10 and 12 are frequently used
in many socially significant problems; however, problems involv-
ing the use of 10, 11, and 12 as multipliers may be solved by
































5. Multiplication and division combinations are easier if they
can be seen as well as heard.

the multiplication process when children know the basic multi-
plication combinations. Facts involving these larger numbers
are of special interest to children who enjoy manipulating num-
bers and establishing relationships. Although such explorations
should be encouraged, emphasis should be given to the develop-
ment of the basic multiplication facts.
Because children grow in interests in their expanding en-
vironment, they will need to use multipliers of more than one
place before they leave the primary grades. The teacher should
take the initiative in solving these problems but should capitalize
on them to the extent that children begin to understand the
peculiar multiplication form and the placing of partial products.
Little understanding of the algorisms or forms of computa-
tion can be gained without an understanding of the idea of place
value and of the action indicated by each of the four symbols of
operation. Much of the difficulty of learning the processes of








multiplication and division could be avoided by helping children
think about the meaning of these operations and about the place
value of symbols within a number.
The ability to multiply with two-place numbers can be de-
veloped through easy stages. The first experience with two-
place multipliers should come in solving problems in which 10
is the multiplier. For example, children may need to find the
number of paper cups contained in 10 packages when there are
12 cups in each package. The addition of 10 twelves should help
the child to discover the idea or generalization that multiplying
a whole number by 10 may be accomplished by adding a zero to
the right-hand side of the number. Two-place multipliers that
are larger than 10 frequently cause trouble, for now there are
two steps in multiplication with partial products to be added.
The procedure in such examples can be made sensible by think-
ing of the multiplier as so many ones and so many tens. For
example, the multiplication of 12 X 24 can be done as follows:
(1) multiply 24 by 2 ones, (2) multiply 24 by 1 ten, (3) combine
the two products to get the answer for the example as a whole.
24 24 24 24
X12 X 2 X10 X12
48 240 48 ( 2 X 24)
240 (10 X 24)

288 (12 X 24)
Later further efforts might be made to emphasize that mul-
tiplication forms represent condensed ways of adding the multi-
plicand singly, in lots of 10, in lots of 100, and so on. Thus, the
product of 324 X 123 may be obtained in the following ways:
Method A Method B Method C
324 324 324
X 123 X 123 X 123
324 324 972 three 324's
324 324 three 324's 6480 twenty 324's
324 324 32400 hundred 324's
324 123 times 3240 ten 324's twice
324 3240 or 39852
324 twenty 324's
324 32400 hundred 324's

39852 39852








In the example just given, 123 X 324 means that children
have to find out how many are in 123 groups of 324 each. In
Method A the multiplicand 324 has been used as an addend 123
times. Certainly, Method A is inadvisable to use in this instance
except to emphasize the meaning of the multiplication process.
Method B is based on the understanding that ten 324's are 3240
and one hundred 324's are 32,400. A hand-operated calculator
may be used effectively to show the multiplication process de-
scribed in Method B. In Method C one is mentally combining
the three 324's which are written separately in Method B, the
twenty 324's, and the hundred 324's. A still more concise method
may be that of omitting the zeros in the partial products.

The use of the zero in the partial products of Method C may
serve to eliminate some of the "zero trouble" the child has in
multiplication. Several methods are used to handle multipliers
having intermediate and termi-
nal zeros. The method shown
here is consistent with Method 324
X206
C above. Later the two zeros _
above may be omitted. 1944 six 324's
64800 two hundred 324's

66744



Skills Analysis Of Multiplication

Multiplication

A. Multiplication combinations whose products do not ex-
ceed 36

B. Two-place multiplicand
One-place multiplier, no carrying
12 $ .12 43
3 3 2

C. Two-place multiplicand
One-place multiplier, carrying
16 $ .16
3 3








D. Two-place multiplicand
Two-place multiplier, without carrying
43 $ .43 40
22 22 22
E. Two-place multiplicand
Two-place multiplier, carrying
25 $ .25
23 23
F. Three-place multiplicand
One-place multiplier, without carrying
413 $ 4.13
3 3
G. Three-place multiplicand
One-place multiplier, carrying
114 $ 1.14 $ 1.04
6 6 6
H. Multiplication combinations whose products do not ex-
ceed 81
I. Three- or four-place multiplicand
Two-place multiplier
224 4,241
15 26
J. Three- or four-place multiplicand
Three- or four-place multiplier
345 4,206 4206
114 514 1643
K. Multiplicand or multiplier involving zeros with emphasis
on multiplier

4266 203 60,000 80,000
540 104 250 200

Dividing Whole Numbers
Young children have many experiences with sectioning, di-
viding, and grouping. Using these experiences in the primary
grades will enable the teacher to build an understanding of di-
vision before the process is introduced.








The introduction to division should occur in connection with
real problems involving concrete objects (sheets of paper, col-
ored pencils, food for a party). Through the use of concrete
objects, the division process can be demonstrated. Then, through
recalling specific multiplication combinations, children may learn
the division combinations. For example, if children already
know that 5 twos are 10, and 2 fives are 10, they should have
little difficulty in discovering that there are 5 twos in 10 or that
there are 2 fives in 10.
The form and method of long division should be practiced
from the beginning. Research indicates that children who use
this form are more accurate and have a better understanding of
the process. Constant use of this form will enable children to
deal with more involved division problems without encountering
great difficulty.
There are two ideas in division that should be understood,
the measurement idea and the part idea or partition. The meas-
urement idea is illustrated by the problem, "How many 3-cent
stamps can be purchased for 15 cents?" Since one stamp may
be purchased for three cents, it is necessary to find how many
3's there are in 15. In this example, 3 is the unit of measure.
The size of each group is known; the number of groups is to
be determined. By arranging the 15 pennies in groups of 3's,
as illustrated below, children can see that there are 5 such
groups.
000 000 000 000 000
The measurement idea of division is related to subtraction in
that the number of groups can be determined by repeatedly
subtracting the smaller group from the total group.
The partition idea is illustrated by the following problem.
"Mother wishes to share 15 candy canes equally among Karl,
Mary, and John. How many candy canes will each get?" In
this case, the number of equal groups is known; the problem
is to discover the size of each group. The candy canes will be
divided into 3 equal groups since there are 3 children. Each
group will contain 1/3 of the canes or 5 canes. The following
illustration may make this idea more meaningful.

/////It will be noted that in each case the answer is five. However,/////
It will be noted that in each case the answer is five. However,








there is a difference in the meaning of each five. In the first
illustration, the quotient of 5 indicates the number of groups of
3 there are in 15; in the second illustration the quotient shows
the number of canes contained in each of 3 groups.
An understanding of the remainder should be made clear
to children through the division of a group of concrete objects.
A bag may contain twelve pieces of candy. After five children
are given two pieces each, there are two pieces left. Problems
of this kind are solved orally and recorded on the blackboard
so that the children associate the division process with the
written form of division.
The process of division can be taught so that it is meaningful
to children. Through the use of materials to represent ones,
tens, and hundreds, children can achieve a great deal of insight
into the long division form. The partition idea of division with
divisors such as 2 or 3 is suggested in this connection. Suppose
Tom and Frank earn 680 and wish to share this money equally.
Six dimes and 8 pennies might be brought out and separated into
2 equal groups. The act of separating should then be related to
the form used:
34
2)68
6

8
8

If the amount 720 is to be divided equally among 3 people, the
child may discover the necessity of changing the left-over dime
into 10 pennies, making 12 pennies to be distributed.
24
3)72
6

12
12

Some teachers use blocks or wooden markers to show ones, tens,
and hundreds. These materials can be manipulated easily to show
such examples as 432 + 3. The series of pictures on pages 55-57
shows various stages of the process of dividing 432 into 3 equal
parts. In teaching children the algorism of long division, atten-








tion should be given to the placement of quotient figures. After
solving the problem 2) 484 with manipulative materials, the
dividend may be separated into parts; then each part may be
divided by 2. Finally all of the partial quotients may be added
200 40 2
together. 2 ) 400 2 ) 80, 2 )T4. Thus 200 + 40 + 2 gives a total
242
quotient of 242. Later, this form 2 ) 484 should be used. Many
4

8
8

4
4

similar experiences will help a child to place quotient figures cor-
rectly and will enable him to see that hundreds and tens are di-
vided in the same way that ones are divided.

