Mathematics for Georgia schools, volume I [1962]

MATHEMATICS
FOR
GEORGIA SCHOOLS
VOLUME I

STATE DEPARTMENT OF EDUCATION DIVISION OF INSTRUCTION CURRICULUM DEVELOPMENT

)

Claude Purcell

State Superintendent of Schools

Atlanta, Georgia

1962

State Board of Education
JAMES S. PETERS, Chairman HENRY A. STEWART, Vice Chairman CLAUDE PURCELL, Executive Secretary

Members

FIRST CONGRESSIONAL DISTRICT SECOND CONGRESSIONAL DISTRICT THIRD CONGRESSIONAL DISTRICT FOURTH CONGRESSIONAL DISTRICT FIFTH CONGRESSIONAL DISTRICT SIXTH CONGRESSIONAL DISTRICf SEVENTH CONGRESSIONAL DISTRICT EIGHTH CONGRESSIONAL DISTRICT NINTH CONGRESSIONAL DISTRICT TENTH CONGRESSIONAL DISTRICT

PAUL S. STONE ROBERT BYRD WRIGHT, JR. THOMAS NESBITr JAMES S. PETERS DAVID RICE FRANCIS SHURLING HENRY A. STEWART LONNIE D. SWEAT MRS. BRUCE SCHAFFER ZACK F. DANIEL

FOREWORD

TO: Georgia Teachers of Mathematics

You wanted a guide for the teaching of modern mathematics. It has been prepared for you. Here it is.

You are teaching an exciting and potentially powerful subject. It is important that we give you every possible aid in teaching it well. Unless our students can grasp the mathematical concepts that will help to give them a clearer understanding of the universe, they will be without one of the most important tools they need to design the future.

I have been gratified at the efficient way this Mathematics Guide has been

worked out. The Georgia Council of Teachers of Mathematics suggested that

a guide be developed that would aid teachers in seeing the teaching of

mathematics as one broad sweep, from kindergarten through the 12th grade.

DR. CLAUDE PURCELL State Superintendent of Schools

The Mathematics Advisory Committee suggested that a group of Georgia teachers be chosen to do this. That group included teachers from rural schools, small town schools, and city schools, and teachers of all grades. They

looked at the nation's best math programs. They considered venturesome ideas of teaching, fitted to an age of

astronauts and yet as basic as adding two and two. They worked hopefully and helpfUlly together to produce

something that would give you the depth and dimension you wanted for teaching mathematics in the modern

moon-going age.

They never lost sight of the idea that mathematics is a daily necessity for people. They grasped the fact that it is not simply the college-bound who need sound foundations in this subject. Youngsters who plan to work in a filling station, or own a farm, or be a homemaker, will need sound grounding in figures. These are important for the purposes of daily living; income tax, balancing a bank account, figuring a crop return, totaling up the grocery bill, measuring a fence, or building a house. Modern mathematics makes mandatory accurate mathematical knowledge. Whether it is John Glenn figuring a race through space or a young couple figuring their budget, they need mathematical accuracy and understanding.
I
There is a new interest in math throughout the nation, a ground-swell of concern that students get the foundations of it. Georgia must be in the fore-front of good math teaching and learning. This Guide is a fine addition to the many things that Georgia teachers and administrators have done to improve the schools so that schools can improve our people.

I hope you find it helpful.

Claude Purcell State Superintendent of Schools

Iv

INTRODUCTION
"Mathematics for Georgia Schools" is one of a series of guides published by the State Department of Education as a result of a basic program for the public schools adopted by the State Board of Education and published in "Curriculum Framework for Georgia Schools" in 1954. This guide was written by a committee selected from elementary and secondary teachers, curriculum directors, and principals throughout the state. It was published in tentative form in the fall of 1961 and used in pilot schools during the 1961-62 school year. An editing committee used evaluations received from the pilot schools to revise the guide during the summer of 1962.
The purpose of the guide is to increase the pupil's ability to think with the ideas of mathematics and to apply those ideas to practical situations. The ability to manipulate numbers and symbols by rote learning of rules is not sufficient mathematics education for students. They must understand concepts, principles, vocabulary, and structure of mathematics. An opportunity must be provided for them to explore and discover for themselves patterns which exist.
Study of the psychology of learning indicates ways in which mathematics programs c:m be improved. Concrete materials, when used in the right way, aid in building understanding. Filmstrips, films, and educational tele vision are helpful. A child's ability to read plays an important role in mathematics just as in other subjects. Proper drill and homework are essential to learning.
The material in the guide is arranged by grade levels for the convenience of teachers using it; however, a teacher may easily adapt it to the needs of individual students by using material from a grade level above or below the one he is teaching. Since the material is developed sequentially, it could be used equally well in non-graded classes.
It must be remembered that this is a guide. It is not meant to replace the textbook. Reference books for students and for teachers will be needed. Teachers may need to participate in in-service courses or summer courses, or do individual studying in order to make the most effective use of this publication.
H. S. SHEAROUSE, Director Division of Instruction
v

MATHEMATICS FOR GEORGIA SCHOOLS
Prepared by the MATHEMATICS CURRICULUM GUIDE COMMITTEE
Gladys M. Thomason, Coordinator Mathematics Education
Georgia State Department of Education Curriculum Development Services Dr. Ira Jarrell, Director

I. Numeration
n. Fundamental
Operations with Whole Numbers ill. Fundamental Operations with Fraction IV. Relations
V. Measurement
VI. Geometry
VII. Charts and Graphs
Vill. Problem Solving
Vocabulary
Books for Children ~uipment and Aids Films
Bibliography
Glossary

TABLE OF CONTENTS

LOWER ELEMENTARY

K

1

2

3

UPPER ELEMENTARY

4

5

6

4

14

26

48

76

102 138

7

16

29

52

78

107 140

8

19

36

60

90

116 145

19

38

61

93

123 155

8

19

39

63

94

125 158

10

21

40

64

96

127 160

10

21

40

65

97

130 163

10

22

42

65

97

131 163

67

165

69

166

70

168

71

167

169

171

vii

KINDERGARTEN

UNDERSTANDINGS TO BE DEVELOPED IN KINDERGARTEN
Arithmetic understandings in kindergarten stem from the child's experiences in a somewhat informal, but not unplanned, program. Informality of the program gives the teacher an opportunity to meet differing needs, and a time to implement experiences which correspond to the short attention span and maturity level of the kindergarten child.
Observing and comprehending the proper sequence of number names as needed; experiencing one to one correspondence in situations where matching quantities are required; exploring the functions of simple tools of measurement; encountering the beginning concepts of fractions and the four fundamental processes; recognizing likenesses and differences in shapes; discovering that objects of different shapes have different names; using quantitative vocabulary on level of communication; solving simple everyday problems with concrete materials--all these experiences blend to form a meaningful back-ground of arithmetic ideas and concepts.
3

",' I
I .I

CONTENT
I. Numeration
A Decimal notation-Learn through daily living experi ences to:

TEACHING SUGGESTIONS

1. Understand there is a proper sequence of number names.

Use finger plays, nursery rhymes, and jingles, such as: "This Little Pig Went to Market" "'Tis all The Way To Toe Town" "One, Two Buckle My Shoe" "Here Is The Beehive" "Baa, Baa Black Sheep" "1, 2, 3, 4, 5; I Caught A Hare Alive"
Example: Five Little Jack-O-Lanterns (using fingers)
Five little jack-o-Ianterns sitting on a gate The first one said, "Oh my, it's getting late." The second one said, "1 hear a noise." The third one said, "It's only some boys." The fourth one said, "Let's run, let's run." The fifth one said, "It's only Hallowe'en fun." Then 0-0-0 went the wind and out went the light And away ran the jack-o-Ianterns on Hallowe'en night.
Use poems, songs, and games: Poems and Songs (Found in State Adopted Texts) "Five Little Monkeys" "The Big Clock" "The Twins" "Two Little Birds" "Five Little Chick-a-dees" "Two Little Kitty-Cats" "Ten Little Pennies" "Ten More Miles" "Five Angels" "Five Black Horses" "Three Blue Pigeons" Games "Simon Says" "Hide-and-Seek" "Miniature Bowling" "Hopscotch" Example: "Fox and Chickens" Children spread out. One child is the fox. Chickens say "Who let you in our yard?" The fox replies, "No one, and I'm goir-g to catch you!" The chickens reply, "You can't catch us ... " and they run to "Chicken House." Any chickens caught become foxes for next game. Play until only a few, the winners, are left. At the end of game count the number of foxes and chickens.
Use objects for counting such as blocks, crayons, tables, chairs, beads, paint brushes, toys.
Be alert to everyday living experiences that give opportunities for counting such as:
Children present or absent Colors used in painting Lunches needed Candles on birthday cake Steps on the stairway Members of family Pennies for cookies Rhythm band instruments

4

2. Explore the idea of one to-one correspondence between the object and the counting names.

Swings on playground Numbers on clock Days of week Characters needed to play stories (3 billy goats, 1 troll, 1 mother bear,
1 father beaT, 1 baby bear, 1 little girl, three beds, three tables, three spoons.)
Examples: "Who is not here today?" "I will write the names on the chalkboard as you name them. Let us count the names. How many children are not here today?" "At dismissal time, Joe may touch the heads of 5 children at a time to get their wraps." Take advantage of centers of interest in science, language arts, and social studies as media for developing functional counting such as: A rock, leaf, or shell collection (counting different sizes and shapes-the number added to the collection from day to day). A feeding station for birds (counting total number of birds -counting kinds and the number of each kind, etc.) Building and furnishing a play house (counting number of rooms, pieces of furniture, etc.)
Provide specific opportunities for auditory counting.
.Examples: Count the beats of the drum, sticks, triangles, etc. in the musical rhythms: one, itwo, three; one, two, three; or one, two three, four; one, two, three, four. Repeat a sound the same number of times that the teacher or another child has made (tapping, clapping, bouncing ball, etc.) Count the repeated phrases or sounds in songs, games, nursery rhymes, and poems. Count motions to make and steps to take: Clap,Clap,Clap! Slide, Slide! Stamp, Stamp, Stamp! Slide, Slide!
Explore ways for children to count forward by ones to find the number in a particular group and backward to find how many are left each time, such as arranging for:
Children to manipulate such objects as blocks, spools, and toy cars. Children to observe forward and backward counting in personal experiences, musical rhythms, games, poems, and finger plays. Examples: When riding elevators in hotels, hospitals, and stores observe the changes in lighted numbers over the door going up and coming down.
When following directions in dramatizing number situations observe the counting as:
Skate forward 5 long slides counting 1, 2, 3, 4, 5. Skate backward to the starting place counting 5, 4, 3, 2, 1.
When and if a semi-concrete situation is needed: Make 5 large charts with large numbers 1 to 5 at the top. Collect pictures distinct in number concept, so number of objects is easily dis tinguished by the children. Show the pictures and suggest that the children count and determine on which chart the picture should be pasted.
Encourage children to discover: How many sheets of paper, balls of clay, or scissors are needed for the children at a particular table. How many children are needed to set the tables for lunch, if there is one table setter at each table. A child may select the helpers by touching
5

3. Recognize 2 and 3 objects without counting.

each child selected and counting.
Provide the following experiences for children: Putting jigsaw puzzle together. Counting paint containers on one easel and placing a paint brush in each container. Counting children at art center and distributing a paper towel to each child.

Give children many opportunities to observe and manipulate objects using the group concept (2 and 3) such as:
Following directions and answering question by making a choice.
Examples: "Bring the box with two blocks-three blocks." "Choose two library books for our story time." "Get 3 sticks, 3 triangles, and 3 bells for our musical rhythms." "String the beads using 2 beads or 3 beads of one color together." "Place pegs on the peg board in 3's by choosing 3 pegs of same color." "What items have two feet; two legs?"

Dramatizing stories, poems to identify 2 or 3 groups:
Examples: "Three Little Kittens" "Three Billy Goats Gruff" "Three Little Pigs" "Three Blind Mice"

Observing number of:
boys or girls together toys or blocks together blooms on a plant wheels on a bicycle wheels on a tricycle items that come in pairs
(gloves, ear muffs, shoes, socks, etc.) items that come in sets
(set of dishes, set of buttons, set of beads.)

Observing blocks arranged in different positions:

Example: Arrange blocks in 2 or 3 groups. After the children view the grouping they may close their eyes while someone removes the grouping. In turn they may recognize the blocks as they first saw them.

o 000 o
o

o

o

o 0 DO

4. Know age, address, and telephone number.
5. Use numbers to locate objects in a series through "fifth."

Play games so that the children may have an opportunity to give information about themselves, such as: ''Who am I?" or ''Lost and found."
Use a play telephone to dial telephone numbers.
Locate birthdays on the classroom calendar.
Use number names to locate objects in a series when: Discussing work or play period, such as: first period, second period, third period, etc. Giving directions or asking questions: "Put the toys on the first shelf." "Move the second row of blocks." "You may take the first turn." "Who is third in line?" ''Who is last in line?" ''What is the second song on the program?"

6

6. Read and write number symbols 1 to 5.

Dramatizing or playing a story: "What happened first in the story? Last?" "Which little pig built a house of brick?"
When children are ready to read and write number symbols the teacher should have some "standards of expectancy." The correct method and techniques should be taught from the beginning. In learning to read the number symbols observing the picture, reading the number, then writing the number may be the correct procedure, as:
picture: 0; symbol: 1; word name: one.
Learning to write number symbols involves a correct perception of the number forms. The child needs to know where to begin in writing each symbol and the direction in which to go. Number symbols may be cut from wood for the children to hanule and manipulate. The "Emery Kinaesthetic Number Cards" may be made to use as an important teaching aid. The directions for teaching the correct formation of the nUhlber symbols can be found in the teacher's edition of many first grade state adopted text-
books. Ex~ple: Starting point:..&:; direction: -+

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l~ I

., ;r~-,
XI \
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/1
I

x

1/-' fl I~

- x' ) '4'1 I

-, )(1-_1-_

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'-"":! I

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I

II. Operations With Whole Numbers

Explore by experiences of putting things together and taking things away-Use the language of sets (symbolism: all oral expressionavoid formal drill.)

A. Addition and subtraction.

Encourage the use of vocabulary: More, more than, add; less than, take away.

1. Understand that addition means to put together.
2. Understand that subtraction means to take away.

Plan for children to engage in some of the following experiences:
Playing Housekeeping: Put more plates on the table; put more chairs in the room. Playing Store: Put more apples in the basket; Add more pennies in the cash register; We need more shelves for the cereal.
Playing games: adding one more and one more as in, "Farmer's in the Dell."
Participating in community fund raising campaigns: adding pennies to the Red Feather, Red Cross, and Polio Funds.
Observing combination of objects about the classroom: there are five chairs in the library center; 2 large, 1 medium and 2 small ones; 'add 2 chairs in our circle for visitors.
Listening for combination of numbers in poems and stories.
Using flannel board, arrange objects to tell a story as: three ducks were swimming and two more swam up. How many ducks were there in all?
Separating a larger group into a smaller group.
Observing a number of like objects in a group; close eyes while someone removes some objects. Estimate, ''How many are left?" Prove the answer by counting.

7

B. Multiplication and division.
1. See that groups can combine to form other groups.
2. Understand that multipication means putting together groups of equal size.
3. See that groups can be separated to form other groups.
4. Understand that division means separating a whole (total) into groups of equal size.

Develop understandings through first hand experiences in daily living.
Examples: Dividing cookies, candy bar, crayon, or fruit to share with another person or group.
Dividing sets of musical rhythm instruments, set of blocks, set of beads, and sets of flannel board objects.

III. Fundamental Operations With Fractions

Common (without symbols)
1. Understand that an object or a group can be divided into smaller parts.
2. Discover that two halves of an object make one whole object.
3. Recognize and understand that one of two equal parts of an object is one half of an object.

Develop understanding of mathematical terms, such as: divide, share, and equal by planning some of the following experiences:
Dividing fruit, cookies, cake, pie, work materials, etc., to share equally with another child or group.
Observing instances of sharing in first hand activities as: "My milk is half gone." "My glass is full." "This is a half circle."
Helping children discover in game playing and musical rhythm instances where smaller groups are made from larger groups, as in:
"Squirrels in Trees" "Swing, Swing" "Animal Trap"

IV. Measurement

Standard or English-Use the processes and become acquainted with the appropriate tools and vocabulary of measurement:

1. Linear-inch, foot, yard, ruler, Plan experiences for children to judge lengths, widths, and thicknesses with-

long lines, short lines.

out consciousness of measuring process.

Show yard stick and foot ruler. Teach their names.

Use the yardstick to measure paper for various activities; to measure space needed for different centers of interest; to measure materials or play house.

Plan for children to observe and answer questions, such as: "How many sticks long?" "How many sticks wide?" "Which one is longer?" "Which one is shorter?" "What size paper do you need for your Valentine or Christmas cards?"

Provide space with tape for children to mark their height-compare height with another child's height.

When taking a walk or going on a trip observe the meaning of "one block," "a short distance," or "a long way~"

8

2. Weight (avoirdupois)understand meaning of scales and pounds.

Help children discover and understand the meaning of "scales" and "pounds" by:
HavIug SCIIOOI nurse weigh and measure children. PlaYlllg StUre or supermarket to discover the meaning of pound of butter, pound of meat, pound of cheese, etc.
During development of activities throughout the day call the children's attention to objects that are heavy or light. Help them compare heavy and light objects; long and short objects; thick and thin objects.

3. Time a. General Understanding Time to begin Time to stop b. Clock Understand that time of day is indicated by the position of the hands of the clock. Recognize the position of the hands of the clock related to daily activities as: getting up time, bed time, lunch time, rest time. Recognize and become aware that the day may be divided into parts as: morning, noon, afternoon, night
c. Calendar-understand that the calendar is used to indicate days of the week and months of the year.

Have available in the classroom a large, functionary clock, and at least one toy clock which children can manipulate. The toy clock may be compared to the regular clock.
Give the children some concept of the passing of time by setting the classroom clock for juice time, lunch time, play time, and T. V. program time.
Provide a jig saw puzzle clock (preferably wooden) for children to put together in order to become familiar with clock symbols. They can observe the classroom clock when attempting to assemble the puzzle clock.
Use nursery rhymes, songs, and creative jingles about the clock, such as: "Hickory Dickory Dock" "A Dillar A Dollar" "Telling Time"
Example: "Our Clock" Tick, tock, tick, tock Come look at our clock. Is it time for play? Is a cookie on the way? Tick, tock, tick, tock. Happily sings our clock.
Dramatize home and school situations to help children understand the parts of a day, such as: "How do we greet family and friends in the morning?"
"What do we say to family and friends at bed time?"
"What part of the day do we usually have lunch?"
Make a large cardboard calendar for each month. As the teacher fills in the calendar day by day, the children have an opportunity to observe the passing of days, weeks, months, and the approach of birthdays and holidays.
Plan experience for children to make the following observations: "What is a calendar?" "What does it tell?" "Show a day-a week-a month." "Find your birthday." "What days do we come to school?" "What days do we go to Sunday School?" "Fino Christmas Day."

d. Scasor.s
4. Money a. Understand that U. S. coins have same shape but different names.

Make tcference to current season in conversation, stories, pictures, art activities, science activities.
Give children opportunity to identify, handle, and discuss real money by:
Counting lunch money. Playing store. Purchasing materials needed for party.

9

b. Understand that coins have different values; the size of the coin does not determine the value.
c. Understand that money is related to "things."

Counting money brought for different activities or functions at schOOl. Discussing the relative worth of the coins, as: Which will buy more, nickel and penny, nickel and dime? Make play money. Using play money to make change in "Play Store."

V. Geometry

A. Understand that objects of different shapes have different 'lames.
B. Recognize the forms of: line, circle, triangle, rectangle, and cube.

In reading or telling stories one should give attention to the shapes of objects mentioned, "The little round ball, or the square box."
Observations to make during the experiences of the day: "What shape of paper did you use?" "Joe played the triangle in the band today." "Make a big circle for our game today." "Place the chairs in a circle." "Our doll house is square." "Which block shape would be better to use next, a square block, or a rectangular block?"

Children may use modeling clay to see how many different shapes they can make and identify by name.

Children may play with boxes filled with cubes or empty ice trays filled with ice cubes.

VI. Charts

Make charts for daily attendance, heIght, weight, daily temperature, number of children eating in lunchroom, and weather changes.
Make charts of recipes for cooking by pasting the picture of the ingredient on paper and drawing the number of cups, teaspoons, or tablespoons by the picture.

VII. Problem Solving

Through oral discussion discover, Encourage child to "compose" number stories with colored beads on a string

become aware of, and solve sim- or colored pegs arranged in a peg board. For example, the child strings 2 red

ple problems as they arise in beads, 4 green, 3 blue, and 2 yellow. Then he tells a story about them to

daily activities.

other children.

Other suggested daily activities:

Furnish a playhouse.

Plan a party.

Plan a trip.

Buy toys at the play store.

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Plan a program. Construct objects with blocks observing that: more are added; some
are taken away.

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Dramatize stories.

i

10

j

FIRST GRADE

_ r ~'I l'

UNDERSTANDINGS TO BE DEVELOPED IN FIRST GRADE
The first grade pupil develops arithmetic concepts needed in his world rather than isolated mechanical skills. The understandings introduced in kindergarten are reviewed with emphasis on comprehending both the spoken word and the written symbol. First graders begin to understand the use of ten as the base of our number system and to master number groups through five without counting. They understand there are many ways of counting. Grouping of objects helps children understand addition as a process of combining groups and subtraction as a process of separating groups. This grouping builds a foundation for understanding the relation of whole numbers. Experiences are given to learn to use numbers in connection with units of measure, time, and money as the need arises in actual classroom activities. Careful study of needs and abilities of individual pupils is essential in the transition of skills from concrete and semi-concrete to abstract situations. Problem solving is extended to include written problems about classroom situations.
13

I. Numeration
Decimal notation 1. Understand place-value of ones, tens

Use place-value board when children begin counting past ten to show that eleven is 1 ten and lone.
Make place-value chart to show the relationship of tens and ones that make the teen family.

0... 0 00

0 0..o



0. 0 o



0

0

0 0 ,'... ..

0 0. ...

0

o.

0 0 .o..... ...

0 0 .. o

00 0"

0 0 ... ' .." III 0

THE TEEN NUMBERS
tens ones

OR 1

1 - 11 ELEVEN

OR 1

2 - 12 TWELVE

OR 1 OR 1 OR 1

3 -- 13 THIRTEEN 4 -- 14 FOURTEEN 5 -- 15 FIFTEEN

OR 1 OR 1

6 = 16 SIXTEEN 7 -- 17 SEVENTEEN

OR 1

8 = 18 EIGHTEEN

OR 1

9 -- 19 NINETEEN

2. Read and write two digit numerals

Make number line for counting to 100. Use strip of paper 2" wide by 200" long.
1 2 3 4 5 6 7 8 9 10 11 12

Make a hundred chart with the children. (Supply paper ruled into 100 squares 10 by 10)
Show the children correct formation of numbers. "Today we need 7 pencils. How do we make a good 7 so the store clerk will know how many pencils we need? Where do we begin to make a 7? (What is your house number? How do we write it?)"
Use kinesthetic number patterns to help children with reversals. Let children trace numbers in sand or on sandpaper to feel correct form of numbers.

3. Recogmze groups to ten.

Tell story of "Three Bears." Use cutouts of bears on flannelboard. Children see and feel three bears. How many ways can we arrange this set to make three?
(2 + 1 = 3; 1 + 2 = 3; 1 + 1 + 1 = 3)
Use blocks, beads, buttons, or counters from box to help children see and form sets (groups).
Show picture cards to help children see how different sets (groups) are arranged.

4. Discover sets (one-to-one correspondence)

Give one pencil to each child at your table. Show picture cards to help children see how different sets (groups) are formed. Make a tail on each kite. Match chairs with children or books with children.

5. Count forward and backward by ones, twos, fives, and tens

Count coat hooks. Count beads on a string. Arrange in groups of 2'8, then 5's, etc. Buy candles for birthday cake or take books from shelf. Count fingers on each hand in classroom by 5's. Count dimes in lunch money by 10's. Count 2's on number line. Count number of children present and number of children absent. Count "Little Cowboys" backward from 10 by 1's. Count "Little Cowboys" backward from 10 by 2's. Show number line. Ask what comes before 6 and after 6 or what number is two less than 17.

!S. Read and write number names through ten.

The teacher holds up 3 sticks. How many sticks do I have? Let some child

draw the sticks, another write the numeral and the teacher writes the number

name.

3

three

picture

symbol

word

Play "Postman."
Pictures of ten little houses are placed on the board. A figure [4J to reprepresent the house number is written above the door. A child is stationed at each house. Several children are given cards bearing number names [fourJ from one to ten. They are postmen and must deliver their mail to the right house. The child at the house checks the mail he receives.

Use charts with pictures, words, and symbols to ten:

t',"o

/--/

/--/

2

/-/
//

six //

//

-/ 6 -/ - -/- -/

7. Recognize ordinal names through tenth to clarify ideas of order and position.

Take turns in games. (Who will be first, second, third?) Plan the order of steps in an experiment. (What is the first thing to do? second?) Point out on the calendar the first day of the week, or second day. Make construction paper train. Cars have ordinal numbers to tenth on them.
15

8. Distinguish between even and odd numbers.

Count children in rows and designate their position. (Who is the fourth person in a row?)
Choose partners for trees to play "Squirrels in the Tree." After partners are chosen the ones left will be squirrels.
Count by twos to 100, beginning with an odd number.
Make the odd numbers black and the even numbers red on the hundred chart.
Let child stack sheets of construction paper for booklets. Stack by twos crisscross so each child can take two sheets quickly. There is one sheet left over. It is an odd sheet. Put it back in the drawer.

II. Fundamental Operations With Whole Numbers

A. Addition and subtraction (with symbols)
1. Combine groups through ten
2. Separate groups through ten

Help each child make an envelope of construction paper containing cards with numbers 1 to 100, ten cards with pictures of groups 1 to 10, and ten cards with number names. Each set is held by rubber band. For number recognition the children hold up the number called by another child or teacher. Children can arrange their cards to show serial order. For addition drills the children hold up card to show the answer.
Play "Number Ladder." Paint a large ladder on the floor of classroom with tempera paint (quickly wiped away with damp cloth). Can use masking tape for making the ladder on the floor. The ladder has ten steps, and each step is numbered. Children can
add or subtract by taking steps on the ladder. (5 steps + 2 steps = 7 steps).
Child can take 5 steps and then 2 more steps would make his answer 7. To subtract the child would back down the ladder a number of steps.
10
9
8
7
6
5
4
3
2 1

Use pictures where an animal or child is leaving or joining the group. Encourage each child to keep a box in his desk for hunkie sticks, paper strips, tooth picks, jelly cups, pop beads, etc. They may combine or separate group objects on their desks. Play "What's Your Score?" Each child uses a bottle cap. One at a time they throw the bottle cap at the target. They tell the number you would add to this number to make nine. Make a fine manipulative device for separating or combining groups by stringing colored spools on coat hanger wire.
16

Play "Fish Pond"
Prepare number fact cards (6" long) in a fish shape and attach a paper clip to the mouth of each fish. Place fish in a blue box with a 3 foot long wood pole to which has been attached a short string and magnet. Pupil fishes and if he knows the correct answer to number facts he keeps the fish. Rules can be varied according to interest and needs of pupils.
= Let children color six disks on manila paper (3" x 5''). Using scissors they
may separate (6 - 4 2)









3. Understand commutative law and that subtraction is the inverse to addition

Make "Number Fans"
Fasten ten strips of cardboard about I" x 8" with a paper clip at one End so that strips can be spread as a fan. Pupils use number fans to separate groups to ten.
Change the order of adding the objects as:

Example A
o 0 0 0

Example B

3"

3 4

o 0 0 .

t. Understand the use of more,
more than, less, less than, take away, and the symbols
+,-,=.

Let pupils experience the commutative law as they group and regroup objects (blocks, pop beads, hunkie sticks) from their box. Be sure these facts are written on chalkboard so pupils can see the reversed order.
Use picture cards to help children see that the order of addends may bf' changed without changing the sum.
Use number fact cards to show children that subtraction is the inverse to addition.
Use pegboard, chalkboard, and flannelboard to show children that these are the signals for addition: plus, sum, total, altogether, both, how many in all. The following are the signals for sUbtraction: minus, take away, left, more than, less than, difference.
Use the above terms in helping children solve their very own everyday prob. lems such as: Jack made 6 balloons on the board. Carl made 4 balloons. Who has more balloons? Who has less?
Tell "Bluebird Story."
"Ten little bluebirds swinging on the line, One flew away, and then there were nine. Nine little bluebirds that were always late, One hurried off, and then there were eight. Eight little bluebirds, blue as the heaven, One went bug hunting; now there are seven. Seven little bluebirds, doing birdie tricks. One tumbled down and left only six. Six little bluebirds, please don't go away, Stay on the line, and let us watch you play."
Teacher removes from flannelboard one bird at a time as the children say the rhyme. Children may create their own take-away story for the six remaining bluebirds.

17

Dramatize "Squirrels and Acorns."
Six little acorns (six children) are under a tree. Two little squirrels (two children) are storing acorns for winter. As they take away acorns they count the number left.
Use similar procedure for "Owls in a Tree."

5. Check addition by adding in opposite direction. Estimate sums and remainders.

Encourage children to estimate the answer to everyday problems: T'1e ham-

ster ate 4 peanuts. Sue gave him 3 more. How many peanuts did he eat?
= I n After estimating, write this mathematical sentence on the board (4 + 3

to build understanding for equal. Let children check problem by adding in

= / opposite direction. (3 + 4

I).

Estimate the answer: Nell and Tim have 10 marbles together. Tim had 6 marbles. How many did Nell have?
(6 + / / = 10)

6. Learn two concepts of subtraction: a. To find how many are left b. To find how many are needed.
7. Use number line for adding and subtracting.

Play. number stories with pegs on pegboard to explore the two concepts of subtraction. (to find how much is left and how much more is needed)
Play "Follow Me." On a large number line across the chalkboard one child says "follow me" and tells his number story. Six boats take away 2 boats leaves 4 boats.

8. Read and write addition and subtraction facts in both column and equation form (mathematical sentences).

Teach equation form (left to right) first. Children enjoy mathematical sentences. They are sometime called arithmetical sentences.

5+4=/ /

8-/ /=5

4+//=9

//-2=1

Explore column addition of three one place numbers. It can be challenging to first graders.

B. Multiplication and division (without symbols)

Let children count by 2's using pegboards, hundred board, or partners.

1. Understand that multipli-

cation means putting together Use felt cut-outs to represent groups of two and three, and let children respond

groups of equal size.

orally to:

2. Use counting on by twos and threes to form other groups.

5 twos are / / 3 twos are / /

4 twos are / / 3 twos are / /

Have children cut pairs of paper dolls, mittens, hats, etc. They begin a chart with a pair of ('ut-outs. Place another pair beneath them and add two more, bringing the total up to four. In another row they put down two pairs and add two more counters, bringing the total up to six, etc.

3. Experience regrouping.

Explore regrouping with ten objects. Let children regroup to discover 2's and 5's give even groups and 3's give an odd group.

4. Understand that division means separating a whole group.

Separate a hundred chart into 10's by cutting with scissors.

18

-

III. Fundamental Operations With Fractions

A. Common fractions
1. Understand and write one half of a whole object.

Cut an apple into two equal parts.
Fold sheet of paper through the middle to make booklet.
Divide children into two equal groups, one half to paint and one half to see filmstrip.

2. Understand and writ~ one fourth of a whole object.

Cut each of 9 bars of candy into fourths so there will be 36 pieces of candy (one for each of 36 children).

Fold art paper into fourths for making pictures of four groups.

= Discuss fractions of the dollar. Use lunch money to indicate lh
lf4 25.,..

504- and

B. Decimals
1. Recognize decimal point in money

Record lunch money by writing cents on right side of decimal point and dollars on left side of decimal point.

IV. Relations
Understand equivalence and inequalities.
V. Measurement
Standard 1. Linear Use inch, foot, yard
2. Liquid Discover cup, pint, quart
3. Weight Use pound
.. Time a. Understand hour and halfhour

Understanding of equivalence has been built into the entire program with special attention given to mathematical sentences.
Play "Pin the Tail on Donkey" and "Pin Nose on Jack-O-Lantern:' Use measurement to determine which tail is pinned nearest; which nose? Provide ruler, yardstick, tape measure, so children may find answers to such questions as:
How long is this pencil? How long is our ITain? How tall am I?
Collect lh pint, pint, quart containers and measuring cup to dis~over: How much milk do we have for lunch? How much water will fill our aquarium?
Play "Can You Guess?" Child holds paper bag in each hand. One bag contains a pound of sugar. Compare weight of a book, piece of wood, or brick to the pound of sugar. Let children guess which will weight more; then place article in other bag. Answer this question: Does it weigh more than a pound or less than a pound? Weigh children.
Let children shut their eyes and estimate the length of a minute. When they think the time is up they raise their hand. Then all watch the minute hand on the clock to determine the length of a minute. Make a time chart:

19

7 o'clock 8 o'clock 9

A DAY WITH THE CLOCK Time to get up Time to eat breakfast

I 10 11
-----
12 1 2 3 4 5 6

-----

Make sure children understand that the long hand tells the minute and the short hand on the clock the hours. Move the long hand on the clock face while the children count by 5's.
Call attention to the number of hours in a day. Point out that there are 12 hours on the clock and that the hour hand has to make 2 trips around. Explain that when we go to bed the day is only half over. While we sleep the clock works. Twelve o'clock at night is midnight.
Race with the clock to clean up in 15 minutes.

b. Name, number, and read days of week, month of year, and seasons.

Keep a record of the days it takes seed to sprout. Learn the number of days in a week.
Observe the number of weeks in a month and how many days over. Record the number of days it took eggs to hatch.
Create interest and help to clarify the calendar by saying this poem after the children have learned the names of the months:
"Thirty days hath September, April, June, and November. All the rest have thirty-one Except February, which alone Has twenty-eight, which we assign Till Leap Year brings it twenty-nine."
Make charts of the four seasons.

5. Money
a. Recognize all coins.
b. Understand relative value of one cent, five cents, ten cents, and twenty-five cents.

Use real coins as children identify penny, nickel, dime. Select the penny. Let the children see that its name one cent is found on the coin. What is another name for a nickel? Why is a dime sometimes called a ten cent piece?
Make a collection of five cent articles. Let children make a toy store. Help children discover coin equivalents of nickel and dime by making a chart of coins mounted with scotch tape. Ask children: "Could you bring 5 nickels for lunch money instead of a quarter?" "What other coins would make a quarter?" Have a penny taped to the door one day. Children become curious. Next daY
a second penny joins the first. Third day three and so on until there are teD

20

6. Quantity Understand pair, dozen, half-dozen, tablespoon.
7. Temperature
Explore thermometer

pennies. Children count pennies, and teacher replaces penmes with a dime. The dime will buy a new toy for their class. This experiment will arouse children's curiosity about money.
Let each child have the opportunity to be clerk for supplies from the store.
Encourage children to bring egg cartons, and discuss how many eggs these will hold. Let children cut these cartons into 2 dozens and pairs.
Make use of room thermometer. Place it so that the children can see it at eye level.
Help children to make a large play thermometer with a movable red and white tape. After the room temperature is read, pupils may adjust the ribbon on the large thermometer to show current temperature.
Keep aquarium at correct temperatures.

VI. Geometry
Explore the following geometric shapes: line, circle, square, triangle, rectangle, cube.

Draw a circle on chalk board. What things do we know shaped like a circle? (ring, wheel, or penny) Follow this procedure for square, triangle, rectangle, and cube.
Play "A Strange Cat." Draw cat on chalkboard. How many triangles do you see?

VII. Charts and Graphs

A. Develop the idea that a chart is a means of organizing and recording information and presenting ideas.

Make individual arithmetic skills chart:

a

[J
c

b.O
...s8.:.:.

::3

u0

c;~ saa:: 0'"'

I KNOW MY ARITHMETIC

e~ ~ CI) "'"0' <'t:l

~i

18J....... i.t ell -'ts:::l c.> uClI

'ts:::l
ell
.~ S ~s::8
~ C'' CS_ ~ M .
CI) !-'o
P:: :;0 .....