Another method of solving division problems that has been
tested experimentally is that of successive subtractions. In using
this method, the children are told to discover how many fours
there are in 16 by successively subtracting four. As children
first attempt to use this method, their work may appear long
and tedious. As they gain insight through many experiences,
they will discover short cuts that indicate more mature responses.

The following illustrations show how the work of children
may be recorded as they progress from immature to mature ways
of using this method:

4)16 4)16 4)16
4 1 8 2 16 4

12 8 0
41 82

8 0 4
4 1

4
4 1

0 4








SEventually, children will be able to. use this method in solv-
ing every division problem regardless of' size of divisor, dividend,
or quotient. The advantages of using the method of successive
subtractions in solving division problems are: (1) its simplicity
enables young children to solve their problems involving divi-
sion; (2) the number of skills needed to obtain quotient figures
is greatly reduced since the method is based on one general
idea, that of successive subtraction;' (3) the relationship between
division and subtraction becomes clear.
The teacher should carefully control the introduction to the
use of two-place divisors. During the solution of real problems
in which a dozen or ten of something is considered, children
should encounter little difficulty using 10 or 12 as divisors. The
next step in the use of two-place divisors is one which demands
close supervision by the teacher. Easy divisors like 21, 22, 23,
31, and 44 should be used. Since the children will not know
division combinations containing these numbers, the teacher
must explain how multiplying between the quotient and the
divisor is accomplished. For example, in the problem:

2
21)42
42


the child may not know that 2 twenty-ones are forty-two and
may need to multiply each number of the divisor by 2. Confu-
sion as to which number of 21 is multiplied first may result
unless the teacher calls attention to the fact that the multipli-
cation begins with the number in the ones column just as in
the case of any other multiplication problem.
Throughout the intermediate grades, when using practice
material, the teacher needs to control the selection of divisors.
Divisors ending in a 0, 1, 2, or 3 are easiest to use since they
rarely call for a change in the trial number in the quotient. Di-
visors ending in 9, 8, or 7 are easier than divisors ending in 4,
5, or 6 since the quotient number is likely to be the same as it
would be if the divisor ended in 0 and were in the next decade.
In general, children should be able to find correct quotient
figures by rounding down or by rounding up. For example, two-
place divisors just above the decade, such as 11, 12, 13, 21, or







22 may be rounded down to 10 or 22 while divisors approaching
a new decade such as 18, 19, 28, or 29 should be rounded up to
20 or 30. Systematic drill in finding trial quotient figures when
the divisor ends in 4, 5, and 6 should not be engaged in until
the children have confidence in their ability to divide and have
reached the stage where extra manipulation of numbers to find
the correct quotient becomes stimulating and enjoyable.


Skills Analysis Of Division
Division (Long Division Method)
A. Division combinations with dividends up to 36:

B. One-place divisor
Two-place dividend
Without remainders, without carrying
Without zeros


2)64


3) 96


6. The problem is to divide 4 hundred-blocks, 3 ten-blocks,
and 2 single-blocks into 3 equal groups. (The circles in-
dicate the three groups.)












I

3/432


7. The first step is to divide the 4 hundred-blocks into 3
groups. Three of these blocks have been used. One block
has been placed in each group; one block remains. The
1 hundred-block cannot be divided equally among the 3
groups in its present form.


I

3432

S3 c


8. The 1 hundred-block has been changed to 10 ten-blocks.
The 10 ten-blocks have been added to the 3 ten-blocks
making a total of 13 ten-blocks.














I J +
If
?1 '32

43
i2
1^


9. The 13 ten-blocks have been divided equally among the
three groups. Four ten-blocks have been placed in each
of the groups. Thus, 12 of the ten-blocks have been used
and I remains unused.


13 I
12


10. The 1 ten-block that was not used has been changed to
10 single-blocks and added to the 2 single-blocks making
a total of 12 single-blocks.




















13
12
I?
f2
11. The 12 single-blocks have been divided equally among the
three groups. Four blocks were placed in each group. It is
now clear that each group contains 1 hundred-block, 4
ten-blocks, and 4 single-blocks or 144.


C. One-place divisor
Three-place dividend
Three-place quotient
Without remainders, without carrying
4)848 3 )663
D. One-place divisor
Two- and three-place dividends
Remainder without carrying
2) 47 4 )49 3) 367
E. One-place divisor
Two- and three-place dividends
Carrying, without remainder
4)96 3) 72 3)432
F. One-place divisor
Two- and three-place dividend
Carrying with remainder
Same number of places in quotients as
3)47 4)59 2)51


4)449



4)524



in dividends
3) 742







G. One-place divisor
Three-place dividend
Two-place quotient
Without carrying, without remainders
2) 126 4) 128 6) 126
H. Division combinations with minuends up to 81
I. Repetition of B, C, D with zero difficulties
2) 60 4 )804 4) 120
J. Two-place divisor, two- or three-place dividend
One-place quotient, without remainders, with or without
zero difficulties
Without adjustment in trial quotient figure needed
12)24 21) 105 32) 192
K. Two-place divisor
Two- and three-place dividend, remainder
Without adjustment in trial quotient figure needed
32)69 24)49 31)203
L. Two-place divisor
Three- or four- or five-place dividend
Two- or three-place quotient difficult to estimate
48) 1546 49) 3793
M. Three-place divisor
Four- or more-place dividend
Quotient easy and difficult to estimate
195 )4052 267 ) 16,000











CHAPTER 4


Fraction Understandings And Skills

V ERY EARLY IN LIFE children begin to use fractions.
Studies have indicated that many children are able to dis-
tinguish between one-half, one-third, and one-fourth before they
enter the first grade. Other first-grade children, however, de-
scribe fractional amounts less accurately and are content to
speak of three pieces in describing thirds or two pieces in re-
ferring to halves or the "big half" of an object cut into two parts.
In dividing objects into halves, thirds, and fourths, these children
are less accurate in drawing and cutting and are satisfied if the
correct number of pieces is obtained.

Children have had experiences with fractions before they
deal with symbols for those experiences. They have observed
the use of common fractions when Mother was following a
recipe, when Dad was sawing a long board into shorter equal
pieces, or when they had to divide a candy bar into smaller
parts. The fact that the denominator may be small makes the
common fraction form easier than other fraction forms for chil-
dren. It is easier to visualize a candy bar being divided into 2,
3, or even 4 pieces than one that is being divided into 10 or 100
equal pieces. Furthermore, the fact that the symbols for both
the numerator and the denominator are visible may facilitate
the learning of the common fraction form. As a result of many
experiences, children learn that a fraction may represent part
of a group or part of a single unit. Later, they learn that what
has been expressed in the common fraction form may also be
expressed in decimal and percentage forms.

Whenever possible, children should have opportunities to
become familiar with the three fraction forms on both oral and
written levels. In the primary grades the emphasis will be on
the oral use of such amounts as one-half, one-third, two-thirds








in relation to such things as cups, papers, fruit, candy; fifty per
cent and one hundred per cent in relation to attendance and
participation in school activities. The children become familiar
with the forms as the teacher writes amounts expressed on the
chalkboard. These figures may represent amounts in recipes,
per cents of children riding the school bus, or the per cent of
children eating in the school cafeteria. In the intermediate
grades children will continue to develop meanings for the frac-
tion forms and will have systematic instruction in the compu-
tation of common fractions.
When children first begin working with fractions, many dif-
ferent kinds of manipulative materials should be used regularly.
As the work progresses, the use of these materials may decrease,
but as new concepts are introduced manipulative materials
should be employed again. Too constant reference to any one
device or aid tends to limit the child in his understanding. Only
as children deal with a variety of materials will they be able
to develop generalizations that can be applied in new situations.
Easily constructed devices will help substantially in building
sound concepts. Wherever possible, children should be encour-
aged to assist and actually make such materials with teacher
guidance in order to insure accuracy. Commercial materials may
also be available but should be selected wisely.
In the beginning, a pupil cannot be expected to understand
all the vocabulary used with fractions. As he grows in his
understanding of fractions, his vocabulary should also grow.
The technical vocabulary -numerator, denominator, mixed
number, improper fraction, and proper fraction should become
refined as knowledge of this vocabulary can be used.
Work with fractions centers around two ideas: the number
of parts and the size of parts. In each of the three fraction forms,
a method is used to present these two ideas. In common frac-
tions the counting (how many) number is called the numerator,
and the name number is called the denominator and indicates
the size of the fractional unit.