(i)
@ :>0s:::;
::0a

"....6..
;>,
""....c0..:.
i:'Q"'

...--- ...---

...---

21

a. Graphs tell a number story.

WE GROW TALL

SUSAN

BILL

FAYE

JIM

VIII. Problem Solving
A. Through oral expression formu- Charles has 9 airplanes. Sue has 3 airplanes. How many more does Charles late creative story problems. have? (Count on number line from 3 to 9, or group one to one to see how many remains.)

1 2 3 4 5 6 7 8 9 10 11 12 13

I

I

B. Solve oral and later written problems involving the mathematical operations taught at this level

The clown is 3 feet tall. Will he fit on the bulletin board that is 5 feet tall? How much space will be left for his balloons?
Children should not be rushed into reading a problem and giving an immedi ate answer. They should be given time to think through and analyze the problem thoroughly. Some ways to help the child analyze and understand the problem are dramatizations, use of representative materials, and the number line.

22

SECOND GRADE

UNDERSTANDINGS TO BE DEVELOPED IN SECOND GRADE
The second grade pupil extends his understanding of our number system to counting to 200 by 1's, 2's, 5's, and 10's. He begins to develop a distinction between the odd and even numbers; to use ordinals through 15; and read an" write Roman numerals through XII.
Building on the foundation laid in grade one, the pupil begins systematil' work on the addition and subtraction facts. He extends his understanding of place value through hundreds, also addition and subtraction of thrdigit numbers without regrouping.
The second grade pupil uses the fractions 1/2, 1/8, 1/4 as they occur in claMroom activities. Simple charts and graphs are developed around classroom situations; He recognizes geometric concepts of figures of familiar classroom situations.
Learning activities are arranged in many ways so that the pupils have frequent opportunities to use instruments of measurement in a natural setting and make reasonable estimations.
The pupil uses situations that may occur in the classroom, home, and community as a basis for developing problem solving skills involving addition and subtraction through oral and written mathematical sentences. The idea of inequalities is introduced ~ simple sentences.

CONTENT
I. Numeration
A. Decimal notation
1. Review and extend experiences by counting: to 200 by 2's, 10's, 5's; to 30 by 3's, to 20 by 4's.

Check attendance; count boys present; count number of papers, number of books, number of pencils, etc., that need to be passed out; take a walk and consider number symbols seen and number of objects seen; count by couples or partners; count pairs of shoes, gloves, and skates.

Example:

ATTENDANCE CHART

Boys Girls Totals

I Tens 1 1 2

Ones
7 2 9

Boys

17

Girls

12

29

The 100 Chart is another useful device in the development of the decimal concept. This consists of a ten-by-ten array numbered as the following figure. This chart may be more useful when made of plywood. Small finishing nails may be driven into the wood. Then removable numbers can be hung on the nails. The child should think of a full line as a single ten and also as ten ones. This chart will be useful in teaching the number names by tens to one hundred.

1 2 3 4 5 6 7 8 9 ill
I---l---Ii---I---I--- ---1---1----+---1---1
11 12 13 14 15 16 17 18 19 20
n 1 - - - 1 - - - 1 - - . , - - - 1 - - - - - t - - - { - - - - j - - - I - - -
22 23. 24 25 26 27 ~ 29 30
I-----------~--r__--
31 32 33 34 35 36 37 38 39 g
1-----1-
-41- -42- 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
------------I--------I---j----I
61 62 63 64 65 66 67 68 69 70
~: I~: : : ++; ~: ~: :

m n ~ ~ M ~ 00

00 ~ ~

Have children count backwards from 30 to 10. First have the class try it silently; then the boys may count it aloud; later another group may count.
The counting frame is an excellent device for teaching the decimal aspect of the number system. The frame consists of ten lines, ten counters to the line.
Best results are pbtained if the room is equipped with a large demonstration

26

frame, large enough to be seen all over the room, and each child has his individual smaller frame.

txD

Y

T TT Y

I
TTl I I

T TT T

l' J

YT "'(TT T I

. T 1111'
TT TTTI I

TT T J I I
I I I ITT
rII I I

By counting by 5's or 10's find out how many paper clips are in a box. The teacher may count to 120, and then ask the class to begin to count from 120 through 300.

J. Extend reading, writing, and meaning of number values through 200.
a. Read and write three digit
numerals. "Learn ordinal numbers
through 15 (Use others beJoDd if needed.)

Write house numbers and page numbers in a big book; number pictures of houses between 75 and 199; read even numbers in books and magazines; read and write telephone numbers of the boys and girls in the second grade; read and write numbers of automobile tags.
Use intensive work designed to improve the quality of numeral writing. The following method may be used to provide the setting. Let children look at their writing of numerals that has been done previously. Point out the importance of legibility when writing numerals. Discuss ways of improving the writing. Then direct class to write numerals 1 through 10 on a piece of paper, and remind them to make the numerals large. To get started, the teacher may write the numerals on the board while the pupils write on the paper. When the papers have been checked, have the pupils concentrate on writing 3's and 4's. Then tell them to write from 33 to 44 on several lines. At the end of the period each child may make a writing sample of these numbers. Give a limited time to do the sample, in order to improve the speed as well as the shape of the numerals.
Make a chart of the numbers from 121 through 221. Use Place-Value Chart or pocket to show 3 place numbers. To illustrate the number 562, place 5 bundles of hundreds in the hundreds pocket, 6 bundles of tens in the tens pocket, and 2 bundles of ones in the ones pocket.
Billy is the tenth boy to come into the room. Identify the day of the week (e.g. 3rd), and the month of the year from the calendar such as May 3, 1961. Make it clear that all ordinals do not end in "th."

27

S.Readand write number word8 through twenty.
8. Extend llkfll in distinguishing between odd and even nllJDbers.
7. RecognIZe that each even number can be separated into two equal groups. An even number is one that is evenly .divisible by two. Recognize that there Ja always one left over when we try to separate each ode! number into two equal lI'Oupa.

Bead and write stories that use number words. For example:
Millions of Cah, Chicken Littl. Count to Ten, Hundreds and Hundreds of Pancakes, The Five Chinese Brothers, My Shoe. Come in Twos, Ten Big Farms, The Four Rider., Over in the Meadow.
Show relationship of a number word to its use in the number system, u foul"teen equals one ten and four ones.
Divide into two teams, odd and even. Read the even numbers in a story book. Count to thirty-three and let the children write the odd numbers on a piece of paper.
The numbers from 9 through 33 may be written on the chalkboard, and children may take turns drawing circles around the even numbers.
Nine boys want to play a game. They may work together trying to divide into equal groups. If they select partners it becomes evident that one boy will be without a partner. Therefore it becomes clear that 9 is an odd number.

8. Repeat and extend the vocabulary and grasp concepts studied in the first grade.
a. Demonstrate objects to be used in the set.
b. Begin to use symbols in an Informal manner.
e. Become aware of sets by
grouping physical objects together.

Provide practice circles that draw attention to adding and subtracting with an odd number.
Draw the wheel on the chalkboard or on a chart. Children add the center number to each of the numbers in the second circle. Answers go in the third circle. Ask questions to stimulate the thinking of children. For instance: "What do you notice about the numbers?" (All are odd numbers.)
"What 'happens when you add 2 to each number in the second circle?" (You get the next odd number. The answer is two bigger. The answer is an odd number.)
The teacher can begin the discussion of the topic of sets by displaying a set of three or four objects on her desk. For instance, a set of dishes, a collection of stamps, a set of circles, a set of triangles, a set of books, or a set of toy animals may be used.
The children may select a set of letters to place on the flannelboard. Then the class may talk ,about other sets; namely, sets of vowels, sets of names beginning with R, sets of words ending with sh.
At another time the class may relate the various groups to which the members of their family belong; to illustrate: Cub Scouts, Church Choir, P. T. A., Rotary Club.
Review and extend set vocabulary by making sentences using worda of set language; namely, one-to-one correspondence, even, odd, group, member, equal, less than, greater than, likeness, difference, collection, and position.
Pair numbers of one set of objects with another set; for example, desk for every child, pencil for each tablet, set of girls in second grade, first twelve counting numbers which mark the hour on the clock, set of six-year-olds in the first grade with set of seven-year-olds in the second grade, the five liqers on one hand with the five fingers on the second hand.

28

9. Strengthen understanding of place value concept and
the ten-ness of our number system.

Any two sets that may De placed in one-to-one correspondence are said to be equivalent to each other.

Place Value: 11 is ten and 1 more Example:

, \
IIIIIIIIII I
r------.,
IIIIIIIIII II
, \
IIII1 I1111 1I1II

Think

Write

II
ten and 1 more or

tens

ones

1

1

I ten and 2 more or tens ones

1

2

Read 11 eleven
12 twelve

I ten and 5 more or tens 1

ones 5

115 fifteen

2!) is 2 tens and 5 ones

25 is 1 ten and 15 ones

25 is

25 ones

10. Strengthen understanding and meaning of zero.
B. Roman numerals Read and write Roman
numerals through :xu

Give pupils opportunity through keeping score to use zero as a number, and
help them discover and generalize that this addend has not changed the sum. Have pupils write such numbers as 103, 120, 200, to illustrate other uaes
of zero.
7 + 0 = 7; 13 + 0 = 13
The sum of zero and any other number is that number. U this generalization can be taught, then it will not be necessary for the child to memorize the zero facts.
Make Roman numerals on a cardboard clock. Write children's ages using Roman numerals.

II. Fundamental Operations With Whole Numbers

A. Addition and subtraction 1. Review and reinforce the work of addition as outlined in the first grade, empha. sizing:
a. Adding is counting on or moving right on a number line.
b. Subtracting is the inverse or undoing of addIng or moving left on the
.1IDe.

Emphasize the meaning of addition and subtraction by using the followiq:

Buy two articles at toy store for 5 cents or less; buy two articles at toJ stores marked 3 cents or less.

Extend pupil's knowledge by showing on a number line:

Adding is counting on or moving right on the number line.

,

\

1 2 3 4 56 7 8 9 10 11 12 13 14 15 16

3

4

'----.....\..'- - - '

SUbtracting is counting oH or moving to the left on the number line.

i

\

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

7-4=3
Ii 2345678910'
The number runner can be used with success to develop meaniDgfal arithmetic concepts. It can be extended along one or more walla to go u far u possible ueyond 100. When the children count by 1'1 to 100, by touchinl the

29

1, '1 I,I'

JI

i

stars and recogmzmg their number names above, they are ready to count across the runner by 2's, 3's, and 4's. Have pupils count to the left or backward to subtract.

2. Introduce the concept of mathematical sentences.
a. Addition is combining like groups and subtraction is the inverse of addition

Combine like groups in' equation form
5 + 0=8
0+ 3 =8 8 -0=2

0+6=15 A+O=15 15-0=A

A group of 0 rJ 0 0 0 and a group of 0 0 when put together equal
a group of 000000 O.
5+2=7 2 and 5 equal 7

b. Work toward the mastery of easy basic facts under the operations of addition and subtraction.

The addition and subtraction facts of 8 are introduced when the class has gained thorough understanding of the facts of 7. Other combinations follow in order until material as outlined for the year is covered. By the end of the second year the children should have mastered the combinations through 10. Combinations above 10 may be introduced as the ability of the class may determine. Some ideas used to introduce basic facts are chairs at a table, girls on a slide, boys in the game, birds in the tree, boys in the class, and books on the shelf.

c. Recognize hard basic facts.

9 chairs + 2 chairs = 11 chairs 2 chairs + 9 chairs = 11 chairs 11 chairs - 9 chairs = 2 chairs 11 chairs - 2 chairs = 9 chairs

Use number line to show that answer is correct.

A game exercise that lets children see how fast they can add is listed here

To play this game, let one child represent the East and another, the Wesl They may stand in the front of the spaces at the board on which have been written the numbers 43Hi and 5134. When one child at the seat says .. number, the two children at the board turn to face the numbers on th. board. They add the number called to each digit of their number at th. board. The first one who turns to face the class and has the correct sum ir
the winner.

Example: Add 3 to the digits of the numbers.

West East
I 4315 I 5134 I 7648 8476

Teacher may record some addition questions on tape. After each question she may hesitate a few moments and then give the answer. The pupils try to write the correct answer before it is given on the tape.

Introduce hard basic facts with oral problems. For example: The children played a ball game. They made h'l'o teams. There were 7 boys on each team. To write the addition fact for 7 + 7, one might think of it as ten plus four.
... ... ... . + . . . .
1 2 3 4 5 6 7 8 9 10 11 12 13 14
14
IT~n, IO~" I

1 2 3 4 5 6 7 8 9 10 11 12 13 14

'--

.-J"'-

---J

7

7

30

Many actIVItIeS may be used to find the answers to hard baSIc !acts. l'UpU8 may use number line, dots, objects, an abacus, and a place-value pocket.

d. Recognize grouping, regrouping and related facts.

A picture of an abacus showing 215 is pictured here.

hundreds
Ir---&

Itens

ones
~I

= 215 2 hundreds, 1 ten, 5 ones
The principle of place value is important, and children should be given many opportunities to use it. Place value charts to demonstrate the groups of tens and ones in a number should be used.

Show that 12c may have 2 groupings.

Dimes

129'

1

12

Pennies 2 12

Show 2 different groupings of 15.

Tens

I15

1

15

Ones
I5 15

Show 2 diferent groupings of 34.

34 34

H= Tens

3

2

14

Regroup a number into the simplest form.

Problem

Hundreds I Tens
1 I6
Answer
Hnudreds I Tens
1 I7

I Ones
I 15
I Ones
I5

In the second grade the pupils need to see small numbers of things at once by groups. The groups may be two, three, four, five, six, seven, eight, nine, or ten. Experiences may include the following:
Checking number of cars on a toy train Checking books on a shelf Checking eyes, ears, arms, legs Checking pennies for ice cream
Exercises in counting by groups may help children see things as a group; for instance:
Immediate recognition of five pennies Rapid counting of beads on a frame Grouping for using see-saw, slide, swings, or other playground equipment Identification of number of fish in a bowl

81

8. Extend addition of one, two, and three digit numl.lera.

Examples of regrouping:

TI +22
TI +22 ~

2 tens 2 tens
4 tens

7 ones 2 ones
9 ones

4 tens and 9 ones = 49

Related facts in numbers make learning easier. Typical examples are illustrated below:

Dick said, "Number families help me learn number facts. If 1 know
that 5 + 2 = 7, 1 know that 2 + 5 = 7, 7 - 2 = 5, and 7 - 6 = 2, because all four facts in the number family use the same number.. That is why they are called a number family."

= John
easy.

said, "I You just

use add

doubles to help two numbers that

me are

ltehaernsamoteh,elrikfeac3ts+.

Doubles are 3 6. When

you subtract, the answer ia the same as the number you took away,

like 6-3 =3."

A double has only two facts in its family, and each fact has two numbera that are the same.

Bill made up a problem with near doubles. Three apples and two apples are how many apples? Tom knew that 2 and 2 were 4, and 1 more would be 6.

In making the transition to addition, help the child relate the process to
counting. Counting becomes a succession of steps of adding one. The sign +
must be associated with the process of combining groups. The following
examples are illustrative of the combining process.

4+1=5
5+1= 6

Addition of two and three digit numbers ia developed through foundatIon number experiences such as the following:
Use place-value frame to teach addition of two digit numbers. Diagrams A and B indicate how two numbers may be entered on the pocket-type frame.

21 +57

A.

Ten One

2

1

B.

Ten One

5

7

2 tens and 1 one 5 tens and 7 ones 7 tens and 8 ones, or 78
Some children may use blocks to count by tens and then count the totals. Another child may use the number line.
Continue to use oral problems, and ask pupils to find way8 to check the
ailBwera.

32

Mrs. Brown has 453 white ducks and 323 black ducks. How many has she in all?
Use dots to show each number on place-value frame.

hundreds
... 7

tens
0.. 7

ones
0... 6

Use another way to find the answer.
4 hundreds 5 tens 3 ones 3 hundreds 2 tens 3 ones 7 hundreds 7 tens 6 ones

4. Extend subtraction of one, two, and three digit numbers without regrouping.

Help pupils learn the different ways to subtract. For example:

12345 6 ****xx

6 -2 '-4-

123456

Cover 3 blocks; Cover 4 blocks
0000000
= 7 blocks - 3 blocks 4 blocks
7 blocks - 4 blocks = 3 blocks

We had 25 sheets of paper and used 8 of them. Count backward to see how many sheets of paper we have left. 25 is 1, 24 is 2, 23 is 3, etc. Tom may write the numbers on the board as each statement is given.

The grocer sold 567 pounds of sugar. He had 898 pounds before he sold any. How much sugar did he have left?

898 -567

8 hundreds 9 tens 8 ones 5 hundreds 6 tens 7 ones
3 hundreds 3 tens lone

To subtract ones, use subtraction fact: 8-7=1
To subtract tens, use subtraction fact: 9-6=3

To subtract hundreds, use subtraction fact:
8-5=3 898
-567
----m-

33

I' ...I

5. Develop the concepts of the Commutative and Associative Laws of Addition.
a. A sum is unaffected by the order of the addends.
b. A sum is unaffected by the grouping of the addends.

Children may gain understanding of the laws of addition oy usmg counten to show amounts in different arrangements. For.' instance:

o 0 000

000 o 0 o 0 000

or

000
o0
000

o0
000
o0

The following arrangement shows the Commutative Law of Addition.

7+8=8+7
Tell children that the sum is the same with addends in any order, if addends are combined in different ways.

Ii il (2+3)+5=2+(3 + 5)
This is an illustration of the Associative Law of Addition.

6. Discover the three major types of subtraction. a. How many are left?
b. Compare two amounts.
c. How many more are needed.

6-4=2 0000 00

7-3=4

0000000
J,H
000

8-0=5

r----.
00000000

Use concrete objects such as red and black checkers to discover the above processes of subtraction by using the following type of questions:
"How many are left?" ;'How many more are needed?" "What is the difference?"

7. Introduce equation form for addition and subtraction facts.

Use oral.problems to introduce the sign (+). Explain that the sign means the same as the word plus. Have children illustrate the action (combining 4 objects with 4 to make a group of 8). Pupils may come to the front of the room in groups of four to show the sum of 8. Teacher may write the following equation on the board:
4+4=8
Ask one child to read the problem and show the action with markers or other objects.