In decimal fractions the numerator is expressed with a nu-
meral; the denominator is expressed in tenths or multiples of
tenths and derives its name and value from the position of the
fraction in relation to one's place. In our decimal number sys-
tem, one's place is the center rather than the decimal point.








For example, in the number 126.05, the six is the center, not
the decimal point. In percentage, all denominators have the
name hundredth. Although different in form (/, .75, and 75%),
common, decimal, and percentage fractions convey the same
ideas of part-whole relationships.
Children need to develop an understanding of the relative
value of unit fractions (fractions whose numerators are 1).
Children need to recognize that 1/2 is larger than 1/3, and 1/6
is smaller than 1/5, and that 1/100 is much smaller than 1/10.
After many experiences in which children deal with parts of
things which they can see and feel, they will discover that the
smaller the denominator, the larger the size of the part and the
larger the denominator, the smaller the size of the part.
A more detailed treatment of each of the three fraction forms
- common fraction, decimal fraction, and percentage follows.

Using Common Fractions
Although children begin to use fractions early in life, they
rarely have a need for solving a problem in which common
fractions with larger denominators are involved. For this rea-
son, it seems advisable that practice with common fractions be
limited to fractions whose denominators are common to problems
of everyday life. Research shows that halves, fourths, fifths,
tenths, thirds, sixths, eighths, and twelfths are the most fre-
quently used fractional parts of a whole.

Adding And Subtracting Common Fractions
The processes of addition and subtraction of fractions give
meaning to each other and are usually taught together. By the
time children are in the intermediate grades, they will be able
to solve orally many problems whose solution depends on adding
and subtracting fractions with life denominators. Next steps
will consist of learning to write problems containing fractions
in their proper form, to find a common denominator, to reduce
the answer, and to borrow in subtraction. If children continue
to have experiences using a measuring cup and ruler, cutting
up fruit and paper, marking off play areas, and the like, they
will understand that any fractional part has other common frac-
tion equivalents. Many experiences pointing out this fact should
be shared with children before an attempt is made to teach








them to reduce the answer, find the common denominator, or
borrow (regroup).

Working with concrete or semi-concrete materials will help
in developing an understanding of equivalent fractions in a
meaningful way. By using charts and diagrams, children can
understand the relationship between parts and smaller parts.
They can see that 1/2 equals 2/4, or 4/8, or 8/16 and that 1/3
equals 2/6, or 4/12, and other equivalent fractions. Through
many experiences of seeing, measuring, and comparing parts,
children should develop an understanding of the following gen-
eralizations: (1) the numerator and denominator of a fraction
can be divided by the same number without changing the value
of the fraction and (2) the numerator and denominator of a
fraction may be multiplied by the same number without chang-
ing the value of the fraction. On page 69 is a photograph of a
group of children who are developing an understanding of these
two generalizations. It will be noted that these children are
using symbols in dealing with fractions while meanings are being
developed through the use of manipulative materials.


Skills Analysis Of Adding And Subtracting Fractions

In the following analysis of skills in adding and subtracting
fractions an effort has been made (1) to illustrate the develop-
ment of skills by using circles and diagrams and (2) to show the
movement from the use of semi-concrete materials to the use of
abstract symbols.


Adding Like Fractions

When children begin work with adding fractions, it is helpful
to begin with words such as: one-fourth and one-fourth are
two-fourths, or 1 fourth and 1 fourth are 2 fourths. This approach
of using the denominator as a name of the parts has a tendency
to strengthen the understanding that the numerators are added,
but the denominators remain the same. When the denominators
are alike, the sum may be a proper fraction, a mixed number, or
an integer as shown in Cases 1, 2 and 3 respectively:














































Adding Unlike Fractions
Adding fractions whose denominators are not alike creates the
problem of finding a common denominator. Case 1 demonstrates
that the denominator is one of the expressed denominators and
can be determined by inspection as follows:


Case 1

+ ri


1 fourth
2 fourths + t = or +
3 fourths


Case 3




3 fourths
+ fourth +I=t=1 or
+I
4 fourths
~or 1








Notice that there
are two parts, a and b,
but these two parts have
S / different names. When
f[a = putting them together
b = in the addition process,
b s it is necessary to change
the half to fourths be-
Sketch 1 Sketch 2 cause the principle of
likeness is basic in all
aspects of adding and
subtracting. By inspection, the relationship of parts a and b can
be established and shown as in the second sketch.

In other instances there are fractions whose common denomi-
nator is not one of the expressed denominators as in Case 2. In such
instances, a fraction chart may be used to help children see rela-
tionships.

Case 2

1 1






A study of the chart will enable children to see the relationship of
thirds and halves to sixths. In the same way, the relationship of
thirds, fourths, and eighths can be developed by inspection.

After many such experiences it is appropriate to help
= W pupils reach the conclusion that the common denom-
i = I inator can be found by taking the product of the two
denominators.
5

In some instances the common denominator is found
by multiplying the largest denominator by 2, then by 3, 4, and so
on until the common denominator is found as in Case 3. This is
sometimes referred to as "doubling" or "tripling" the denomin-
ator. Again, a fraction chart will enable children to see fraction
relationships.








Case 3


+

By a similar study it can be observed that is equivalent to -
and that is equivalent to 3. In this example, the common de-
nominator is found by doubling the largest denominator.
1-- 2
1 3

Adding Mixed Numbers
Adding mixed numbers is based on the assumption that children
have the ability to select a common denominator as well as to de-
termine what a mixed number is. Preliminary work should be
designed to insure that children have achieved these abilities.

Case 1


(Like denominators
(A) 24
+ 3V


(B) 1I
+ 3,
410 == 52 = 51


Case 2
(Unlike denominators
(A) 2- = 2-
+ 31 = 31

5 4


(B) 1 = 1
+ 7=

-1-3 = 21


Subtracting Fractions
It is good practice to use the same terminology in teaching the
subtraction both of fractions and of integers. Again, the use of illus-
trative teaching material is advisable.








In subtracting fractions or mixed numbers it should be remem-
bered that:
1. Fractions must be expressed in terms of a common denom-
inator.
2. If the fraction in the subtrahend is larger than the fraction in
the minuend, the fraction in the minuend must be changed
as illustrated:
21 = 15
3- 3

1 or 11
(In the illustration it is to be noted that the minuend has been de-
composed. This is consistent with the method used in subtracting
whole numbers.)
The problem of finding the common denominator is the same as
in addition. Therefore, three similar situations are illustrated:
1. Like denominators-no reduction
2. Common denominator-one of the expressed denominators
3. Common denominator-unlike either of the expressed de-
nominators

Case 1
(Like denominators
Mary has % of a pie. She gave to her sister, how much did she
have left?


A. B. (Amount given C.
to her sister)


Case 2
(Common denominator one of the expressed denominators)
John found I of an inch on his ruler; Bill found of an inch on his.
Bill discovered that John's part was longer than his. How much
longer? By counting on their rulers, they can readily see that V


W 96f' i








of an inch is equal to 4 and that of an inch is g of an inch longer
than or 4. We must subtract from W, but must be changed
to eighths. How many eighths?

W = By counting, both Bill and John can readily see that
- 1 = 7 the answer is i.
1

Case 3

(Common denominator unlike either of the expressed denominators
Mother had I of a yard of ribbon. She gave Alice
.I of a yard for her doll's dress. How much ribbon
did Mother have left?
In the illustration, A represents the amount of
ribbon Mother had-- yard. She gave Alice I yard
which is shown in B.

A.


B.' i t

,---v----------------
Amount given Amount Mother had left
to Alice

< _____ ^^------------------
C i 1' l 17 :I T" "" I:.- 1-
C ....... ..:......


After Mother had given Alice of the ribbon, she had | and one-
half of a sixth left. It is necessary to halve each sixth making -s's
in order to have common units, as shown in illustration C. By
counting the number of ''s, it is clear that Mother had -'s left.