8. Comprehend simple column of three addends; sums of 18 or less for most.

Some addition problem situations arising in classrooms are as follows:
"A Game of Ninepins"
Write the names of the four players of ninepin" on the chalkboard so that all members of the class may keep the score. Before the game begins set up rules for playing. For example, the first person who gets a score of 12, wins. The rest of the class may take turns playing. Add the pins knocked down after three tries.

Bill Sue Jack Dick

2

2

5

3

4

5

1

1

1

3

2

4

'7 10 -8- S

34

B. Multiplication and division

It should be stressed that, when adding three or more numbers, only two numbers can be added at the time.

Example:

Method A

Bill's score 2 4 1
Method B

Think

or

2 4
6and 6 1
''[

Think
4 1 Sand 5
2
'1

2

Think

4

2 + 4 = 6, then 6 + 1 = 7 or

1

1 + 4 = 5, then 5 + 2 = 7

Stress the idea that (2 + 4) + 1 or (1 + 4) + 2 results in the same anawer.
As the child becomes proficient he is able to do this equation form mentally.

Pass out counters. Show two and two and two on the desk. Find out how many three 2's are.

1. Discover by manipulati~n and from pictures the products of 2 ones up to and including 6 ones.
2. Add twos and doubles.

Use objects to illustrate:

0 0 Q9 -......-..-

0--....0---

2 + 2 +2

2 2
62
'6

3. Develop concept of combining equal groups, like two 4's, two 5's, two 3's, two 6's, etc.

Two 3'. are 6 Three 2's are 6
2X3=6 3x2=6 Use games and objects to discover answers to doubles. For example: Draw objects:

DODD DODD

4+4 2 fours = 8

\
1 2 3 4 5 6 7 8 9 10 11 12

::... Give understanding of separating two's and doubles.

Mark two 4's on the number line. Help children discover that multiplimt10D is a quick way of adding the same number several times.
Direct class to put six pennies on the desk and diVide the coins into two groups. There will be three pennies in each group.
000 000
lq lq lq I, lq If:

5. Strengthen understanding of grouping to find two equal parts of 6, 8, 10, 12.

Let children discover that one divides to find the number of equal parts and to find the size of one of the equal parts. Help pupils to see that division is a quick way of subtracting the same number several times.

4

-2

1

-2-

2 2/4

-2

2

-0

Two 2's are 4.

To find out how many 2's are in 6, keep taking 2 from 6 until none remain, and then count to find how many groups are used.

6 -2 1 -4-
-2 2 -2-
-2 3 -0-

3 2/6
Three 2's are 6.

Lead children to discover how many 2's in 4, 6, 8, 10, or 12.
123 4 56
~'---V-"_,~~~-.---~
II III II IIIII IIIII
1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16

6. Build the following concepts: combining equal groups (multiplication); separating to have equal groups (division).

Give readiness experiences for division, as: dealing cards in equal groups for class game, dividing eight sheets of paper equally among sixteen boys, sharing ten bars of candy with five boys.

7. The commutative property for operations of multiplication means that numbers may be multiplied without regard to the order of the factors.

This is an illustration of Commutative Law of Multiplication:
5X2=2X5
Five twos is another way of writing two fives.
After children have worked with oral problems of multiplication, they should realize that in multiplication, as in addition, the order of the terms does not affect the answer.

III. Fundamental Operations With Fractions

A. Common
1. Re-teaching and extending the meaning of one half of a single object; the terms half and one half.
2. Help children to associate the idea, the name, and the symbol.

Use many kinds of diagrams to show that J,2 of an Qbject is one of two equal parts; also 4 of an object is one of its four equal parts:

Teach that J,2 and 4 are the symbols used for the name of a certain part of an object or a group.
36

(oooo)f-
0000

Help children to become familiar with one half of a group, one fourth of an object, and one third of an object.

The Idea

The Name The Symbol

onethird

one-half

1/2

one-fourth 1/4

Use measuring parts.

cup

to

show

1/2,

1/4,

1/3.

Break

i'lndy

bar

into

two

equal

lIltroduce the decimal point with money expressions.

Une dollar, or lOOe, is usually written $1.00. The sign $ is the dollar sign. The dot is always written between dollars and cents.
Teach children what the decimal does: "It keeps the dollars on the left side and the cents on the right side."
LR
$1.29 Emphasize the use of the word and in reading as;
One dollar and forty cents Say and to separate the whole dollar and part of a dollar.

37

IV. Relations

A. Equivalence
Guide child to have better understanding of equivalent values of money and measurement.

Reinforce the idea that mathematics is primarily a study of relationships.
Help children apply their knowledge of the principles of base ten numeration to understand a collection of coins. Count nickels to find that two nickels are equivalent to a dime.
Extend pupils' experiences with measurement.
Extend pupils' experiences with money.
3 + 4, 5 + 2, 1 + 6, 8 - 1, 9 - 2, and 6 + 1 are six ways of saying the same
numerical quantity.
Develop the pupils' ability to recognize the equality of the number in two groups: Mrs. Brown's chickens and Mrs. Smith's chickens; the same number of chairs and children; the marbles in this group and the marbles in that group.

B. Inequality
Extend the concept of order between numbers.

Provide opportunities for children to learn that 5 is less than 6, 4 is less than 5, and 2 is less than 3 by showing on a number line. Adding is moving right on the number line.
Show that 6 is greater than 3
o 1 2 3 4 5 6 7 8 9 10

Subtracting is moving left on the number line.
o 1 2 3 4 5 6 7 8 9 10

C. Order of decimal
Develop the understanding of grouping and regrouping.

2 is less than 3
Help children discover that in our number system each figure in a number has value according to the place it holds in the number.

Hundreds

Tens

Ones

Place-value chart showing 235
I $~ 7 -----

dollar

dimes

.....-
pennies

Place-value box showing $1.64
Use a place-value chart in which ones, tens (bundles of 10 ones), and hundreds (bundles of 10 tens) are placed in labeled pockets on a place-value chart. Have the pupils use a bundle of one hundred counting sticks and bundles of ten sticks to build a three-place number having a zero in onel place; for example, 180. They will represent this number with a bundle of one hundred, eight bundles of ten, and no extra counting sticks.

38

D. Ratio
Provide readiness for concepts of ratio by:
a. Identifying equal num. bel's of similar objects.
b. Distributing equal numbers of objects to equal groups.
E. Proportion
Extend understanding of ability to use correctly 1/2 1/3, 1/4 as they occur in classroom activities.
F. Percentage

Correspondence can be set between objects that are quite different. Thus suppose there are only 3 toy airplanes for a group of 6 boys. Divide the airplanes into groups of two and show that two airplanes will go with, or correspond to, each 3 boys.
Review the study of fractions. Have children draw pictures of three pies and divide the pies into the following proportions: halves, fourths, and thL."dI. Help children cut or fold paper to show the relationship of fourthl, halTeI, and whole, and of thirds and whole.
Plan readiness experiences, such as checking attendance and recognizing when 100% are present. Learn that if the class is separated into two equal groups for a game, each group will represent 50% of the class.

v. Measurement
Standard
1. Linear. Demonstrate and give understanding of use and relationship of foot ruler, yardstick, and inch line.
~. Liquid measure. Verify simple equivalent measures using cup, pint, quart, and gallon sizes.
3. Weight: Learn concept of pound and half puund.
4. Time: Demonstrate understanding of relationship and use of minute and hours, days of the week and months of the year, seasons
.' aBO use of the calendar.

Measure room and hall using yard stick, place 12 cardboard inch squares on masking tape to measure one foot long.

Use units of measure in classroom situations as; handful, jarful, pitcher, pail, cup.
Use cup, pint, quart, gallon to fill an aquarium, to water flowers, to measure milk, to compare capacity of collected containers. Use examples to indicate
that 2 pints = 1 quart, 1 pint = lh quart, and 1 cup = lh pint.
Find out how many pints of milk are needed to serve the class.
Compare weights of bag of apples, bag of sugar, book, several books; find own weight.
Make clocks using paper plates; collect pictures of clocks; collect a few old clocks; set clocks for various periods of the day; read classroom clock as time activities change; also discuss time activities in the home such as Dad's coming home, postman's arriving, and bed time.

11 12 1

L 10

:4

3

8

4

7 65

Use the calendar to indentify days of the week, seasoDs of year, and months of the year.
39

5. Money: Maintain and extend understanding of value of penny, nickel, dime, quarter, half-dollar, dollar. Make change from the above coins to one dollar. Read money amounts with use of t/ and $ signs.
6. Quantity:
7. Temperature: Begin temperature reading.

Count money collected for lunch, for party, for milk; make a list of articles that can be bought with each coin; make a chart displaying coins, build a toy shop, and put price tags on toys. Compare a quarter, a dime, a nickel, and a penny as to size and appearance. Discover that 4 quarters make one dollar and two quarters make a half dollar. Discuss situations in which it is necessary to get change to have coins for telephoning, vending machines, and parking meters.
Discuss articles that may be bought by the dozen. Visit a grocery store to buy doughnuts, oranges, eggs, apples or rolls.
Make charts showing pictures which illustrate a dozen.
Cut out pairs of items from a magazine.
Draw some pairs of objects.
Count objects by twos.
Arrange children in pairs as partners.
Read and record temperature in a room and outside; note if temperature is greater in the room and how much; make a weather chart for each month; compare temperature in January with temperatures in other months.

VI. Geometric Concepts
A. Recognize in classroom the simple geometric forms as: square, cone, cube, circle, rectangle, a straight line, a square corner, triangle, sphere, and cylinder.

6
D

C[b\)
o bJ

B. Develop habit of calling geometric forms by their geometric names.

Plan for children to observe geometric forms in everyday life. Recognize shapes into which these things are made: wheel, cake, cookies, cracker, window, clock, door, table top, etc. Play games in circle formation. Use toothpicks to arrange geometric forms. For example: A square is formed by 4 lines of equal length. A triangle is formed by 3 lines.
Handle and identify solid figures, as cubes (blocks), cylinders (jars, pipes), cones (ice cream cone), and rectangular solids (milk cartons).
Draw a row of' circles on the chalkboard. Let one child put an X on the third circle. Ask another child to put a square around the first circle. Have the next child draw a triangle in the fifth circle.
Cut shapes from colored paper and put together to make designs.
Have children develop the habit of saying "in the shape of a circle," "in the shape of a square." Encourage children to describe in their own words objects that are round, square, or rectangular. For instance: A ball is round. This box is square. The milk carton is rectangular.

VII. Charts and Graphs

A. Practice making very simple graphs about daily experiences. Make a chart of odd and even numbers.

Have children make simple charts. Write the odd numbers in red and even numbers in green. Make one chart with rings around the odd num

40

, I,

B. Use the following: comparing charts; progress records; chart showing equal parts; chart With coins.

Keep progress charts. Have each child keep graph of his own progress in arithmetic, spelling, or other subjects. Paste a strip of colored paper on chart for each library book read. Record child's weight on chart several times a year.

C. Find and interpret information from a chart.

Use comparing charts. Two charts are needed for this activity - one for numbers 1 to 49, the other for numbers 100 to 149. The teacher says, "Find 14. Find the number just before it. Find the number just after it. Now on the other chart find 114. Find the number just after it." The teacher records the numbers on the board as the children find them. The teacher should give enough practice to estalillish the idea of relationship between 2 and 3 place numbers.
Charts similar to the following may be provided for pupils who need additional help in reading and interpreting charts.

WHO COLLECTED :MOST TOYS

Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6

xxxxx x :xxxxx:xxxxx x xxxxx xxxxx xxxxx xxx xxxxx xxxxx xx xxxx

First:
Second: Third:

Fourth:

_

Fifth:

_

Sixth:

_

Pupils who express interest in charts may be helped to make a chart showing the number of pupils in each activity group. They may be encouraged to use a certain symbol for each group of ten pupils and another symbol to represent individual pupils, as follows.

NUMBER OF PUPILS IN EACH GROUP

o Means 10

x Means 1

Group 1 Group 2 Group 3 Group 4 Group 5 Group 6

000 DO DOD DO DO OOiOO

x xxxxx xx xx xxx
X

D. Develop an awareness of the
function of a chart as a way
of recording materials for future use.

Learn to use a chart to gain information. A record of temperature for a
school week might be kept on a chalkboard. Write the days of the week in a
column and have the temperature that is reported each morning and afternoon recorded beside each day under the headings A.M. and P.M.
Monday, 9 A.M. 400

120

110 90 70
50 30 10

100 80
60 40 freezing 20

5 20
40

10
SO

~

41

VIII. Problem Solving

A. Relate all four processes in oral and written problems.
1. Use many oral word prob lems to provide setting for study of: a. Addition b. Subtraction c. Multiplication d. Division

Relate all four processes in oral and written problems. In oral problems be sure to change names and articles to make problems real situations for children involved.
Addition readiness: Four children brought flowers to school. They brought six violets, three pansies, one aster, and five petunias. How many flowers were brought to school?
Subtraction readiness: Dick has five pennies. He knows that he does not have enough for a ten cent ice cream cone. He wants to know how many more pennies he needs to buy the big cone.
Multiplication readiness: Mother is going to cut two pieces of cake for each guest at Jane's party. If Jane invites 5 friends, how many pieces of cake must Mother cut?
Division readiness: Tom has 8 apples that he wants to divide with four friends. How many should he give to each boy so that the apples are divided equally?

2. Discuss oral word problems with the class.

Draw a simple diagram on the chalkboard of the blocks leading to the school. Let children find how many blocks they live from the school. Find out which child lives nearest to the school. Determine which child has the longest way to come to school.

3. Present quantitive situations by using word problems.

Present quantitive situations more effectively by using word problems. For example: Twelve balls were found on the Little League Field by the second grade boys. There were already four balls in the classroom. How many balls have they now?

4. Learn to read and work problems by using several methods of getting answers.

Learn to read and work problems by using several methods of getting the answer. For example: Find the answer

4+5=1

Make a guess and then count.

Counting: Count by ones

Partial Counting: Count one group only

Grouping: Visualize grouping patterns

Relationships: Sense abstract relationship to a known fact.

= = = 4
4

++

4 5

is

8, 1

so 4 + 5
less so it

is

9 or 9.

5

+

5

10;

Counting by multiples:

A combination of counting and grouping, and relating to another process such as subtraction fact from an addition fact from a multiplication fact.

Encourage pupils to think of approximate sizes, for example:

There were about 30 pupils in a bus;

Last week we used nearly 100 bottles of milk.

5. Read and interpret three steps:

Read and interpret three steps. Word symbols: Bill had three nickels. His Uncle John gave him two more nickels. How many nickels did he have then?

a. Word symbols b. Representative material c. Number symbol

Representative m2.terial: Put three nickels on the desk, and then put two more nickels on the desk. The total number of nickels is seen immediately.

42

6. Use games to show need for mastery of basic facts.

Number symbols: 3 + 2 = 5
Use number game to enable pupils to feel a need for responding with reasonable speed. For example: Draw a ten-inch circle on the chalkboard with one number in the center and other numbers around the circle. One child may point to a number on the outside of the circle. Another child may add the number on the outside to the center number. Play the game with five or six individuals or with teams. Although this game indicates addition, a similar game may be used for subtraction, multiplication, and division.

B. Help children identify processes to be used in solving problems.

Use original problems that have been recognized by the teacher and pupils in the classroom during a certain period of time. For example:
Have pupils imagine that a group of 5 girls combine with a group of 3 girls. To demonstrate their understanding have pupils use objects (buttons, corks, etc.,) to show two groups.
Milk is 3 cents a bottle. Ten boys are going to buy milk. How many pennies are needed to buy the milk?
Ask 5 boys to go to the front of the room. Then let 2 of the boys sit down. Have children observe the number of boys at first, how many are going away, and how many are left.
Seven birds are in the oak tree, and three birds are in the pine tree. How many more birds are in the oak tree than are in the pine tree?

I
I'

Ned has ten dimes. How much more does he need to buy the dollar drum?

Get 10 blocks. Arrange them in groups of 2. How many groups do you have?

DDDDD DDDDD
t/t/ 0 o

2 2 2 2 2
10
5 2/10'

43

C. Dramatize the story problem.

Take the pupils to the post office to buy stamps to mail a package to a class-
Illate.
Have each pupil make his own picture book of basic facti.

D. Help children discover hidden facts in problems.

Teach the pupils to look for hidden facts in the problem.
Apples are 40 cents a dozen. How much are 6 apples? The word dozen means 12 things. Six apples cost 20 cents.

E. Give children opportuntty to Sid wants to measure a cover for his booklet. He wants it to be six inches make up problems about money, long and 10 inches wide. measurements, calendar.
Betty is seven years old. Five candles are on the cake. How many more candles does she need on the cake?
Mary has a dial telephone. Tom learns to dial 86-150.

F. Use matching or other imznature procedures to ah01W correct answers.

Children who are not sure of the correct answer may make stars for each number. One child may take pieces of paper to make combinations.
Use beadlines to show addition facts. As the teacher indicates numbers on the chalkboard the child may use a number beadline worksheet for practice of different combinations.
Pupils may frequently use matching or other procedures to show that answers are correct. These procedures not only provide the beginner with a good basis for understanding, but also give the pupil who already knows the answer a better grasp of the fact and the process.

G. Encourage znental solutions without use of written coznputation.

In studying any number family such as that of seven, the teacher writes the

numeral 7 on the chalkboard and asks many different ways as be can. Some

the pupil which he

to state this amount Inigbt suggest are 3

in
+

as 4,

'T + 0, 9- 2, 10 - 3, and 5 + 2.