7
? == T




























12. Children use objects and diagrams to gain understandings
about multiplying and dividing fractions.
Multiplying Common Fractions
The use of many objects and diagrams will help children gain
understanding in multiplying common fractions. After children
gain an understanding of multiplication, short cuts may be intro-
duced. (It should be made clear that cancellation is reduction
before multiplying rather than after.) An understanding of the
use of the word "of" to indicate the multiplication process will
need to be developed carefully. The following illustration may
help the child to see the reason why "of" means times in mul-
tiplication.
1. 3 X 12 means 3 groups of 12 or three twelves.
2. 2 X 12 means 2 groups of 12 or two twelves.
3. 1 X 12 means 1 group of 12 or one twelve.
4. 1 X 12 means of a group of 12 or one of the three equal
groups of 12 or of 12.

0000 0000 0000)
Case 1
(Multiplying a fraction by a whole number
The teacher should help children observe that "1 half + 1 half







+ 1 half equals 3 halves." Thus, children can see the relationship
of addition to multiplication. Out of seeing this relationship, then,
children learn that i used three times is the same as 3 X -.




S+ Q+ oo-

Case 2
(Multip'ying a whole number by a fraction
I X 12 means ( of a group of 12 or 2 of the 3 equal groups of 12
or I of 12.

(000 (oooo 0000


Case 3
(Multiplying a fraction by a fraction
Using real material will facilitate the understanding of multi-
plying a fraction by a fraction. A piece of paper may be folded
in thirds. A third may be cut off. By folding this third in two,
children can observe that one of these folded pieces is one-half of
the third or one-sixth of the total piece of paper.



S1 X4 or ofi=
1

Case 4
(Multiplying mixed numbers
Meaning can be given to all types of examples in which mixed
numbers are employed by using actual measurements or drawings
in solving real problems. Work with concrete materials should lead








to the general rule that holds for all cases: when multiplying two
numbers (integer, simple fraction, or mixed number) express each
as a simple fraction; then multiply the numerators together to
get the numerator of the product and the denominators together
to get the denominator of the product. When possible, cancella-
tion may be used. It should be clear to children, though, that
cancellation is a way of reducing fractions before they are mul-
tiplied.
1. A mixed number multiplied by a whole number
Problem situation: Jack earned 1- dollars each Saturday.
How much did he earn in 3 Saturdays?
3 X li means 1 three times. Addition can be associated with
this problem. Jack can see that he has 3 one-dollars and 3
halves. He can also see that 3 and -'s are 4-.

1 3 X 1i= 3 X T= 4= 4

33 = 41


2. A whole number multiplied by a mixed number
Problem situation: Jane's recipe for cupcakes calls for 4
cups of flour. Jane wishes to make 21 times as many cakes
as the recipe makes. How many cups of flour will she need?
After actual measurements have been used to solve the
problem, children should see that a whole number may be
multiplied by a mixed number in two ways: (1) by mul-
tiplying the whole number by the parts of the mixed num-
ber and (2) by changing the mixed number to an improper
fraction.
4
21
2 X4= X4=-=10
2 ( X 4)
8 (2 X 4)

10

3. A mixed number multiplied by a fraction
Problem situation: John walks 1- miles to the beach. When
he has walked | of the distance, how far has he walked?








Using a scale drawing, John can see that when he has walked
two-thirds of the distance he has walked two half-miles or
one mile.


2 x 1 1 2 V 3 1
1 1 1



4. A mixed number multiplied by a mixed number
Problem situation: How much will 2- yards of lace cost at
124 cents per yard?
12'c 12c 61c 121
--------e ----- --2
121
1 yd. 1 yd. yd. 6C

314
By using the scale drawing, children can apply the process
of addition in finding the total cost of the lace. They can
also see that 24 yards of lace will cost 2' times as much as
one yard costs. Then the computation may be done by
changing the mixed numbers to improper fractions and mul-
tiplying numerators together for the numerator of the prod-
uct and the denominators together for the denominator of
the product.
24 X 121 = X = -a = 311

Dividing Common Fractions
In the discussion of division of whole numbers two ideas were
treated: measurement and partition. Work in the division of
fractions is also based on these two ideas. Although division of
fractions is interpreted as comparison or ratio, the purpose of
comparison is twofold: to compare a larger number with a smaller
number and to compare a smaller number with a larger number.
In comparing a larger number with a smaller number, the quo-
tient is thought of as a how-many-times number. For example,
in a situation in which two pies are to be divided into halves, the
question is, "How many halves are there in two pies?" The quotient
4 means that there are 4 halves in 2 whole pies. It also means that
2 is four times as large as ".
In another situation two children are to share of an apple,
what part of the whole apple will each child get? The quotient
T shows what part of the apple each child will get. As children








work through many problem situations they should develop an
understanding that the dividend is how-many-times the divisor
or a part of the divisor. Through many experiences children will
develop the ability to know when the quotient is a part of the
number and when it is how-many-times the number. They will
discover that measurement is used when the dividend is larger
than the divisor, that the part idea or partition is used when the
dividend is smaller than the divisor.
Children frequently have difficulty understanding why the quo-
tient is larger than the dividend when the divisor is a proper frac-
tion. To help build and strengthen the understanding that as the
divisor becomes smaller the quotient becomes larger the
following illustration may be used.
1. 8 8 = 1 How many 8's are there in 8?
2. 8 4 = 2 How many 4's are there in 8?
3. 8 2 = 4 How many 2's are there in 8?
4. 8 1 = 8 How many 1's are there in 8?
5. 8 +- = 16 How many i's are there in 8?
6. 8 = 32 How many I's are there in 8?
Work with division of fractions should be centered around real
situations and begin with the division of fractions by integers
and integers by fractions. Simple problems with logical conclu-
sions will help the child to understand division of fractions.

Case 1
Dividing whole numbers by fractions (Measurement idea)
1. 4 I means "How many halves in four?"
By counting the halves on a ruler: "1, 2, 3, 4, 5, 6, 7, 8"



1 2 3 4

Thus, children see that there are 2 half-inches in one inch.
In 4 inches there are 4 times as many half-inches or 8 half-
inches. Children need many experiences of this type in order
to discover the principle of inversion.
2. 6 -t = means "How many I's are there in 6?"
This particular problem can be solved by finding how many
bows of ribbon each I of a yard long can be cut from a piece
6 yards long. Using an illustration like the one below, chil-








dren can see that there are 9 pieces (each 2 of a yard long)
in 6 yards of ribbon.
1 2 3 4 5 6 7 8 9






Teaching children to divide by the process of inversion can be
related to former experiences. Children have learned in dealing
with the part-idea of division of whole numbers that 8 divided
by 2 is the same as of 8. They have also learned that factors
may be reversed in multiplication, for example, T X +-= X Y.
Drawing upon these understandings, children will be able to see
that in the example given above 6 -2 and 6 X 3 will give the
same results.

Case 2
Dividing fractions by a whole number (Partition idea)








In the illustration above, of a pie is divided into 2 equal parts
-one part for Dick, one part for Tom. Children can readily see
that Dick and Tom would each receive 4 of one whole pie. There-
fore, + 2 I= They can then observe that i + 2, and X i give
the same results.

Case 3
Dividing a fraction by a fraction (Measurement idea)
1 2


te i t








The use of a ruler will help children find the number of fourths
there are in three-fourths. By asking how many fourths there
are in three-fourths, the child can count and see that there are
three. (The three means 3 one-fourths in 3.) By applying what has
been previously learned, it is possible to see that
-- ==- X 4t= =3

Case 4
Dividing mixed numbers
Meanings can be given to the different kinds of division ex-
amples in which mixed numbers are used either as divisors or
dividends by using real problems and audio-visual aids. In the
examples below problem situations are given to illustrate the kinds
of problems that may be used in approaching the teaching of
division of mixed numbers.
1. A mixed number divided by a whole number
Problem situation: Mary wishes to cut 3 streamers from
li yards of ribbon. How long will each streamer be? By
using a diagram, children can see that one-third of 1 yards
is yard.

yd. yd. Iyd.
Later, children may work the example by changing the
whole number and mixed number to fractions. Then by
inverting the divisor, they can proceed as in multiplication.
1 +3= + 3 3 =- X = ---
2. A mixed number divided by a fraction
Problem situation: Nancy is making dresses for her doll.
She uses I yard of cloth for each dress. How many dresses
can she make out of 1y yards?
By actually measuring the cloth using I yard as the unit of
measurement, Nancy can discover that she can make 6 doll
dresses out of the material. Later, the example may be
worked by changing the mixed number to a fraction. Then
by inverting the divisor, the numerators can be multiplied
together to get the numerator of the product, and the de-
nominators may be multiplied together to get the denom-
inator of the product.
1i --- =-+ =iX = -=-c -








3. A mixed number divided by a mixed number
Problem situation: John's aquarium holds 7- gallons of
water. He fills it with a bucket that holds 1I gallons. How
many bucketfuls will he need in order to fill the aquarium?
By actually counting the number of times the bucket is
emptied into the aquarium, children can find the answer
to the problem. Later, the process of inversion may be
used in making the computation.
3
1 1 15 5 _/ 4
7- l--3- 2 ----X-=6
2 4 2 42X
1
Work with visual aids should lead to the general rule that
holds for all cases: to divide with fractions, first change
whole numbers and mixed numbers to fractions. Then, in-
vert the divisor and multiply. Cancellation may be used
when possible.

Decimal Fractions
Many abilities developed while working with whole numbers,
money numbers, and common fractions, as well as an understand-
ing of the decimal number system, prepare pupils for working
with decimal fractions. While working with whole numbers,
children learn that each position in a number has a value ten
times as great as the one immediately to its right. They learn
that this holds true for decimal fractions as well. As pupils work
with decimal fractions-using illustrative materials, building
and analyzing decimal fractions just as they did when taking
apart and putting together whole numbers-they will observe
that decimal fractions are but an extension of the number system.
Essential to an understanding of decimal fractions is the fact
that one's place is the center of the number system, not the deci-
mal point. For example, in the number 25.5 the first number to
the left of one's place represents tens, or 20 in this particular
number; the first number to the right of one's place represents
tenths, or 5 tenths in the number given. The value of a digit is
determined by the place or position in which it is written, its
place in relation to one's place. The decimal point is used to
indicate the end of the whole number part of a mixed number.
For example in the number 52, the 5 means 5 tens; in the num-








ber 0.52, the 5 means 5 tenths. Each place or position in a number
has a name which indicates its value.
Children analyze decimal fractions in much the same way
that they analyze whole numbers. The decimal fraction takes
on more meaning as children learn to do the following:
1.24 = 1.00 + 0.2 + 0.04
1.24 = 1.00 + 0.20 + 0.04

IT= 1 16 + Toa
1 = + oI + 4

Comparing common fractions with decimal fractions makes it
possible for children to observe (1) that the numerator of the
decimal fraction can be seen and (2) that the denominator is
indicated by the position in which the numerator is written. In
either case, common or decimal fractions, the numerator shows
how many parts there are in the fraction, and the denominator
indicates the size of the fractional unit.
In previous work with money numbers, children have learned
of the decimal relationship between pennies and dimes and be-
tween dimes and dollars. It is appropriate to emphasize the
reverse relationship at this point, namely, a dime is one-tenth
of a dollar; one cent is one-tenth of a dime or one one-hundredth
of a dollar.
Experience shows that most of the errors made in multipli-
cation and division are "unreasonable" errors which should be
detected quickly through a process of mental approximation.
Children in fifth and sixth grades should develop this habit under
the direction of the teacher. The exercises that follow are indica-
tive of the kind of computation that should be done mentally.
The symbol means "approximately equals."
(1) 28.04 X .49 28 X /2 = 14
(2) 5.08 X 6.98 5 X 7 = 35
(3) 32.61 X 48.62 30 X 50 = 1500
(4) 4.18 .49 4 % = 8
(5) $39.44 $9.86 $40 $10 = 4
(6) 21/8 _- 4/8s 20 4 = 5
Experiences with decimal fractions should not be limited to
the sixth grade. Children in kindergarten and first grade regu-
larly use small sums of money. Teachers of young children can








share records with members of their class. By the time children
are in the fourth grade they probably will have observed changes
in air pressure and temperature; they will have seen many
gauges in use-thermostats, oven thermometers, odometers, gas-
oline pumps. They may have kept records of growth. In addition
to these experiences, they will have used various scales in
weighing activities, rulers, tape measures, and yardsticks.
Many understandings and skills develop as a result of these
observations. Through these experiences children develop the
understanding that measurement may be expressed in both the
common or the decimal fraction form. They learn that one-tenth
of a mile may be written .1 or 1/10, and on an automobile is rep-
resented in still a different way. Likewise, .5, 5/10, 1/2 are also
the same. Occasionally, the children will hear someone say a
number like 3.15. At this time the teacher should explain that
the other fractional form is 3 15/100 and point out the simplicity
of the decimal form. It is at this time that children begin to
understand some of the advantages of each form. They learn that
when the whole is divided into few parts (when the denomi-
nator is small), the common fraction form is usually adequate.
When very fine differences are to be measured, the decimal form
is used. For example, the diameter of a screw and the expansion
of mercury are measured in decimals. Scientific instruments
usually measure in tenths, hundredths, thousandths. Before chil-
dren reach the sixth grade they should have had many experi-
ences with numbers expressed as decimal and common fractions
and know that an amount expressed in one form has its equiva-
lent in the other form.

Adding And Subtracting Decimal Fractions
When children understand the place value of numbers and
the use of zero as a place-holder, they will be ready to add and
subtract decimal fractions and mixed numbers; they will, how-
ever, need to learn to read and write decimals. In order to write
decimals correctly, they will need to form the habit of listening
for the denominator as it is spoken so that they can visualize
the number of places beyond the one's position. If the writing
of decimals is practiced carefully, children will find that adding
and subtracting decimals is not much different from adding and
subtracting whole numbers. They will also find that they carry
and borrow in adding and subtracting decimal fractions just as









they do when adding and subtracting whole numbers. Relation-
ships between common and decimal fractions are emphasized.
By using both fractional forms during initial stages of instruc-
tion, relationships can be noted.

Adding Decimal Fractions
1. Mixed numbers with the fractional part expressed in
tenths, with varied whole numbers
Without carrying
a. 2.5 + 3.1
b. 2.5 + 31.4
c. 4.3 + .5 + 24.1
2. With carrying
a. 2.8 + 3.4
b. 2.7 + 31.4
c. 4.3 + .5 + 24.4
3. Mixed number with the fractional parts expressed in
tenths and hundredths, varied whole numbers
No carrying
a. 2.5 + 3.15
b. 2.5 + 3.15 + .04

Subtraction Involving Decimal Fractions
1. a. Mixed numbers with fractional part expressed in
tenths
No borrowing
6.5 2.3
b. Fraction from mixed number; no borrowing
Fractions expressed as tenths
6.5 .3
c. Whole number from mixed number; fraction ex-
pressed in tenths
6.5 3
2. a. Mixed number with fraction in minuend expressed
in tenths, and fraction in subtrahend expressed in
hundredths; borrowing occurs only once and at the
extreme right
6.7 2.51
b. Mixed number with fraction in minuend expressed
in tenths, and fraction in subtrahend expressed in








hundredths; borrowing occurs in both columns of
fraction
5.3 2.47
3. a. Mixed number from whole number with remainder
a mixed number
6 3.5
b. Mixed number from whole number with remainder
a fraction
4 3.5
4. Varied mixed number from varied whole numbers
$64 $3.56 164 3.56

Multiplying Decimal Fractions
In multiplying decimal fractions it is important that children
learn how to place the decimal point. This sometimes involves
adding zeros. This need occurs very rarely and is of little con-
sequence in the elementary grades. However, the proper placing
of the zero requires an understanding of the decimal system.
Children can discover why the decimal point is placed where it
is in the product. They can do this by noting the relationship of
multiplication of decimals to the multiplication of fractions. For
example, in this illustration, X 5 = 15 the children
observe that tenths times tenths gives hundredths in the product.
From many similar observations, they can develop the idea that
the product should have as many decimal places as there are
in the multiplier and in the multiplicand together: .3 X .5 = .15.
In working with whole numbers and common fractions, chil-
dren learn that factors may be reversed in multiplication, usually
for the sake of convenience. They learn when working with deci-
mals that the same principle applies. It is important for children
to discover that the product must be less than 1 when tenths
are multiplied by tenths.