44

THIRD GRADE

-

UNDERSTANDINGS TO BE DEVELOPED IN THIRD GRADE
In the third grade the pupil extends his concepts of the tenness of our system of numeration. He develops the understanding that each digit has a value in accord with its position in the numeral. This concept is extended to include numerals of four places. He counts to 300 by l's, 2's, 3's, 4's, and 5's, and to 1000 by 10's and 100's. He understands the distinction between odd and even numbers and uses ordinals through thirty. The pupil acquires an understanding of Roman numerals through XXX.
The 81 basic facts of addition and subtraction, and the basic multiplication and division facts through 6 x 6 are mastered in grade three. The pupil develops the understanding that addition and subtraction are inverse processes, and multiplication and division are inverse processes. The generalizations concerning zero are introduced in the basic processes. He becomes aware of the commutative law of addition and multiplication, and the associative law of addition.
The pupil is able to add both vertically and horizontally. He masters the column addition using one, two, and three digit numbers with four addends which involves regrouping. He masters the subtraction of two and three digit numbers with zeros which necessitate regrouping. He is able to multiply with ease a three of four digit number by a one digit number. He begins to estimate sums, differences, products, and factors. Mathematical sentences with various symbols are used to acquaint the pupil with inequalities.
The third grade pupil makes use of the standard units of measure. He recognizes simple geometric figures, he makes and reads simple charts, graphs and maps. The pupil develops various methods of solving one-step problems. He understands the meaning of fractional parts, and sees the relationship of simple fractions.
47

-

CONTENT

TEACHING SUGGESTIONS

I. Numeration
A. Decimal notation 1. Extend understanding of kinds of numbers.
a. Count to 300 by 1'5, %'5, 3'5, 4's, and 5's; Count to 1000 by 10's and 1OO's; Count backwards from 100 by 1's, 2'5, 5'5 and 10'5.

Help children to distinguish between whole numbers and counting numbers. Lead them to understand that no matter how far one counts, there are still more !lumbers. There is no largest number.
The set of whole numbers may be written:
Set R = {O, 1, 2, 3, 4 ...}
The set of counting numbers may be written:
= Set S {1, 2, 3, 4, ... }
The set of even numbers may be written:
Set M = {2, 4, 6, 8 .}
The set of odd numbers may be written:
Set N = {1, 3, 5, 7, .}
Expand instruction to include several aspects of countinlt by use of a variety of teaching aids, as: Counting chart
Prepare a large demonstration chart for experiences with multiple counting.

1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100

How to make a counting chart:
Take a piece of plywood about 24 inches by 24 inches and screw into it 100 utility elL" hooks (from any ten cent store) 2 inches apart putting 10 hooks in each of the 10 rows. Use price tags or plywood tags to hang on these hooks. Put the numerals 1 to 100 on the tags. The other side of the tag may be used for percent, therefore it is more valuable to make plywood tags.
This chart can be used to help pupils gain an insight into the nature of our number system. Interesting facts can be brought out by such questions as: (1) Is there anything unusual about the ninth row? (2) What do you find when the first and the ninth columns in each row are added? When the second and eighth are added? When the third and the seventh are added?

48

b. Engage in mental arithmetic by beginning with various numbers and pro- gressing by 2's, 3's 4's and 5's.

Number strips and charts.
Prepare number strips for practice in counting by 2's, 3's 4's and 5's.
Example: The numbers that are missing in the strip below are the ones to say when counting by 2's from 32 to 40.

31 ? 33 ? 35 ? 37 ? 39 ?
Example: Prepare number charts with circles or squares around every 2nd, 3rd, 4th, 5th, or 10th number for counting by these numbers. Chart for Counting by 4's:

CD 1 2 3

567

9 10

11 @ 13 14 15 @ 17 18 19 @

21 22 23 @ 25 26 27 @ 29 30

'C. Review and reenforce distinction between even and odd numbers.

Note: Prepare the chart above to include numbers through 100.
Number line
Use the number line for teaching multiple counting. Pupils will discover that in counting by 2's every other number is omitted, that in counting by 3's two numbers are omitted and so on. Encourage pupils to begin with an odd number to count by 2's or 4's and to begin with an even number to count by 3's or 5's.
1 2 3 4 5 6 7 8 9 10 11 12 13

d. Use ordinals through thirty.

"Quick Change'
To help pupils identify the ordinals as a means of understanding numerieal position play the game, "Quick Change." Arrange pupils in sets of 10. Nine pupils take ordinals for names and the tenth child is the leader who calls out, "Seventh change with fourth." Pupils having these names must not only change places but also change names to correspond to their new seating order. If a pupil does not respond to his ordinal name promptly, he must give up his place to the leader.

2. Place value
a. Understand the grouping Help pupils visualize the difference in value l,Jetween lone, 1 ten, and 1 of tens in our system of hundred by using paper marked off in lh inch squares. numeration.

1 hundred
49

o
1 ten lone

PI

b. Extend understanding of Tape together ten sheets of IOU squares to represent 1 thousand. base ten in value through thousands.

c. Understand zero as a digit which means "not any" in the place it occupies and that it helps determine the value of other digits by serving as a place holder.

Have pupils write numbers containing zeros from Qlctatlon. ~uggest the following arrangement.
4=4ones 40 = 4 tens 400 = 4 hundreds
4000 = 4 thousands
Use place value chart:

Th I H

T Ones

1,230

1 I2

3

0

360 201

-

1

I2

3
1

-

6
0

-

-

0
-
I

-

d. Recognize zero as a place of beginning on the thermometer.
e. Regroup numbers through hundreds place.

Locate zero on the Fahrenheit thermometer. Help pupils discover that weather temperature is measured "above zero" and "below zero."

Use bundles of 100 sticks, bundles of 10 sticks, and single sticks to help pupils see that:

329 is 3 hundreds and 2 tens and 9 ones

25 is

2 tens and 5 ones

Provide practice in regrouping as: 888 means:

8 hundreds 8 tens, 8 ones or
8 hundreds, 88 ones or
88 tens, 8 ones or
888 ones

3. Reading and writing num bers.
a. Read and write four-digit numerals.

Use chart to develop understanding of place value for four-digit numerals.

thousands place

3 thousands or 3000

hundreds place

5 hundreds or 500

tens- place
ones. place 3 5 2 4~

2 tens 4 ones

or 20
or 4 3524

b. Read number names through one hundred and write number names through fifty.

Provide experience in use of dates and telephone practice in reading and writing larger numbers.
Have pupils copy ten or more number names in a column and fill each with the figure that matches the name. Encourage pupils to make each f better and to circle the best figure in each row.

50

c. Review the language of sets introduced in earlier grades. Extend understandings to include use of abstract symbols to indicate members of sets, term and symbol for empty or null set, and use of braces in expressing arithmetical ideas.

Make charts with the number and name. Use flash cards with number names and have children give the corresponding numeral.
Review language of sets by using classroom situations.
Examples: To review one-to-one correspondence and two-to-one correspondence match 9 books with 9 children and match 10 books with 5 boys (2 books for each boy).
To review term "members of sets" plan examples, as: How many members are there in the set of girls in this classroom? Write the set of even numbers found on the face of the clock on the teacher's desk. How many members does the set of even numbers that you wrote have?
How many members are there in the set of odd numbers found on the face of the clock?
Provide opportunity for use of a variety of abstract symbols to indicate members of sets:
Set A =

Set B =

What member does Set A and Set B have in common?

Develop the idea that sometimes a set has no members, as: The set of live tigers in the classroom; the set of eight-year-old girls who weigh 500 pounds; or the set of boys in the classroom who are ten feet tall. Bring out the idea that the name of a set that has no members is the empty set or the null set, and that there is a special way of writing this idea. Let K represent the set of live tigers in the classroom. This idea can be written as:

= { Set K

Point out that the marks used to enclose a set are called

L

}

braces. Point out, also, that there is another symbol used to indicate an empty or null set. This symbol is </>.

(If more material is desired on sets refer to Fourth grade IT, A, 10.)

B. Roman numerals 1. Explore uses of Roman numerals.
:a. Discover additive and sub-
tractive principles of this system of numeration.
Bead and write Roman nuIIlera1s through XXX. Rec0IDize L and C.

Make lists of uses of Roman numerals, as: Numerals on faces of some clocks, in preface and introduction of books, in outlines, in the Bible, the V on some old nickels and in sections of newspapers.
Help pupils to generalize that: Each "I" coming after I, V, and X adds 1 to the value of the number and each "I" coming before V and X subtracts 1 from the number, as:

12 3

5

6

7

8 10 11 12

I IT ill V VI VII vm X XI XU

and

4

9

IV

IX

Have pupils make the following chart.

Arabic Numeral

Number Name

Roman Numeral

1

One

I

2

Two

IT

3

Three

ill

4

Four

IV

5

Five

V

6

Six

VI

51

it

Play a game by asking: What letters did a Roman boy use to write the numbers 15, 21 or 19?

Use toothpicks to form such numbers as XV, XXI and XIX

Ask children to change tooth picks to make a true number sentence.

False: ill-II=IV

= True:
VI - II IV

False: XI+II=IX

True
= IX + II XI

II. Fundamental Operations With Whole Numbers

A. Addition and subtraction

1. Master. the 81 basic addition and subtraction facts.

For some pupils it will be necessary to reteach the 45 basic addition/and subtraction facts. Use manipulative aids to lead pupils to discover the 36 remaining basic addition and subtraction facts. Encourage mastery of the 81 basic addition and subtraction facts.

Develop a variety of charts showing the 81 basic addition and subtraction facts.

2. Demonstrate the correct use of zero in addition to and subtraction from a number with the original number as the answer.
3. Understand subtraction with zero as the remainder.

Lead pupils to discover the generalizations: n + 0 = nand n - 0 = n Note: "n" represents any number

Lead pupils to discover the generalization:

n-n=O

Example: 5 -5 -0-

4. Read and write addition and subtraction facts in both equation and column form. Use a literal number to stand for the variable (unknown quantity) in an equation.
5. Understand that: a. Changing the order of the addends does not affect the sum.
b. Addends may be grouped in any way without affecting the sum.

Examples:
n= 10-2 n+3=7
o
-4 -6-

15-7=n 8+n= 14
o
+7 -9-

Examples:

5 + 4 = 9 or 4 + 5 = 9

5 +4 -9-

4
or +5
9

(Commutative Law of Addition)

Examples:

7 + 1 + 4 = 12 or 7 + 1 + 4 = 12

'--y--J

'----y---'

8 +4=12or7+ 5 =12

(Associative Law of Addition)

52

F

6. Discover that addition and subtraction derive mean ing from each other as that subtraction is the inverse of addition.
7. Use the number line for addition and subtraction.

Examples:

2

7

+5

-5

7

~

3

8

+5

-3

8

5"

5+4=9

5

7

+2

-2

7

5""

5

8

+3

-5

"s

3"

9-5=4

Examples:

0

5

I

I

10

15

I

I

3+4= 7

14

r-----~'------__.,

o

5

10

15

=5 ''-----------~
14 - 9 5

8. Master recognition and solution of the three different types of subtraction situations:

a. Find out how many are left.

Use the "how many are left" type subtraction situation while pupils are developing the subtraction facts.
Example: Tom had 5 balls. How many balls did he have left after he gave away 2?
had

b. Compare two amounts.
e. J'ind how many more are needed.

000 had left

00
gave away

"5 take away 2 leaves 3"
= "2 from 5 leaves 3"
"5 - 2 n"

Comparative subtraction should involve only the basic facts that the pupils have already learned. Employ dramatization and other visual techniques to develop an understanding of the "how many more" or "how many fewer" situations. The fundamental concept involved here is a one-to-one correspondence of the objects in the two sets (groups). The unmatched objects constitute the difference or the excess.

Example: Jane has seven pencils and two erasers. How many more pencils does Jane have than erasers?

IIIIIII

more
I I I-I
"7 is 5 more than 2" "The difference between 7 and 2 is 5" "5 more than 2 is 7" "2 is 5 fewer than 7"

Introduce the "how many more are needed" type situation late in the year. Provide many experiences with objects to help pupils visualize the total quantity, to understand that it is as if the original group had been removed,

53

and that only the group that was added remains. This will help the pupils to see the reason for subtraction in finding the solution.

Example 1.

Sue needs 8 books for her reading group. She has 5 books. How many more books does she need for her group?

DDDDDDDD

1

2

3

4

5

6

7

8,

more needed "5 and 3 more make 8" "3 more are needed to make 8"

Example 2. Mary has 7 pieces of candy. She has 10 friends. How many more pieces of candy does she need for them?
7 + 0 = 10

Examples:

9. Understand the relation of sum to addends, relation of remainder or difference to sum, and relation of unknown addend to sum.
10. Understand relation between ba$ic addition facts and higher decade addition; add endings without and with regrouping.

3.}} addends
+ 163
201 J sum

505 sum -83
422 difference or remainder
505 sum - 83 known addend
""422 unknown addend

Begin systematic instruction in addition of two-digit numbers after the pupil has an understanding of the 81 basic facts and generalizations for the use of zero. Introduce higher decade addition with examples that do not require regrouping in the one's column and then use examples requiring regrouping. Encourage pupils to solve examples by adding on the number line, by using the place value chart or the abacus, and by using the number
method.

Example: Miss Smith asked Bob to get a new pencil for each of the 16 b()ys and 18 girls in the class. How many pencils should Bob get?

tens ones

1

6

1

8

3

4

= 16 _ 1 ten and 6 ones
+18 1 ten and 8 ones

34 _ 2 tens 14 ones or 3 tens and 4 ones

11. Master addition of columns of one-, two-, and threedigit numbers with four addends including internal
zeros.

Examples:
7 6 9
+5
27
Write in column and add: 22+9+10+78
Write ina column and add:
104 + 7 + 82 + 56'7
54

12. Master subtraction of twodigit numbers and threedigit numbers without and with regrouping and with internal zeros.
13. Check addition and subtraction
a. Adding in opposite di. rection. (addition is com. mutative).
b. Check subtraction by ad. dition.
c. Estimate sums and remainders.

Examples:
= 39 3 tens and 9 ones = -16 1 ten and 6 ones
23 - 2 tens and 3 ones

= 63 6 tens and 3 ones or
-47
---ur
5 tens and 13 ones -4 tens and 7 ones
1 ten and 6 ones

709 = 7 hundreds and 0 tens and 9 ones
-126

--sB3'

or

6 hundreds and 10 tens ani 9 ones

-1 hundred and 2 tens and 6 ones

5 hundreds and 8 tens and 3 ones

Provide experiences for pupils to add a column by adding down; check by adding up the column.

~ } (9 + 7) + 8. Check:
+8

9
+~ } (8 + 7) + 9

~

24

Guide pupils to discover that the inverse process will "undo" the original

operation. Use the vocabulary associated with the process.

14 sum - 9 known addend --5- unknown addend

Check:
5 Add the two addends
+ 9 to get back to the
--r4 sum.

When numbers are rounded off, sums and remainders can be estimated with greater ease.

Estimate the sum of 27, 13, and 8:

27 is about 30 13 is about 10 8 is about 10

Estimated sum:

Exact Sum

30

27

10

13

10

8

50

48

Point out that often it is very useful to know the approximate answer.

Estimate the difference between 810 and 187. Use a number line marked in units of 100.
o 100 200 300 400 500 600 700 800

Estimated difference: 810 is about 800 187 is about 200
The difference is about 600

Exact difference: 810
-187
fi23

55

Pi

B. Multiplication and division of whole numbers.
1. Understand that multiplication is a short method of adding like groups of the same site.

Use oral problems from classroom experiences.

Example: Sam can carry two chairs at a time. How many chairs can he move in 3 trips?

hhhhhh

2

+

2

+

2

6

2

2

or 3 twos = 6

+2

-6-

2. Use horizontal form of multiplication with proper vocabulary and signs.
3. Use theverticaJ form of multiplication as another way of writing the same facts.

3 3 3 3
12

4 threes = 12 or 4 X 3 = 12; The sign "X" means times. The one factor 4 indicates the number of times the group is added. The factor 3 indicates the size of the group. The product 12 is the size of the new group resulting from the operation.

4 X 3 = 12 is the same as: 3 X4
12

Note: The size of the group is written first. Under it is written the number which indicates the number of times the group is repeated. The size of the combined group is written under the line. Encourage use of the following terms:

factor X factor
product

4. Check multiplication by adding and by reasoning.

4X2=8or2 2 2 2
8

2 or 2 X5 2
10 2
2 2
10

5. Discover and master the basic multiplication facts with twos, threeS, fours, fives, tens, and the sixes through 6 X 6 = 36.

Pupils should manipulate objects arranged in equal subsets (subgroups).
Help pupils understand that as long as they count by equal groups to find the answer, they are using the long method, addition. When they can respond immediately (as 6 X 3 = 18) they are using a short method called multiplication. A "table" arangement of multiplication facts should not be made until the facts have been discovered and understood.

6. Understand that factors may be interchanged without affecting the product (multiplication is commuta-
tive).

As the multiplication facts are developed help children discover that as one learns each new fact, one learns an extra one, as: If 3 X 2 = 6, then 2X3=6.
Note: Help pupils understand that tht! sense of the two examples is different. however, 3 X 2 means adding 3 twos; and 2 X 3 means adding 2 threes, although the products are the same.

7. Discover the generafuation that whenever zero is a factor, the product is zero.

6 X 0 = 0 and 0 X 6 = 0 nxO=OandOxn=O
Note: "n" represents anyntimber
Relate this concept to addition.

56

8. Discover the generalization that 1 times any number results in the original num ber.

Example: The team played four games without scoring a single point. Write the team's score.
o o
00r4xO=0
+0
-0-
Note: "n" represents any number

9. Understand that division is a short way of subtracting equal groups.

10. Discover that division is the inverse of multiplication. Verify division by multiplication.

11. Master the division facts

through

6

6)36

Use objects and let pupils dramatize problem situations.

Example: From a stack of 15 books ask different children to get 3 books until there are no books left. Ask them to find the number of times 3 was subtracted by counting the number of children who were able to get 3 books.

Use number symbols to explain how the books were divided.

000000000000000 000000000000 000000000 000000 000

15
-3 12"are left
-3 -9-are left
-3 ~are left
-3
-3-are left -3 -O-are left

Help children to see the short way to find how many 3's in 15.

5 Think: How many 3's in 15? Use multiplication facts to determine 3)15 and verify answer.

5 X 3 = 15 Point out that if one knows that there are 5 threes in 15, one
can subtract them all at once, which is the division process. 5
3) 15 15

Develop the division facts about the 2's, 3's, 4's, 5's, and 6's through product of 36 with concrete materials after the multiplication facts have been developed.
Use the number line to divide a number by moving to the left from it in equal steps.

12 + 3 = ?

012
0(
4 threes

345
0(
3 threes

678
0(
2 threes

9 10 11 12 13 14
0(
1 three

Provide practice in writing the family of 4 facts for each multiplicationdivision combination, as:

3X5=15

15+3=5

5 X .3 = 15

15+5=3

Emphasize the relationship of the multiplication and division processes by

57

finding the missing number in an equation, as:

2X3 ?

or

2X3= n

2X?= 6

or

3Xn= 6

? X3=6

or

n X3= 6

2xn=6

or

2X3= 0

12. Use vocabulary pertaining to division.

factor Known factor 1 product

13. Discover the generalizations:
a. Any number divided by 1 results in the original
number.
b. Any number divided by itself results in 1.
c. Zero divided by any number results in zero.

Examples: 5...;.- 1 = 5
4...;.-4=1 0...;.-4=0

38 ...;.- 1 38
53,,,,;,- 53 1
o ..;.- 19 = 0

1 n~

n

n ...;.- n = 1 0 n= 0

14. Understand that division is used to find:
a. The number of equal groups (measurement or comparison).

Use problems related to pets, games, or books. Since drawings for measurement division problems are easier to construct, the basic division facts should be developed with measurement type problems.
Examples:
Jane has 12 new insects for her insect collection. If she puts 3 insects in a box, how many boxes does she need? The pupils should think: "How many 3's in 12?"
4
3 1'""'12
Help children to generalize that in this type problem the unknown factor (an abstract number) indicates the number of equal parts.

b. The size of the equal groups (partition).
15. Multiply two-, three-, and four-digit factors by a onedigit factor. a. Without regrouping.

Bill has 15 apples that he wishes to share equally with three friends. How many apples will each friend receive? The pupils should think: "The 15 apples need to be divided into 3 equal groups."
5 3 1----r5
Help children to generalize that when finding one of the equal parts group, the factor is an abstract number telling the number of parts.

Before teaching the traditional multiplication algorism found in most text books, help pupils understand that tens and hundreds are multiplied as ones are multiplied.

32

3 tens + 2 ones

x3

X3

Point out the relation to addition:

32 The size of the group being repeated is a large group (thirty-two) and 32 it is added 3 times. 32

Note: Some third grade pupils can visualize and discover the Distributive LaW while working with this type example.
= 21 4 X 2 tens 8 tens
X 4 4 X lone = 4 ones
84
or

58

o. With regrouping

= 4 X 21 (4 X 20) + (4 X 1) =

80+

4=84

Develop multiplication involving hundreds in the same manner:

143 2 X 1 hundred = 2 hundreds x20r2x4 tens =8tens
2 X 3 ones = 6 ones or
2 X 143 = (2 X 100) + (2 X 40) + (2 X 3) =

200+

80+

6=286

Guide pupils into the realization that the plan of regrouping used in addition applies in multiplication.

46

46

X2 means +46

2 sixes are 12. The 12 is regrouped as 1 ten, 2 ones. Put the 2 enes in the ones place. 2 forties are 80 and 1 more ten makes 90. The product is 92.
Note: When three-digit factors are multiplied by one-digit factor it may be necessary to review the place value chart as well as to stress the procedure for regrouping (carrying) in the tens and hundreds places.

16. Divide two, three, four, and five digit products by one digit factors.
a. Without regrouping.
b. With regrouping and without and with remainder.

Although division with unknown factor larger than one-digit is not found in some third grade textbooks, the use of unknown factor of more digits than one gives children a better insight into the process of division than working only with the basic facts.
121 4 / 484
1 hundred + 2 tens + lone 4 / 4 hundreds + 8 tens + 4 ones
Use the place value chart to regroup in division, changing thousands to hundreds, hundreds to tens, and tens to ones.

53

"3

50

5 tens + 3 ones

3 / 159

or

3 / 15 tens + 9 ones

150

-9

9

211 factor -1
10
200
4 / 845 800 (4 X 2 hundred) 45 40 (4 X 1 ten)
5
4 (4 X lone)
1 difference or remainder
or
59

F

17. Estimate products and factors.

4 / 845
800
45
40
5
4
-1

200
10
1 211 and a remainder of lone

Encourage pupils to estimate products by thinking from left to right as:

431 X 2 Think "2 times 400 is 800. The answer must be more than 800.
2 X 500 is 1000, so the answer will be between 800 and 1000. Since 431 is closer to 400 than 500 the answer will be closer to 800."

Encourage pupils to estimate unknown factors by reasoning and by use of understanding of tens.

III. Fundamental Operations With Fractions

A. Common Fractions 1. Understand that a fractional part is one of a given number of equal parts of a whole object or of a group.
2. Master the concept that the fractional parts into which a whole object or a group may be divided are equal in size.
3. Compare fractional parts to determine relationships between 1/2, 1/3, 1/4, 1/(l' and '/8'

Discuss instances in which pupils have used lh. List these in two columnsinstances in which lh is applied to a single object and instances in which lh is applied to a group. Extend this exercise with 1/3 and 1/4, Use a magnified number line showing fractional parts.

o

1

2

o 1/42/43/44/45/46/47/48/4

To study equal parts of a whole object, have pupils fold and cut squares, circles, and rectangles of paper into halves. Fit the parts to show that parts of each figure are the same size. Fold halves again into fourths and see that these fractional parts are equal.

Equal parts of a group may be studied by the activity of sharing objects-

marbles, pencils, candy bars, etc. Divide a group of six pencils into two equal

parts by giving one at a time to each of two girls. Have pupils count to see

how many each one has and that each has the same number. Extend this

= = activity with dots to find 1/2 of 10, 12,
1/4 of 4,8,12,16, and 20; 1/2 of 10

14, 16, 18, 20;
0; 1/4 of 0

1/3

of 3.

3,6,

9,

12,

15,

and

18;

To demonstrate how parts of things vary in size use the same sized paper plates cut into halves, thirds, fourths, and eighths. Superimpose one '/3 of a paper plate on lh of a plate. Let pupils discover in this manner that '/3 is smaller than lh; that '/3 is larger than lf4; etc.

Make charts to compare size of thirds and fifths of the same object.

60

-
Show :lh and lf4. of a pie.

Measure liquids to show that 1 quart is lf4 of a gallon and 1 pint is :lh of a quart, and 1 cup is lf4 of a quart.

4. Discover that dividing by abstract numbers is another way to find equal parts of a whole object or of a group.

Use objects to show that:

= :lh of 12 means 12 divided by 2. 12 -;- 2 6

'/3

of 9 means 9
9 -;- 3 = 3

divided

by

3.

= % of 8 means 8 divided by 4. 8 -;- 4 2

5. Use the symbol for fractions with understanding and know the proper use of the names of the parts of the fraction
6. Demonstrate with concrete objects addition and subtraction of like fractions.

Make sure that pupils understand that the symbols :lh, '/ 3 , lf4, '/ 5 ' and V8 must represent a half, a third, a fourth, a fifth, or an eighth of an object or group.

Emphasize the meaning of the names of the parts of a fraction.
1 ~ numerator represents the number of equal parts selected for

_

special notice

3 ~ denominator represents the number of equal parts into which

the whole group has been separated. It gives the size or name

of the fraction.

Use flannel board with circles and squares cut into halves, thirds, or fourths, and fit together fractional parts of the same sized paper plates.

B. Decimal Fractions
Understand that the decimal point is used to show which numbers represent parts of a dollar, or cents.

Although formal instruction in decimal fractions is introduced in the fifth grade, pupils should be provided many experiences in reading and writing dollars and cents as a background for an understanding of decimal fractions.

Relations

Relations between sets of elements constitute one of the significant tools of the mathematicians. In elementary mathematics, the word relations concerns a scheme whereby it becomes possible to answer a specific question with regard to specific items. In fact, the understanding of relations governs the child's ability to work successfully with numbers, signs, and operations.
The earliest idea of equivalence classes is presented to the child when he understands that a natural number can be designated by many different symbols, as: (5 + 1), (1 + 5), (4 + 2), (2 + 4), and (3 + 3) all belong to the same equivalence class 6, a natural number. He also understands that
= = 2 + 4 4 +2 (Commutative Law for Addition) and that (1 +2) + 3
1 + (2 + 3) (Associative Law for Addition).
61

= Use concrete materials to help pupils see intuitively that 214 112 and that
= 4/ R 2/ 4 or 1/ 2 ,

B. Inequality Introduce symbols for less
than, <; greater than, >

Review the idea of one-to-one correspondence to emphasize the concept of equivalence. Sets that match or that are equally numerous are said to be equivalent.
Use relative value of coins to further emphasize the concept of equivalence.
Guide pupils to a discovery of inequality by using counters to show that some numbers are less than or greater than other numbers. Examples using whole numbers:
3 + 0 < 7; 2 + 0 > 3; 2 + 3 ? 7
Point out that on a horizontal number line the larger of two numbers is the one on the right of the other number.

c. Order of decimals
D. Ratio and proportion Understand how a correspondence can be made between the objects of one set and the objects of another set.

Provide for reading and writing dollars and cents in social situations. Stress the use of the word and in reading: $2.36 as "Two dollars and thirty-six cents." Help pupils see that one says and to separate the whole dollar from or dollars from parts of a dollar.
Strengthen the understanding of order of decimals by reviewing the decimal nature of our number system. Use place value chart. Point out that in writing money every-numeral on the right of the decimal point represents an amount less than one dollar.

An informal approach to the idea of rates symbolized by a pair of numerals called ratios; and proportions symbolized by a pair of equal ratios, may be used to help some children understand the basic structure of problems dealing with multiplication and division.

Examples:

If pencils cost 3 each, how many pencils can Sue buy with 15 pennies? Ask a pupil to arrange the 15 pennies in groups of 3 and place a pencil beside each group. The rate can be expressed as "3 cents per 1 pencil." Sue is to spend 15 cents for pencils at the same rate. Therefore the same rate may be expressed as "15 cent~ per N pencils." Thus this problem calls for the solving of this proportion which is made up of two equal ratios.

3

15

1 1r

Help the pupils see that the process needed for the solution of this problem calls for d i v i s i o n . '

The Cub Scouts want to spend the weekend at their camp. The camp has four cabins. Only three boys can sleep in each cabin. How many boys in all can sleep in the four cabins?

Ask the pupils to draw a diagram of the 4 cabins with 3 boys in each cabiJI.

The rate is expressed by "3 boys per 1 cabin." It may also be expressed ill

the form "N boys per 4 cabins." This problem calls for the solving of thiJ

proportion:

3

N

'1=4

62

E. Percent

Help pupils understand that the process needed for the solution of this problem is multiplication.
Relate to classroom experiences: 100% of the pupils in the class eat in the lunchroom everyday. 50% of of the pupils in the school contributed to the March of Dimes.

v. Measurement

A. Units of measure
1. Rediscover the concept that distance, time, and quantity can be measured and compared.
a. Use non-standard units in measuring.

Suggest that pupils use a stick to measure the distance that the "class champion" can jump. Return to classroom, mark this distance on board by applying the stick the correct number of times. Have a pupil measure how many jars full of water the fish bowl will hold. Have pupils estimate how long it takes the class to walk to the lunchroom. Verify the time by a watch.

b. Use appropriate nonnumerical measurement vocabulary.
2. Understand need for and use of standard units of measurement for finding:

Near, far, not too far; more time, too long, late, almost; full, nearly full, too full; about the same, etc.

a. Length
b. Quantity c. Weight d. Time

Provide a variety of situations in which pupils select the device for measuring
= = foot, yard, inch, half-inch, and quarter-inch.
Guide pupils to discover these relationships: 12 in. 1 ft.; 3 ft. 1 yd.; 36 in. = 1 yd.; 2 half-inches = 1 inch; and 4 quarter-inches or 4 fourth-inches = 1 inch.
Use problem situations with half-pint or cup, pint, quart, and gallon.
= = Help pupils discover the equivalent measures: 2 pt. 1 qt.; 4 qt. 1 gal.; = = 8 pt. 1 gal.; and 4 cups 1 qt.
Use scales to help solve problems that develop understanding of ounces, pounds, half-pound, and quarter pound. Develop by lifting the approximate weight represented by one pound.
Extena earlier concepts to include the use of circular number line in such problems as:
If it is 11:00 o'clock now and school closes in four hours what time will it be when school closes?

Develop ability to tell time to five-minute and one-minute intervals and to the quarter-hour and the half-hour. Help pupils discover that there are 30 minutes in a half-hour, 15 minutes in a quarter-hour, and 24 hours in a day.
Use actual calendars for helping pupils extend their abilities to telling the day of the week on which the month begins or ends, the date of the first Sunday and other Sundays, to find special dates of interest and write the days of the week and months of the year in order.
63

.,

e. Temperature

Present a number of situations to help pupils interperpret symbols on the thermometer. Help them discover that the column of liquid in the thermometer rises as it gets warmer and falls as it gets cooler. Emphasize that temperature is
measured in degrees CO) and point out that 68 to 72 degrees Fahrenheit is
normal room temperature. Help pupils discover that temperature changes
with the seasons.

B. Money 1. Extend an understanding of the use of the decimal point in writing numbers that represent United States money. 2. Extend an understanding of relative value of coins and denominations of paper money.
3. Make change as needed.

Use classroom or other experiences for problems that involve reading and writing money using four figures, as: $36.14.

Develop with pupils a chart of equivalent values of coins from counting real money:

1 nickel 1 dime
1 quarter

5 pennies 10 pennies or cents 2 nickels 1 nickel and 5 pennies 25 cents
5 nickels 2 dimes and 5 pennies 2 dimes and 1 nickel

Continue to the dollar in coins and to five dollars in paper money.

Extend skill to making change from one dollar. Start with amount of purchase
and count to the nearest multiple 0}5. Then count nickels, dimes, quarters,
or half-dollar to the amount ($1.00) given to the clerk.

decimal point in easy problems of addition, subtraction, multiplication, and division involving money.

Examples:
June bought rolls for 23, a dozen eggs for 51, and a chicken for 78. How much should she pay in all?
$ .23 .51 .78
Jake bought a fountain pen that cost $2.75. He gave the clerk a 5 dollar bill. How much change should he get? How much will 3 dozen apples cost at 52 a dozen? Martha paid $2.58 for 2 yards of cloth. How much did one yard cost?

VI. Geometry
A. Extend recognition of simple geometric shapes
B. Reproduce simple geometric shapes

Provide experiences to help pupils learn to identify the following geometric forms:

Line figures

Solids

circle square triangle rectangle

cube sphere cone cylinder

Suggest experiences that will prompt pupils to use their own ingenuity in illustrating various shapes. Frame their drawings or paintings on mats cut in squares or circles. Prepare an exhibit on spherical objects as globes, balls, apples, oranges, grapes, and marbles. Make a mural of geometric shapes found on the school grounds. Make cone-shaped hats, Indian tepees, and flower containers. Arrange a display of cube shaped objects.

64

VII. Charts, Graphs, and Scale

A. Make simple bar graphs
B. Use picture charts of various types
C. Draw maps of neighborhood, school, and classroom.

Prepare large graph showing attendance for one week or longer. Individual pupils might make bar graphs of their scores on spelling tests.
Picture charts that tell the story of the teen numbers could be used to strengthen the pupils' knowledge of the basic fundamental operations.
These would not be very precise but would be useful in helping pupils establish relationships.

VIII. Problem Solving

A. Develop a meaningful procedure for solving problems.
B. Solve on&ep problems.

Each pupil should be helped to formulate some logical procedure, as:
Analyze problem. a. Find what it tells. b. Find what it asks.
Decide the process to use-addition, subtraction, multiplication, or division. Show process with number sym!:;ols as writing the number sentence. Estimate the answer. Make the computation to find the answer. Check or verify the computation. Compare estimated answer with computed answers.
Use problem situations from classroom experiences, as:
There were 12 books on the shelf. Mary took out a books for her reading group. How many books were left on the shelf?
Analyze problem.
a. Problem tells (1) There were 12 books. (2) a books were taken
away. b. Problem asks how many books were left? Decide process. This is a subtraction problem because it asks you to take away a set
of a from a set of 12.
Show process with number symbols.
= = = 12 - a 0 or 12 - a ? or 12 - a N.
Estimate answer. Pupils may offer various solutions, as: Counting backward from
12 by 1's to a takes four steps, 80 the answer is 4o 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Make computation.

12 1 ten and 2 ones or 12 ones

-8

- a ones

--4-

4 ones

Check or verify answer. 12 is the known sum.
a is the known addend.
? is the unknown addend.
This information expressed in a number sentence is:
= ? + a 12 or 0 + a = 12 or N + a = 12.
Some pupils know the answer as one of the basic addition facts. Other pupils
may need to use counters or to count forward from a to 12 on the number
line.

The principle involved is: Subtraction is the inverse of addition.

65

p

C. Learn to read and work problems by usir.g several methods.
D. Develop general procedure for problem solving.

Use only one-step problems except for verbal problems requiring. choice of process as readiness for two-step problems.
Another illustration: If apples were selling 5 pounds for thirty cents, how much would ten pounds cost? Method 1: Find the cost of each pound and multiply by 10.
= 30 -;- 5 6
lOX6=60 Method 2: If 5 pounds cost 30 cents, then 10 pounds would cost
two times as much, or 2 X 30 cents = 60 cents
Method 3: Finding the answer by proportion.
3 10
D ~30=
Draw diagrams. Use illustrations to supply missing facts.
Use analogous problems.
Make original problems. Suggest solutions of problems without actually solving them.
Use many oral word problems. Develop concepts and teaching skills, vocabulary, signs and symbols necessary since the language of mathematics is unique.