Skills Analysis Of Multiplication
1. a. Multiplying sums of money by whole numbers with
product less than one dollar
21X X 2 $.22 X 2
b. Multiplying sums of money by whole numbers with
products more than one dollar
$2.53 X 2








2. Multiplying mixed number by whole number
4.5 miles X 2
3. Multiplying mixed number by mixed number, varied
fractional parts where it is not necessary to add a zero
in the product before putting in decimal point
3.2 X 4.55 12.03 X 2.5


Dividing Decimal Fractions
Division of decimals is particularly useful in finding per cent,
averages-rainfall, grades, money, population, production-and
in dividing money and financial responsibility among several
people. In dividing money, children learn to place the decimal
point in the quotient by estimating and by checking the reason-
ableness of the answer. The use of the words "tenths" and "hun-
dredths" enables children to place quotient figures accurately
when dividing a fraction by a whole number. For instance, by
writing the example of 3 ) .9 as 3 ) 9 tenths, children can see that
the quotient 3 tenths or .3 is both obvious and reasonable. When
dividing a whole number by a fraction, children need to under-
stand the generalization that the divisor and the dividend may
be multiplied by the same number without changing the value
of the quotient. In the example, 4 divided by 1/, or written as
a decimal, 4 divided by .5, the question becomes "How many
.5 are there in 4?" The answer is known to be 8, but in order to
understand the answer, the pupil must think of the dividend as
4.0 or 40 tenths. Then, .5 ) 4 becomes 5 tenths ) 40 tenths. The same
quotient can be obtained by multiplying both the dividend and
divisor by 10. The example then becomes 5 ) 40 or 8.
In finding averages and per cent, it is frequently necessary
to add zeros to make the fractional part of the quotient more
accurate or to add zeros when the quotient is less than a
whole (when a larger number is divided into a smaller). Since
elementary children rarely have a need for dividing by a mixed
number, little emphasis is placed on learning to put the decimal
point in the quotient when the divisor is a mixed number.


Skills Analysis Of Division
1. Dividend a money number







a. Division of money, less than one dollar, into several
parts
750 -5 5)$.75
b. Division of money, more than one dollar, into several
parts
2) $2.48 30) $1.35
2. Dividend a fraction or a mixed number with fractional
part expressed in tenths, hundredths, or thousandths;
divisor a whole number
.24 2 $3.69 12 $36.69 30
3. Dividend a fraction, whole number, or mixed number;
divisor a whole number; zeros added to dividend
12).75 12)212)7.5
4. Mixed numbers
a. Dividend a mixed number; divisor a mixed number;
added zeros not needed
8.5) 297.5
b. Dividend a mixed number; divisor a mixed number;
zeros added
1.22) 12.34
c. Dividend a whole number; divisor a mixed number;
zeros added
1.5) 68

Using Percentage Fractions
In first, second, and third grades, children develop under-
standings about percentage fractions if they hear the teacher
speak of half the class as fifty per cent. Likewise, if forty per
cent of the class are girls and sixty per cent are boys, they should
hear that, too, and come to understand that one hundred per
cent represents all the class. Since monthly attendance is de-
scribed in terms of per cent, children should have the attendance
record before them as they note the effect of illness, stormy
weather, and the like on the class record. When they are older,
children.learn to use per cent in expressing such ideas as popu-
lation decrease and increase, production, and consumption.
While there seems to be no place in the elementary program
for independence in the solution of problems involving per cent,
teachers should take advantage of all suitable opportunities for
developing meanings about percentage fractions .












CHAPTER 5


Learning Outcomes Chart

THIS LEARNING OUTCOMES CHART is both a combination
and a revision of the Meaning and Use of Standard Units of
Measure, the Skills Allocation Chart, and the Mathematical Con-
cepts Chart found in the original arithmetic bulletin. It is hoped
that incorporating them in the text proves practical and that
teachers will find the form helpful. It will be noted that the out-
comes are not separated in terms of grade levels. This arrange-
ment makes it possible for a teacher to see the relation of the
work in a given grade to that of previous and subsequent grades.
The arithmetical skill is identified in the first column. The bar
itself indicates grade allocation of the skill and the height of the
bar indicates relative emphasis suggested for each grade. As
the teacher works with a class or an individual pupil it will be
necessary to make some adjustments for the maturity of the
children involved. The activities and procedures are suggestive
only, and no attempt has been made to include an exhaustive
listing. Moreover, they are not graded and in many instances are
suitable for several levels.

LEARNING OUTCOMES CHART
A. LEARNING TO USE THE DECIMAL NUMBER SYSTEM

Grade
Skill Allocation
Identification -- - Suggested Activities and Procedures
123456
1. To use numbers as Using calendars to find birthdates, important
neededbythegroup holidays, correct date, special information
or child to interpret in which date is important
experiences in a
meaningful way I *>* Using clocks to follow daily schedule; to de-
termine correct time; appropriate task for
time available; television and radio sched-
ules









LEARNING OUTCOMES CHART

Grade
Skill Allocation
Identification ---- Suggested Activities and Procedures
123 45 6


2. To enumerate by
counting objects by
l's







1-10




1-20




1-50






1-100


1-200


Using rulers to measure size of articles, dis-
tances, children, and furniture (feet, inches,
yards)
Using containers for measuring amounts
(cups, pints, quarts, gallons); following a
recipe, setting up an aquarium, making a
large amount of paste or paint
Using scales to weigh pupils, articles (pounds,
half-pounds, ounces)
Using thermometers in adjusting room tem-
perature, reading weather reports, taking
body temperature
Using real and play money to interpret price
tags, advertising pages in newspapers, cata-
logs, foreign coins
Counting children in order to form commit-
tees of three, four, or five members; counting
spots on dominoes; securing a paint brush
for each paint jar; helping set the table,
matching silver to plates; counting pennies
in a dime; counting articles needed by a
group of children; counting number of chil-
dren needed for small group games; count-
ing number of days before a holiday; count-
ing wheels on a car, tricycle, or bicycle;
counting candles on a cake; keeping score;
counting pupils absent.
Counting pennies in two dimes, four nickels;
counting articles needed by a group of chil-
dren; keeping score; counting number of
puzzle pieces and checking with record on
puzzle box
Counting pennies in half dollars or two quar-
ters; using calendar, counting the days of
month, counting pupils present; distribut-
ing textbooks, library books, or other school
supplies
Counting pennies in a dollar, seats for audi-
torium program, number of duplicated
papers or programs when entertaining
adults or other classes, using chart con-
taining 100 squares or circles arranged in
rows of ten (child touches each square or
circle as he counts to 100)


Using experiences described above in count-
ing to 200, using two charts or two abacuses









LEARNING OUTCOMES CHART


Skill
Identification

3. To enumerate by
counting objects by
2's
2-10





2-20


2-50




2-100


2-200

4. To enumerate by
counting objects by
5's
5-50


5-100


5-200



5. To enumerate by
counting objects by
10's
10-100


10-2,00
6. To enumerate by
counting objects by
3's and 4's


Grade
Allocation
- - Suggested Activities and Procedures
123456

Counting by couples or partners, counting
pairs of shoes, gloves, mittens; matching
groups of two pennies with five two-cent
| stamps
Counting chairs at double tables, couples, or
partners; putting away blocks, picking up
two at a time; matching groups of two pen-
nies with 10 two-cent stamps in a defense-
stamp book; counting eggs in carton ar-
ranged in double row
Using similar experiences as for 2-20; distrib-
uting two cookies per child, two pieces of
art paper per child, or rhythm band sticks

Using charts of 100 squares or circles arranged
in rows of ten, each row composed of five
groups of 2 squares or 2 circles; using
abacus with pupil touching or moving two
Sbeads at a time as he counts by 2's

I Using two charts or two abacuses instead of
one

Counting by using nickels; counting visual
materials (cards or charts) arranged in
Groups of fives