66

a about about the ~ame above add addend addition after afternoon age air all all day all gone almost almost half all together A.M. amount and angle answer any apiece approximate are around as as large as as long as as many as as small as at August autumn away
back backward bank base before behind below beside between big bigger biggest block both bottom bought bunch buy by
calendar cardinal cent

VOCABULARY FOR PRIMARY LEVEL

center change check c1rc1e cloek close coin cold colder coldest collection column combinations combine comes compare cone container cost costly count counter counting couple cover crowd cube cup cupful cylinder
date day decimal point December deep deeper deepest degrees denominator diameter difference digit dime disc divide distance division dollar double down dozen draw
each
~arly
earn eight empty enough equal equally

even exact
factor Fahrenheit fall family far farther farthest fast faster fastest February few fewer fewest fifth fiity figure find first fire follows foot for forward four fourth freezing point Friday from front full
gallon gave game get glassful go going great great big greater than group grouping
half half dozen half of group half of one half past half pint half pound halves heat heavy heavier heaviest

height high higher highest hohday hot hour hour hand how many how much hundred hundreds place
in inch inches inverse is
January July June
lack large larger largest last last night late least left length less less than light lighter lightest like line liquid little long longer longest long time low lower lowest
made make many March match May mean means meaning measure

67



member of set middle minus minute minute hand missing number Monday money month more more than morning mest multiply
narrow need next nickel night nine no no more noon not none November now number number line number patterns number stol'1 numeral numerator
oblong o'clock October odd off old older .oldest on once one one fourth ')ne half open over ones place ounce or order
paid pair parallel part pennies penny perimeter periods

piece pile pint place value plus pound price problem process product put
quart quarter
rate read rectangle rectangular received regroup remainder rent right ring Roman Numeral round round off row ruler
same same as Saturday save scales season second see sell 'Separate September set seven seventeen .seventh :several shape share short shorter shortest show side sign six sixth
size slow small smaller smallest sold solid

some soon space speedometer spend sphere spring square square corner start stop story street straight subtraction summer Sunday symbols
tables tablespoon take take away taking tall taller tallest tape measure teens tell telling temperature ten tens tens place tenth thermometer thick thicker thickest things third thirds thousands place three through Thursday time tiny to today together tomorrow
too little too much top total toward Tuesday twelve twenty four twice two triangle
68

under up unlike units use
value
wage wages warm way Wednesday week weigh weighs weight what which who whole why wide winter with write wrong
yard yards yardstick year yesterday young younger youngest
zero

BOOKS FOR CHILDREN
Primary Level
Asimov, Isaac. The Clock We Live On. Abelard-Schuman. 1959 Bael", Howard. Now This, Now That. Holiday. 1957 Barr, Catherine. Seven Chicks Missing. Walck. 1962. Behn, Harry. All Kinds of Time. Harcourt, Brace. 1950. Beim, Jerrold. The Smallest Boy In The Class. William Morrow. 1949. Bendick, Jeanne. How Much and How Many. Whettlesey House. Bianco, Pamelo. The Doll In The Window. Walck. 1953. Bishop, Claire and Wiese, Kurt. The Five Chinese Brothers. Coward-McCann. 1938 Brann, Esther. Five Puppies For Sale. MacMillian. 1948 Bragdon, L. J. Tell Me The Time. Lippencott. Brown, Margaret Wise. Two Little Trains. Scott, William R. 1949. Browne, Georginan K. Look and See. Melmont. 1950. Budney, Blossom. A Kiss Is Round. Lothrop. 1954. Burton, Virginia Lee. The Little House. Houghton Mifflin. 1942. Chaffee, Fish. The Five Jolly Brothers. Wonder Books. 1951. Chalmers, Audrey. I Had A Penny. The Viking Press. 1945. Chalmers, Audrey. Hundreds and Hundreds of Pancakes. The Viking Press. 1946 Cooke, Barbara. My Daddy and I. Abelard-Schuman. 1961. Corcos, Lucille. Joel Spends His Money. Abelard-Schuman. 1961. D'Aulaire, Ingri and Parin, Edgar. Don't Count Your Chicks. Doubleday. 1943. Duvoisin, Roger A. Two Lonely Ducks. Knopf. 1955. Eichnberg, Fritz. Dancing On The Moon. Harcourt, Brace. 1955. Elkin, Benjamin. Six Foolish Fishermen. Childrens Press. 1957. Estep, Irene. Good Times With Maps. Melmont. 1962. Flack, Marjorie. Angus and the Ducks. Doubleday. 1943. Flack, Marjorie. The Boats On The River. The Viking Press. 1946. Friskey, Margaret. Chicken-Little-CountToTen. Childrens Press. 1946. Gag, Wanda. Millions of Cats. Coward-McCann. 1945. Hoberman, Mary A. and Norman. All My Shoes Come In Two's. Little, Brown. 1957. Hogan, Inez. Bear Twins. E. P. Dutton. 1935. Hogan, Inez. Twin Lambs. E. P. Dutton. 1935. Hogben, Lancelot. Wonderful World of Mathematics. Doubleday. 1955. Ipcar, Dahlow. Ten Big Farms. Knopf. 1958. Karasz, Ilonka. The Twelve Days of Christmas. Harper. 1949. Kay, Helen. One Mitten Lewis. Lothrop. 1955. Krauss, Ruth. The Big World and The Little House. Harper, Crest. 1949. Krauss, Ruth. The Growing Story. Harper. 1947. Krum, Charlotte. The Four Riders. Follett. 1953. Lansdown, Brenda. Arithmetic For Beginners. Grossett. 1959. Lawrence, Ann. The Clown's Clock. Fideler. 1946. Leaf, Munro. Arithmetic Can Be Fun. Lippencott. 1949. Malter, Martin S. Our Largest Animals. Whitman. 1958. Malter, Martin S. Our Tiniest Animals. Whitman. 1955. Marino, Dorothy. Edward and the Boxes. Lippencott. 1957. McClintock, Mike. A Fly Went By. Random House. 1958. Meeks, Estherk. One Is The Engine. Follett. 1947. Merwin, Decie. Time For Tammie. Walck. 1946. Minarik, Else Holelund. Little Bear. Harper. 1960. Morton, Alice D. Cooking Is Fun. Hart. 1960. Pine, Tillie S. and Levine, Joseph. The Chinese Knew. McGraw. 1958. Podendorf, rna The True Book of Space. Childrens Press. 1959. Reed, Mary and Oswald, Edith. The Golden Picture Book of Numbers. Golden Press. 1959. Ridgeway, Marian. How Far. McKay. 1949. Russell, Betty. The Big Store, Funny Book. Whitman. 1955. Schlein, Mariam. Shapes. William R. Scott. 1952. Schlein, Mariam. Fast Is Not A Ladybug. William R. Scott. 1953. Schlein, Mariam. It's About Time. William R. Scott. 1955. Schneider, Herman and Nina. How Big Is Big. William R. Scott. 1946.
69



Shapp, Charles and Martha. Let's Find Out What's Big and What's Small. Watts. 1959. Skarr, Grace. The Very Little Dog. William R. Scott. 1949. Slobodkin, Louis. Millions and Millions. Vanguard. 1955. Suess, Geisel T. Dr. McElligot's Pool. Random House. 1947. Suess, Geisel T. One Fish, Two Fish, Red Fish, Blue Fish. Random House. 1961 Suess, Geisel T. The Cat In The Hat Comes Back. Random House. 1957. Thurber, James. Many Moons. Harcourt, Brace. 1943. Todd, Mary Fidelis. ABC 0 and 1 23. McGraw-Hill. 1955. Tresselt, Alvin. Follow The Road. Lothrop. 1953. Tudor, Tasha. Around The Year. Walck. 1957. Tudor, Tasha. 1 is One. Walck. 1956. Watson, Nancy D. Annie's Spending Spree. Vicking. 1957. Watson, Nancy D. What Is One? Knopf. 1954. Watson, Nancy D. When Is Tomorrow? Knopf. Withers, Carl. Counting Out. Walck. 1946. Wooley, Catherine. Two Hundred Pennies. Murrow. 1947. Ziner, Fennie and Thompson, Elizabeth. The True Book of Time. Childrens Press. 1956. Zolotow, Charlotte. Over and Over. Harper, Crest. 1957. Zolotow, Charlotte. One Step, Two Lothrop. 1955.

MATERIAL FOR PRIMARY LEVEL

Equipment and Aids

Abacus

Exhibit case

Acorns

Flannel board and figures

Addition and subtraction flash cards

Flannel geometric shapes-

Animated clock dial

circle, octagon, rectangle, square, triangle

Arithmetic games

Flash Cards

Art supplies

Folding perception cards

Balls

Fraction cards

Bead frames (Demonstration and individual)

Fraction wheel

Beads

Fractional parts (construction paper pies)

Beanbags or ringtoss

Gallon container for measuring

Bingo sets

Hundred chart

Blocks

Hundreds board

Bottle tops

Ice cream sticks

Boxes of crayons

Jack-stones

Buttons

Large counting frame

Calendar

Large fraction board

Cards showing fractions of single objects

Magnet board

Cereal boxes

Marbles

Chalk (full length)

Measuring cup

Checkers

Measuring spoons

Clock dials

Milk bottle tops

Clothes pins

Model of cones and spheres

Colored beads

Musical rhythm instruments

Colored cubes

Number board (From 1-100 and another from 100-200)

Colored straws

Number charts (1 through 100)

Compass

Number fact finders (colored spools on wire hangers)

Counting bar

Number grouping disks

Counting disks

Number line

Counting frame

Number readiness posters

Counting sticks

Cubes of starch and sugar

One-half pint

Cylindrical solids

Pair of shoes, guns, socks, earrings

Demonstration clock face

Paper plates

Dominos and domino cards

Paper clips

I

Egg carton (dozen)

Emery kinaesthetic number cards

I

Paper circles Paper squares

70

r
peg board and pegs
Pencils physical education supplies picture postcards Pint container for measuring pipe cleaners Place value charts Place value sticks (100 sticks) Plain cubes Play clinical thermometer Popsickle stick Quart container for measuring Real alarm clock Real money Rectangular solids
Ruler Scales Scissors Seeds Set of number readiness cards
Shells Slated globe Slide-Hide Calculator

Soda straws Spools Straight edge String Symbol cards (set of 10 number symbols) Tablespoon Tape measure Teaspoon Ten toys for counting (rubber or plastic) Tenpins Thermometer - clinical Thermometer - weather Tinker toys Tongue depressors Toothpicks Toy cash register Toy dial telephone Toy money Toy scales Toy train (to teach ordinals) Wooden spoons Yardstick

Films and Filmstrips
A Day Without Numbers (Wayne University) A Number Family in Addition (Popular Science Publishing Co., Inc.) Addition and Subtraction I (Young America) Addition and Subtraction II (Young America) Addition Is Easy (Coronet) Advancing in Simple Addition (S. V. E) Advancing in Simple Subtraction (S. V. E.) Arithmetic for Beginners (Bailey Films, Inc.) Division Is Easy (Coronet) Helping Children Discover Arithmetic (Wayne University) Introduction to Fractions (Johnson Hunt Productions) Let's Count (Coronet) Let's Measure: Inches, Feet, Yards (Coronet) Let's Measure: Ounces, Pounds, Tons (Coronet) Let's Measure: Pints, Quarts, Gallons (Coronet) Making Change for a Dollar (Coronet) Man and Measure Series (Filmstrip House) Measuring Temperature (Young America Films, Inc.) Measuring Time and Things (Du Kane Corp.) Money Experiences (Curriculum) Money Lessons for Primary Grades (Filmstrip Visual Education Consultants, Inc.) Multiplication Is Easy (Coronet) Parts of Nine (Young America) Parts of Things (Young America) Subtraction Is Easy (Coronet) Telling Time by the Clock (Bailey Films, Inc.) The Calendar: Days, Weeks, Months (Bailey Films, Inc.) The Meaning of Plus and Minus (Encyclopaedia Britannica) The New Elementary Mathematics Filmstrips in Color, Grade 1, Series 1, Directed
by Dr. Bernard H. Gundlach (Colonial) Language of Sets Addition through Nine The Basic Addition Table Subtraction through Nine

71



The Understandings of Ten

Playing with Numbers

The Number Line

Fractions The Number System (Encyclopedia Britannica)

The Teen Numbers (Young America)

The Threes (Popular Science) The Twos in Division (popular Science)

Tick-Tock the Learning Clock (Curriculum)

Time (Modern Talking Picture Service)

Using and Understanding Numbers (S. V.E.)

Using and Understanding Simple Measures (S. V. E.)

Using and Understanding Subtraction (S. V. E.)

Using Numbers (Encyclopedia Britannica)
Counting from 10 to 15 Counting from 15 to 20 Counting from 20 to 40 Counting from 40 to 100

Reading Numbers to 50 Reading Numbers to 100
Writing Numbers to 100 Working with Numbers to 100

We Discover Fractions (Coronet) What Are Fractions? (Instructional Films)

What Is Four? (Young America)

What Numbers Mean (Popular Science)

What Time Is It? (Coronet) Work and Play With Numbers (Eye Gate House, Inc.)

Time and Money Addition and Subtraction Concepts

Work and Play with Numbers 9 and 10

Work and Play with Number 11 Work and Play with Numbers 12 and 13

Work and Play with Numbers 14 and 15

Work and Play with Numbers 16 and 17

Work and Play with Number 18

Work and Play with Number 19

Work and Play with Number 20

Work and Play with Problems

Work and Play with More Problems

Working with Numbers to 100 (Encyclopedia Britannica)

Zero a Place-Holder (popular Science)

72

FOURTH GRADE

UNDERSTANDINGS TO BE DEVELOPED IN FOURTH GRADE
In the fourth grade the average pupil reads and writes nine digit numbers with facility. He counts by sixes, sevens, eights, and nines backwards and forward from 1 to 100. The pupil understands odd and even numbers and their operation and has no difficulty with the use of ordinals through fiftieth. He is able to read and write Roman numerals through eighty and recognizes the existence of the quinary number system (base 5).
The pupil has such understanding of place value and regrouping that he can Perform the operations of addition, subtraction, multiplication, and division without difficulty. The use of the commutative, associative, and distributive properties of numbers with understanding is emphasized. His addition and subtraction operations involve the use of three to five numbers. The fourth grade masters multiplication facts through nines with their reverses, is able to multiply by two and three digit numbers, and divide by a one digit number with a reasonable degree of accurC~.cy. The pupil's ability to estimate answers, find averages, and use signs and vocabulary related to all processes is increased. The pupil becomes acquainted with sets and set terminology. During the fourth grade the pupil develops ability to find simple fractional parts of the group and of the whole object and to compare fractional parts with each other and with the whole object. He has increased understanding of the decimal point in U.S. money.
The pupil understands and is able to perform simple operations with several kinds of units of measure. Many opportunities to become familiar with simple geometric shapes are provided. The ability to make and interpret simple maps, charts, and graphs is developed.
Awareness of problem situations and ability to analyze these situations in terms of the relationship between the known and unknown factors are extended.
75

CONTENTS
I. Numeration
A. Decimal notation 1. Read and write 9 digit numbers.

TEACHING SUGGESTIONS

Collect and read large numbers from newspapers, magazines, house numbers, telephone directory, and other sources.

Make bulletin charts showing pictures of abacus; read and write numbers from abacus.

2

6 5, 3

4

1

2o6

B

n

(1\ ~R

(

l

Use colored strips of paper to illustrate numbers through millions using pocket-type place value frame as with the number 154, 782, 320.

Millions

Thousands Ones

154

782

320

2. Count by sixes, sevens, eights. nines backward and forward from 1100.
3. Extend concept of place value and regrouping.
4. Review set, sets, members of sets, null or empty set, likenesses, differences, less than, more than, same as, equal, one - to - one correspondence, odd, even, compare, and braces.

Arrange numbers according to size; write numbers from dictation using commas to separate.

tJe objects such as grains of corn to illustrate numbers.

Match cards with figures to cards with numbers written out:

I five million, two hundred

II

= fifty-seven thousand, one

5,257,110

I hundred ten

------------

Use hundreds chart, hundred counting frame, and pegboard (See diagram in Grade 3, I).

Count by 6's, 7's, 8's, and 9's to 100 and back to 1.

Play the game "Buzz" using 6's, 7's, 8's, and 9's.

Use place value aids to help children understand the value and necessitY . of regrouping to solve problems.

Help children understand, extend, and continue to use sets.
Let children learn and use symbols: equal to =; less than <; greater than >i'
not equal to ~; not greater than ~; not less than <\:.

'16

Learn symbols < less than, > greater than, =1= not equal,
::I> not greater than, <4: not less than.
5. Use odd numbers and even numbers.
6. Extend concepts of ordinals through 50th.

3=3 5> 2 2<5 4=1=3
Use for counting off for group situations. Use in development of sets. Play games such as "Finding the Odd Number."
Help children to extend their understanding of position as thirty-first day, fortieth person, forty-fifth chapter. However, stress the idea that "th's" end ings are not necessary for the ordinal concept.
"Sit in chair number 7." "This is November 15."

B. Roman numerals
Read and write Roman numerals through LXXX, and recognize D, C, M.

Recognize Roman numerals used in chapters of books, dates on buildings, outlines, Bible, sections of newspapers, clock faces, and V on nickeL

Extend understanding of the value of I and X following and preceding a numeraL

Match cards with Roman numerals and Arabic numerals. Play games such u "Ro-Match."

Write various dates and numbers u Roman numerals.

The volumes of the encyclopedia our group needs today are , , .

The chapter about insects is

.

I wu born in the year

.

C. Numeration in systems with bases other than ten
Become acquainted with base five.

Compare base five with base ten.

Count beyond the square of the base.

Perform simple addition and subtraction in base five.

Base Five

Numeral Word Name

1

one base five

2

two base five

3

three base five

4 four base five

10 flTe one zero base five

11 flv. one, one base five

12 fl.. one, two base five

13 flv. one, three base five

14 fl .. one, four base five

Base Ten

Numeral Word Name

1

one

2

two

3

three

4

four

5

five

6

six

7

seven

8

eight

9

nine

The digits 0, 1, 2, 3, 4 are common to both base five and base ten. The digits 0, 1, 2, 3, 4 mean the same in both systems.

Use 10 symbols in base ten, and 5 symbols in base five.

Group

9

objects,

one

group

of

5

with

4

singles

left

over.

We

write

"14U

n "

and read "one five and four ones," or "one four base five."

34fl ..
+ 23f1Te
112flTe

42f1.. -14f1Te
23u..

77

II. Fundamental Operations With Whole Numbers

A. Addition and subtraction
1. Review. addition and subtraction facts.

Diagnose the need for reteaching any addition or subtraction facts. Let children make and use addition charts and basic sum cards.

front

back

14
@

7+ 7 12 + 2 2 + 12
6+ 8 8 +6 5+ 9 9+5 3 + 11 11 + 3 4 + 10 10 + 4

2. Add 2, 3, and 4 place numbers, regrouping to tens, hundreds, and thousands.

The 14 tells the sum. The 11 tells how many basic addition facts result in the sum of 14 (not including 1).
Use number line to perform addition and subtraction.
Help children increase speed and accuracy in adding and subtracting basic facts through the use of relay games, baseball, race games, detective (see who can find words that suggest adding or subtracting), Follow Me (teacher says, "Add 5 and 3, take away 4, double what you have left, add 2, take away 4 and where are we?"). Pupils hold up number cards with correct answers.