I Using experiences similar to those described
gI above

Using experiences similar to those described
S above

Counting bundles of ten sticks; counting by
using dimes; using abacus with child touch-
ing or moving one row of ten beads as he
counts by tens; using a chart of 100 squares
or circles arranged in rows of tens with child
mE touching each row as he counts by tens


S | Using two 100-charts or two abacuses
Counting total numbers o,f fasteners used in
making booklets; identifying number of
buttons on dresses or skirts; forming groups
of 3 or 4 children when playing a game;
working in committees of 3 or 4, and then


' ~ '








LEARNING OUTCOMES CHART

IGrade
Skill Allocation
Identification Suggested Activities and Procedures
-_I 4R~i


7. To count by rote
by l's

1-10




1-20




1-50





1-100


1-200
8. To count by rote
by 2's

2-10


2-50


2-100
9. To count by rote
by 5's


counting by 3's and 4's to determine total;
S counting eggs for egg hunt using egg cartons
arranged in rows of three and four; picking
S up 3 or 4 jacks at one sweep
Playing games in which pupil counts to ten
while a child hides an object; teacher or
pupils count to see how long it takes to get
a task done; using counting rhymes; count-
ing as child skips, jumps rope, skates
Using same experiences as above for counting
from 1-20, counting in rhythm as 1, 2, 3,
4, 5, 6 (number in boldface spoken with
accent) 1, 2, 3, 4, 5, 6, 7, 8,9, clapping hands
to music, bouncing the ball

Using same experiences as above, counting
* the ticks of a clock or watch, the number of
* pages read, number of inches which describe
Sa child's height
Using experiences similar to those described
above; counting the number of pounds a
pupil weighs, such as 69 pounds; counting
up to the number which .tells.how many
children are in the second grade; counting
with or without the help of a number chart
Shaving number symbols

Using similar experiences as for 1-100 ex-
tending through 200
Counting by 2's as child jumps rope, swings;
counting by 2's as pupil bounces a ball,
counting with or without the aid of a
number chart using only number symbols;
using counting rhymes


S Using similar experiences as for 2-10


l Using similar experiences as for 2-10
Counting by 5's in such games as "Hide-and-
Seek"; counting by 5's as pupil skates,
jumps rope, swings, bounces a ball; count-
*I mg with or without the aid of a number
chart using only number symbols; counting
S rhymes


I I I I I









LEARNING OUTCOMES CHART

Grade
Skill Allocation
Identification --l- Suggested Activities and Procedures


5-100


5-200

10. To count by rote
by 10's




10-100 ;.


10-200

11. To count by rote
by 3's and 4's

12. To recognize
groups of objects
without counting
2-5


6-10

11-18


SI Using similar experiences as for 5-50


Using similar experiences as for 5-100

Playing games in which pupils count by tens
as other pupil hides an object; teacher or
pupil counting by tens until task is done;
counting by tens as child skates, bounces
a ball; counting with or without the aid of
a number chart using only number symbols;
counting by tens up to a dollar without the
Suse of money or visual materials

S Using similar experiences as for 10-100
I through 200

Counting by groups of three and four objects
followed by rote counting; counting by 3's
S to determine cost of five 3-cent stamps


Reproducing, identifying, comparing groups
of objects: teacher-planned experiences
S with systematic follow-through


Identifying correct number in group shown
on chart or card; first pupil will count to de-
termine the correct number; with greater
familiarity, he will be able to identify
groups without counting; the size of the ob-
jects used and the pattern of the group
arrangement help determine whether the
pupil can at a glance recognize a group of
5, 9, 11; recognizing as a group of 11 an
object gone (an egg carton of a dozen with
one empty place, a package of a dozen pen-
cils with one pencil removed); recognizing
groups of 11-18 as one bundle of ten sticks
and so many more; reproducing different
arrangements of group patterns (page 34);
labeling each arrangement with correct
number; matching two different arrange-
ments of the same-sized group; matching
group with number symbol; comparing
group arrangement to determine which is
larger or smaller (readiness for later work
in addition and subtraction)








LEARNING OUTCOMES CHART

Grade
Skill Allocation
Identification --1234- Suggested Activities and Procedures
123456


13. To write
symbols


number


1-10




11-100


Larger numbers
14. To understand
place value:

Unit's place





Ten's place





Hundred's place
and higher






15. To know the place
of number in the
series:

1-10


11-20


Demonstrating the formation of each figure
by pointing out the place of beginning, the
direction of the strokes, and the ending
(Detailed help is available in manuals of
| |I State-adopted texts.)
Emphasizing the writing of figures in the
second decade after children have learned
thoroughly to write the figures in the first
| decade; building a 100 number chart, the
__ first row beginning with 1 and ending with 10

S1 111 Writing numbers beyond 100 after place value
I I is understood
Writing numbers from 1-10 in a column,
teacher demonstrating correct placing of
I the number 10
Correct Incorrect
7 7
8 8
9 9
10 10
Illustrating numbers 11-19 as ten objects and
S|I so many more (so many more ones); learn-
II ing that the 1 in 16, for example, means 1
SI I I ten, and that the 6 means six ones (page 32)
Providing many experiences with money,
sticks, strips of paper, so that pupils can
see that a ten is ten times the value of one,
that a hundred is ten times the value of ten,
II and that a thousand is ten times the value
| 1 I of a hundred. (The understandings that each
I 1 Il place has a definite value and that by start-
ing with units or ones and going toward the
left each place has ten times the value of the
preceding place are essential; this under-
standing will enable children to deal intelli-
gently with large numbers that are not
easily demonstrated with manipulative ma-
terials.)
Recognizing and placing in correct sequence
a series of group patterns; matching each
group pattern with the correct number
| symbol; writing or telling the number that
11 is one more or one less, two more or two
Sml m less


Providing experiences similar to those de-
scribed above








LEARNING OUTCOMES CHART

Grade
Skill Allocation
Identification I- ---- .....Suggested Activities and Procedures
________________ 1j23456 __________________________


21-100 and larger
16. To recognize uses
of zero
Absence of
quantity












Place-holder


Locating correct address by noting that 245
comes after 241 and before 247; dialing cor-
rect telephone number; determining correct
I date (social studies); locating the correct
11 I page in a book; providing experiences sim-
ilar to those described above
Keeping a record of scores in playing ten pins
(zero means no ten pins were tipped over);
3 1st turn
0 2nd turn
S I I 2 3rd turn
5 Total score
Providing many similar experiences in order
to develop the generalizations that:
1. A number plus zero equals the number.
2. Zero plus the number equals the number.
3. A number less zero equals the number.
Observing that 10's are written (10, 20, 30) so
that 0 occupies l's place; that 100's are
written so that zeros occupy both l's and
10's places (The zero is a place-holder just
as all other figures are. The zeros used in
II writing 100, for example, hold the places
of ones and tens and also show the absence
11 l of ones and tens.)


B. ADDITION AND SUBTRACTION


1. To develop mean-
ing for the addition
and subtraction
facts:
2-9


Discovering that addition is a process of put-
ting small groups together to form a larger
group; that subtraction is a process of
taking away a part of a group or of com-
paring two groups; using subtraction pro-
cess to determine:
1. how many left
2. how many gone
3. the other part
4. the difference
expressing the answer orally as it is discov-
ered many times through dramatization and
manipulation of concrete materials; estab-
lishing the relationship between addition
and subtraction by pointing out the related
facts, such as:
3+2=5 5-2=3
2+3=5 5-3=2









LEARNING OUTCOMES CHART


Skill
Identification


10-19
2. To teach facts with
sums and minuends:
1-9


10-19
3. To teach higher-
decade addition
facts:
Sum in same
decade



Sum in next higher
decade
4. To teach addition
with more than two
addends:
Single column addi-
tion with sums
through 9
Single column addi-
tion with sums more
than 10, five ad-
dends






Column addition of
two-place numbers,
without carrying,
five addends, sum
less than 100, in-
cluding irregular
addend


Grade
Allocation

1-i- -- Suggested Activities and Procedures


Engaging in activities similar to those de-
scribed above; emphasizing the meaning of
teen numbers by discovering that each teen
number is ten and so many more; regroup-
ing a combination such as 9 and 4 = 13 in-
to the familiar combination of 10 and 3
(page 36)


I Expressing with written symbols the facts
S |* E that have been discovered orally

II E Engaging in activities similar to those de-
H|| [E scribed above
Emphasizing the relationship of the simple
combinations to higher decade addition
facts:
I 4 14 24 34
S2 2 2 2

Emphasizing the relationship of smaller com-
binations to larger combinations, such as:
lI 15 25 65
9 9 9
mn -
Helping children to add the sum of the first
two addends to the third addend by dra-
matizing the process, by using objects and
semi-concrete materials



Refining the skill of column addition so that
children reach a mature level of operation
through such stages as the following:
1. Thinking and saying, "3 children and 3
children are 6 children, and 2 more chil-
dren are 8 children."
2. Thinking and saying, "3 and 3 are 6, and
2 more are 8."
I 3. Thinking and saying, "3, 6, 8."
S 4. Practicing both oral and written prob-
lems
Finding cost of school supplies: scissors, cray-
ons, notebooks, tablets
Finding cost of admission to two, three, four,
or five events at a school carnival

B |i tI BFinding cost of bus fare for five days


I I I I I


--1-I-I--l--I---


' ' '









LEARNING OUTCOMES CHART

Grade
Skill Allocation
Identification -t-[- Suggested Activities and Procedures
12345 6


5. To teach addition
using two addends,
two-place or more
Two-place upper
addend, one-place
lower addend,
without carrying




Two-place upper
addend, two-place
lower addend,
without carrying








Two-place upper
addend, two-place
lower addend, with
carrying in one's
place









Three-place ad-
dend, three-place
lower addend, car-
rying in one's and
ten's places


Helping children to express the process both
orally and in writing in such problems as,
"We have thirty-one children in our class;
this morning two new pupils were added."
31 children
I+ 2 more children

i | 33 children
Helping children express appropriately such
problems as, "We have thirty-three children
in our class; twenty-three mothers will be
our guests this afternoon. How many chairs
will we need?"
33 = 3 tens and 3 ones
23 = 2 tens and 3 ones

5 tens and 6 ones
II Drawing upon their understanding of place
value children can recognize that 5 tens and
6 ones are 56.
Helping children understand carrying by an-
alyzing such problems as, "Forty-six pupils
ordered plate lunches; twenty-six pupils
ordered milk. How many children bought
food in the lunchroom?"
46 = 4 tens and 6 ones
26 = 2 tens and 6 ones
6 tens and 12 ones
Drawing upon the knowledge that 12 is one
ten and 2 ones children can understand why
S the 1 ten is carried to ten's place and added
to the 6 tens, giving a total of 7 tens and 2
* El ones, or 72.
Analyzing such problems as the following:
"We have 249 girls and 264 boys in our
school. How many attend our school?"
249 = 2 hundreds and 4 tens and 9 ones
264 = 2 hundreds and 6 tens and 4 ones
4 hundreds and 10 tens and 13 ones
Recalling that 13 is one ten and 3 ones will en-
able children to understand why the 1 ten
is carried to the ten's place and added to the
10 tens and that 11 tens are one hundred
and 1 ten:









LEARNING OUTCOMES CHART


Skill
Identification













Four-place upper
addend; one-, two-,
three-, or four-
place lower addend
carrying as needed
6. To teach subtrac-
tion using two-,
three-, or four-
place minuend;
one-, two-,
three-, or four-
place subtrahend
Two-place minuend
one-place subtra-
hend without
borrowing




with borrowing


Grade
Allocation
1-2 -1 I6


oi'
IIaI


.IIl


Suggested Activities and Procedures


1 ten <
249 = 2 hundreds and 4 tens and 9 ones
264 = 2 hundreds and 6 tens and 4 ones
4 hundreds and 11 tens and W2ones
3 ones
1 hundred r 1 ten<
249 = 2 hundreds an 4 tens and 9 ones
264 = 2 hundreds and tens and 4 ones
5 hundreds and y1tens d ifones
1 ten 3 ones
Providing experiences that extend procedures
used above


Helping children understand the subtraction
process; the following illustrations are sug-
gestive:


Teaching the new step involved in borrow-
ing:
45 = 4 tens and 5 ones
-18 =1 ten and 8 ones
Since 8 ones cannot be subtracted from 5
ones, one of the 4 tens must be changed to
ones and added to the 5 ones; the example
may then be re-written:
45=4 tens and 5 ones=3 tens and 15 ones
-18= 1 ten and 8 ones=1 ten and 8 ones
2 tens and 7 ones
or 27


1_1_1_1_1_1_1__________---.----------









LEARNING OUTCOMES CHART

Grade
Skill Allocation
Identification I Suggested Activities and Procedures
1 2345 6


Two-place minuend
two-place subtra-
hend, without
borrowing



. with borrowing





Three-place minu-
end, one-, two-,
three-place subtra-
hend .. .without
borrowing


... with borrowing



Three- and four-
place minuend
with zeros in:

one's place
one's and ten's
place


one's, ten's, and
hundred's places


ii..



l'.
I









5:.
ME
El.E
on





onII

1.
El
son

01
MA


Using a similar procedure when dealing with
large numbers; the following examples may
be helpful:
4124= 4 thousands 1 hundred 2 tens 4 ones
-1482= thousand 4 hundreds 8 tens 2 ones

Step 1:
The 4 thousands are decomposed into 3 thous-
ands and 10 hundreds; the 10 hundreds are
then added to the 1 hundred, making 11
hundreds
3 thousands 11 hundreds 2 tens 4 ones
-1 thousand 4 hundreds 8 tens 2 ones

Step 2:
The 11 hundreds are decomposed into 10
hundreds and 10 tens; the 10 tens are then
added to the 2 tens, making 12 tens
3 thousands 10 hundreds 12 tens 4 ones
-1 thousand 4 hundreds 8 tens 2 ones

Step 3:
Now that the minuend in each place is larger
than the subtrahend, the subtraction can be
performed.
Using similar steps when the minuend con-
tains zeros:
4000=4 thousands 0 hundreds 0 tens 0 ones
--1234= 1 thousand 2 hundreds 3 tens 4 ones

Step 1:
3 thousands 10 hundreds 0 tens 0 ones
-1 thousand 2 hundreds 3 tens 4 ones

Step 2:
3 thousands 9 hundreds 10 tens 0 ones
-1 thousand 2 hundreds 3 tens 4 ones

Step 3:
3 thousands 9 hundreds 9 tens 10 ones
-1 thousand 2 hundreds 3 tens 4 ones
2 thousands 7 hundreds 6 tens 6 ones
= 2766









LEARNING OUTCOMES CHART

C. LEARNING MULTIPLICATION AND DIVISION

Grade
Skill Allocation
Identification 1 213- Suggested Activities and Procedures
123456


1. To develop an un-
derstanding of the
process of multipli-
cation as a quick
way of adding
equal-sized groups
of like quantities












2. To develop an
understanding of
the relationship of
multiplication to
division






3. To develop control
of the multiplica-
tion facts whose
products do not
exceed
36




45



81


Beginning with the addition combinations
that are doubles such as 2 + 2, 3 + 3,
show that two 2's are 4; three 2's are 6, and
so on.

Finding the number of bottles of milk needed
for children at all the tables when there are
two children at each table; four children at
each table.

Finding the number of cars needed to take
children on a trip when 5 children ride in
each car
Determining cost of ice cream for class at
5c for each serving
Examining egg cartons to find that two 6's
are 12; six 2's are 12; three 4's are 12; four
3's are 12
Finding number of articles that can be bought
when cost of each article is 5 cents or less
and child has 25 cents or less
Examining egg cartons to see that in one
dozen there are four 3's, three 4's, six 2's,
and two 6's

Developing chart of multiplication and di-
vision facts and analyzing it for relation-
ships (page 46)


Using counting frame, help children to "see"
as well as to say the combinations by
grouping the beads to demonstrate the facts
being studied, for instance, two groups of 5
beads; 4 groups of 5 beads
Helping children to discover needed facts by
relating the unknown to the known facts,
for example, a child may reason that four
8's are twice as much as two 8's.
Using experiences similar to those described
above; changing units of measure to smaller
units of measure: pints to cups, quarts
to pints, gallons to quarts and pints, bush-
els to pecks