Use number wheels, higher-decade addition game using regrouping as:

25 + 8 =
42 + 10 + 5 =
45 and 9 are
32 + 9 =

21 13
62

Use subtraction questions:

43

How much is 9 from 18? 15 take away 8 is how much? What is the difference between 20 and 12? How much greater is 24 than 1O? 17 is how many more than 9? 16 is how much less than 20?

Develop skill to respond readily to basic facts; add lunch money; keep records of days present and absent; keep records of spending allowances; work puzzles and riddles.

Use place value aids of various kinds to reinforce understandings of regroupings.

Help children recognize when regrouping is necessary and apply the principle of regrouping. When the sum of tens figures is ten or more, change 10 to one hundred and carry the one hundred to hundreds place. Add this one with the hundreds figures.

Practice with exercises such as:

10 tens means

hundred and __ tens

17 tens means __ hundred and __ tens

25 tens means

hundreds and __ tens

78

3. Add columns of 7 one-place numbers, 5 two, three, and four-place numbers.

8 6

.T 1 HITIO TIHITIO

7

72 368 7584

5

59 297 5372

4

48 579 3729

9 3

36 65

2 0/4 783

2867 8072

Thousands 8

Hundreds 0

Tens 7

Ones 0

368 2 1475 2913
8070 1 - 10 (ones)
I'- - 16 (tens)
- - - 19 (hundreds) 6 (thousands)
3682 = 3 thousands 6 hundreds 8 tens 2 oneil 1475 = 1 thousand 4 hundreds 7 tens 5 ones 2913 = 2 thousands 9 hundreds 1 ten 3 ones
8070 = 6 thousands 19 hundreds 16 tens 10 ones or
8 thousands 0 hundred 7 tens 0 ones
Give practice in rearranging horizontally listed numbers to vertical form and adding:
3278 + 459 + 72 + 4685 = 1

4. Extend understanding of Commutative and Associative Laws of addition.
5. Increase skill in addition through the use of mathematical sentences.
.. Extend understanding of the three types of subtraction problems.

Emphasize the Commutative Law of Addition:
= = 6 + 9 15 or 9 + 6 15
Emphasize the Associative Law of Addition:
= (4 + 6) + 7 = 17 or 4 + (6 + 7) 17

Use sentences for finding the variables and comparing addends:

15 + 0 = 32; 0 + 12 = 20
= 24 + 16 = 0; 17 + 23 + [J 50

(42 + 24) > (36 + 12); 66 > 48

(9 + 11 + 23) < (16 + 15 + 30)

43

<

61

(520 + 340) > (520 + 304)

860

>

824

Encourage children to take responsibility for discovering and correcting their own mistakes.

Encourage children to use different methods to find answers to subtraction problems as:
40-32=1 think: 32 + 8 = 40.

Use concrete objects and counters to review the three methods of subtraction:

79

Additive-subtraction problems ask how many more are needed or how much more is needed.
A book costs $.79. Sue has $.55. How much more money does she need?
= .55 + n = .79 n .24 Subtractive-subtraction problems ask how many are left or how much is left.
There were 54 cookies on the table. Tom put 25 of them in a box. How many were left on the table?
= 5 4 - 2 5 = n n 29
Comparative-subtraction problems ask you the difference between two amounts.
Betty weighs 75 lbs. Jane weighs 69 lbs. What is the difference in their weights?
= 69+n=75 n6

7. Subtract 4 place numbers, Use place value aids to develop understanding of regrouping in subtraction

regrouping to 4 places.

using rings to represent groups.

I

o~

2912 2 J It ~ - 1 156 1 146
Thousands
2
1
1

Hundreds 3 1 1

Tens 0 5 4

8. Increase skill in subtraction through mathematical sentences.

Use sentences and equations:
36- 8=0
0-9=41
= 167 - 0 145

N-27=65 82-N=48

Ones 2 6 6

80

9. Develop ability to round off numbers and estimate answers.

Use number line to help in rounding off numbers to nearest hundreds and thousands.
Rounding to hundreds 225

o

100

200

225 would be nearer 200 than 300.

RiHluding to thousands

1500

300 2500

o

1000

2000

3000

1800 would be nearer to 2000 than 1000

Give practice in estimating; help children realize that estimating may give clues to the correct answers:

297 is about 300 2895 is about 3000 698 is about 700

Round to nearest hundreds and estimate sum:

197+393=?

Round to nearest hundreds and estimate difference:

296-192=?

10. Attain proficiency in adding and subtracting as inverse processes; use one to verify the other.
11. Review vocabulary and signs.
12. fucrease skill in the use of zero in addition and subtraction.

Use the number line to review the inverse relationship between counting on and counting off.
13

13 - 5 = 8

o

5

10

15

'----,-----" '---.r------'

8

5

8+5=13

Verify one with the other:

5232

2557

-2675 +2675

2557

5232

+, Make symbols and vocabulary (sum, remainder, plus, minus, -, equms,

difference, addends) meaningful to children through experiences with their use in various situations.

Review the use of zero, showing that it is a place holder and indicates "not any." In the number 302 we have 3 hundreds, no tens, and 2 ones. Since there are no tens, we use zero to hold the tens place so we will not read "32."

By the use of concrete examples, strengthen the understanding of the generalizations involving zero.

When zero is added to or taken away from any number the result is the original number:
= n + 0 n; 0+ n = n; n-O=n
When any number is subtracted from itself, the result is zero:

n-n=O
In subtraction think of a three-place sum with a zero in tens place as so many tens and so many ones:

81

;

13. Continue practice in methods of checking addition and iubtraction.

7 9 17 8 I' , use 1 ten of 80 tens - 3 6 8 and leave 79 tens
439 When this understanding is fully developed, move to problems with two or more zeros in the sum:
5 9 9 10 ~ II II fJ -4277 1723
Help children become conscious of the importance of checking their work.

365

186

179
-ure

+ 179
~

1554 273 116 547 618 1554

j'ifP
18

f80 I
8357

65 26

84 -----uf ones

/!:i?8 21 tens

56 228

223

A magic square is a good device to use in which the child adds vertically, horizonally, and diagonally to get the same sum:
15 15 15

15 15 15

14. Become acquainted with union and intersection of
sets and their symbols
(Un> used to indicate
operations performed on
sets.

Teach the union and intersection of sets. The fundamental operations of addition, subtraction, multiplication, and division give answers to problems. Similarly, union and intersection are the main operations with sets. The symbol for union is U (sometimes called cup), and the symbol for intersection is n (sometimes called cap).
The union of sets is the combining of sets as:
Set A = {I, 2 ,3 ,4, 5}
= Set B {3, 4, 5, 6} = A U B {I, 2, 3, 4, 5, 6}
The union includes all the members of the sets A and B.
= R the set of whole numbers which makes 4 + 0 < 8 a true mathe-
matical sentence as: Set R = {I, 2, 3}
= S the set of whole numbers which makes 5 + 0 < 12 a true mathe= matical sentence as: Set S {I, 2, 3, 4 , 5 ,6}
\
The union includes all the members of the sets Rand S.
= R U S {I, 2, 3, 4, 5, 6}

In the intersection of sets we find a set that includes only the members common to both sets as:
= Set A {I, 2, 3, 4}
= Set B {3, 4, 5, 6}
A n B = {3, 3}
Since Set R = {I, 2, 3} is the whole numbers which satisfy 4 + 0 < 8 and Set S = {I, 2, 3, 4, 5, 6} is the whole numbers which satisfy 5 + 0 < l2,

82

15. Identify subsets

then R n S = {I, 2, 3}
Empty or null sets { }, or <p, are a result of two sets that have no members in common such as:
Set C = {1, 2, 3}
= Set D {4, 5, 6}
Therefore the intersection of C and D or C n D = { } or <p.
= Since Set W + {1, 2} is the set of whole numbers which satisfy 3 0 < 6.
= and Set V {3, 4, 5, ...} is the set of whole numbers which satisfy
2 + 0 > 4, then W n V = { } or t).
Use other examples such as:
= (1) Set P {3, 5, 2}
= Set Q {7, 3, 5}
Does Set I ~ Set Q? Why?
What is the Ulll, 1 of Set P and Set Q?
What is the intersection of Set P and Set Q?
(2) Is Set T = {3, 4} a solution set of whole numbers which will satisfy
A + 0 = 7?
Is Set Y = {5, 2} a solution set of whole numbers which will satisfy the above equation?
(3) Set Z = {6, 4, 2}
Set H is the set of all even numbers less than 10. Which is correct? Set Z = Set H or Set Z oF Set H?
Sometimes we have sets within a setSet A = {1, 2, 3, 4, 5, 6}
= Set B {2, 3, 4}
Set A = {all girls in school}
= Set B {all girls in a class}

B. Multiplication and division 1. Review 2's, 3's, 4's, 5's, 10's

Review facts previously learned by: Children making their own flash cards.

front
8 X4

back
xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx
32

front
4
X8

back

0000

0000

0000

0000

0000

0000

0000

0000

32

83

---------------
Using number wheels such as:

Making multiplication charts

x0 12

89

Jo
3~rK-' 11
----w- 2 I

0 0
0

0 1
2

0 2
4

3O 6 .~ 04_5100_~102iI~1~4_o 8_t r109-8--

-_3_ _T: _o
4 I0

-_3 1I-6_-I._9 ._12_.1_5_1_1_8_1_~24 4 I 8 I12 1jt\20 I 24 I 28 32

I2-7-
36

5 I~= 5 I 10 \ 15 12025 30 f3540 45

6 0 6lulwl~ ~ H a ~ 54
---7I1--0- ---7-~--- H--nrl2-8- ~ a ~ w 63
8 0 8 H ~ ~ ~ W M 72

-9-iO-9- 18 27 '\3"64554163 72 81

2. Increase understanding of relationship of multiplication to addition.
3. Learn 6's, 7's, 8's, and 9's.

Using flannel board and hundred board to show the possible arrangements of equal groups as, 4 equal groups of 2 each:

o0

4 X 2 = 8 or 4 two's are 8

00

4X2=2+2+2+2=8

00

2X4=4+4=8

o0

Using sentences, as:
= 8 X 5 0;
= o X 6 30.

= 7 X 0 35;

Help children to see the basic facts of multiplication as short-cut method of adding equal groups.

12

12

12

X4

12

"""48

12

48

Through the principle of interchanging factors (Commutative Law of Multipli cation) the children can see they have already learned several facts about 6's, 7's, 8's, and 9's.

Strengthen the child's understanding of the Commutative and Associative

properties of multiplication.

Commutative

8X7=7X8
= 9 X 6 6 X 9

84

-

Associative

(6 X 5) X 8 = 6 X (5 X 8) (8 X 6) X 4 = 8 X (6 X 4)

Use the number line to demonstrate factors:

35's

555

,...--A--..,...--A--.. ,...--A--..
.................. ..

o

5

10

15

Make charts of pictures, drawings, and cut-outs to show facts and their reverses.

4. Begin practice in factoring

Continue to use flash cards, multiplication charts, number wheels, flannel board, and sentences to learn new facts.

Let children find all possible combinations of factors that will give a certain product such as factors of 36:

2 X 18; 4 X 9; 3 X 12; 2 X 3 X 3 X 2.

36 X 1; 6 X 6;
2 X 9 X 2;

F'actors of 24 are 6 X 4; 2 X 12; 24 X 1; 3 X 8; 3 X 2 X 4; 3 X 2 X 2 X 2.

5. Introduce prime numbers

Help children to see that a prime number has as its factors only itself and 1.

3 =3 X 1 2 =2 X 1 5 =5 X 1

= 7 7 X 1
ll=llXl
13=13Xl

Extend understanding of sets by writing mathematical sentences for exercises such as:

The set of whole numbers which are factors of 30.

6. Strengthen understanding of generalizations concern ing zero and one.

Any number multiplied by zero gives zero:

o X n = 0;

= n X 0 0

Any number multiplied by one gives the original number:

1 X n = n;

n X 1= n

7. Review dividing by 2, 3, 4, 5; learn dividing by 6, 7, 8, and 9.

Review facts previously learned and learn new ones through the use of:

Individual flash cards;

front

back

12 /48""

4

85

Number wheels;

49

48

Contests, number line, magic squares and circles, flannel board, and place value pockets. Games such as:Quizmo, Quest-O-Light, Buzz, Bingo, relays, and baseball.
Sentences such as:
= (12 + 4) + 2(10 + 5) 0
Grouping objects such as:
Place 16 buttons on a table and arrange in 2, 4, or 8 equal groups. What divisors are being used?
Show equal groups by the use of narrow pieces of paper cut into rectangles, cIrcles, and squares.

8. Increase understanding of relationship of division to subtraction.

Show that division is a short-cut method of subtracting equal groups.

48
-12
36
-12 ~ -12
l2
-12 --0

4 12 / 48

9. Review generalizations concerning one and zero.

= Any number divided by 1 results in the original number: n -;- 1 n = Any number divided by itself results in 1: n -;- n 1 = Zero divided by any number results in zero: 0 -;- n 0
Use problems from school, home, and community experiences.

10. Extend understanding of multiplying involving regrouping and zero.

Review place value in multiplying.

175

= 5 X 5

25

X5

= 5 X 70 350

875

= 5 X 100 500

875

Help the child to see that he multiplies by tens just as he multiplies by oDaTo show that the product means tens, write a zero in the ones place.

35

32

X20 X30

700 960

86

11. Begin multiplying by two digit and three digit num bers.

Help the child to see that he multiplies by hundreds just as he multiplies by
ones and tens. To show that the product means hundreds, write 0 in the ones and tens place.

215 X 200
43,000

643 400
257,200

Help children to see that the number of ones becomes the number of tens when using ten as a factor.

36 x10
360

1 ten X 36 ones = 36 tens or 360 ones

Help them recognize that multiplying by a two-place number consists of multiplying by a one-place number and a multiple of ten.

16 X 13
~
160

= 3 X 16 48
10 X 16 = 160 . 208

208

45 X 27
3I5
900
1,215

7 X 45 = 315 20x45=900
'1,215

Help them understand position and meaning of factors and product.

12. Extend dividing by onedigit number with and without remainders.

TIHITIO
I; X \i I~ ~:~~~~

1

81

1
1

0l)O 8\6r1

Partial

3 6 1 2 I J product

5 5 I3 I8 I6 Product

Use concrete and representative materials to develop the concept of remainders as:

Increase skill in multiply. big and dividing as inverse P1'Ocesses, one used to veri-
" the other.

We have 31 apples. We want to put 6 apples in each box. How many boxes can we fill? How many apples are left?

How many 6's in 31?

5 boxes
6/31
30
-1 apple left

Use objects arranged in groups to enable the child to see, through counting, the groupS of tens and ones left over.

Use problems in which the child knows the size of the equal groups and must find the number of equal groups and the remainders.
Use practical drill as: 52 -;- 6 = 0 and 0 remainder 79 -;- 9 = 0 and 0 remainder

Make children aware of the inverse relationship between multiplication and division. It is useful in computation and problem-solving.

87

.-------------------

= = 7 X 6 42; 42 + 7 6; 42+6=7

88

8"

80

88

The combined group is divided into equal parts.

7 / 616

X7

The combined size of the equal parts should

560

616

equal the size of the original group.

56

56

14. Practice regrouping of the product when necessary in division.

Reinforce children's understanding of the necessity to regroup within the product. The child can determine whether he is dividing into tens, hundreds, or thousands by the factor so he will know where to begin writing his answer.

Use place value aids to clarify and strengthen understandings when needed as: how many 3's in 633?

2 hundreds 1 ten lone 3 / 6 hundreds 3 tens 3 ones
= 633 600 + 30 + 3

211 3 / 633

200 threes in 600 10 threes in 30 1 three in 3

How many 4's in 60?
Tens I Ones
I1 I 5
4/ ~ 0
;\~
I

How many 8's in 584?
I Tens Ones
I7 3
8/58 4 56

15. Attain skill in determining zeros in the factor.

Review previously learned skills.
Help children use zeros as they need them to keep each part of their answer in its proper place. Sometimes they are needed in either or both tens and ones place.
Emphasize that in the first example below there are no ones; therefore a zero is placed in the ones place in the factor.
[n the second example there are no tens and a zero is placed in the factor.
In the third example there are four tens, but there are not enough tens to make a group of seven; a zero is therefore put in the tens place in the [actor, and the tens are regrouped with the ones.

610
10
600
b / 3050 3000
----so 50

401 -1
400 6 / 2406
2400 -6-
6

507 -7
500
7 /3549
3500
"---:49
49

16. Understand division process through the use of partial factors.

Provide opportunity for children to gain a fuller understanding .of 0.. division process through the use of partial factors written to the nght
the product.

88

17. Begin to use tests for divisibility.

The followmg examples are arranged in order of increasing difficulty.

known factor

7 I 3549 I-<-!-p7ro0duct 490 I

71~1400

320805091
4912215091 49

400

30

Partial product

7

7491 100 700
44991 7 507

507 unknown factor

7 I 35491500

3""54090 1 49

7

507

Encourage children to estimate the unknown factor before working out the problem as there are about 600 fives in 3050.

A number is divisible by 2 if the number represented by the right-band digit is divisible by 2.
A number is divisible by five if the right-hand digit is 0 or 5.
Show these in many cases and let children form generalizations.

18. Find averages.

Help children gain an understanding of the meaning and method of finding average. To find the average of a group of numbers they have to be combined and redistributed into equal groups. Sometimes the result is the same as one of the numbers but not always.

Provide experiences that give practice in finding averages as: height, weight, daily attendance, amount saved per week, cost per book, number of books read per month, amount child contributed, scores on a game, number of pupils in a grade.

Relate meaning of average to locating mid-point between larger and smaller numbers.

What is the average between 21 and 43?

21

43

------so- I . . . . . I . . . . . . _,_.-;;I;:-'-.~""'-=--'-'-~--''='="1..:....:..-'-

20

40

50

19. Estimate answers using rounded numbers.
20. R e v i e w vocabulary and signs; multiplication, factors, p r o d u c t s, division.
known factor, +, x.

Give opportunity for children to estimate numbers, a pmcess which serves as a check on the answer and helps in problem solving.
Example: Read the problem, estimate the answer, then find the exact answer.
Mr. Smith bought four tires at $17.00 each. About how much did they cost?
Think: 17 is nearly 20. 4 twenties would be 80. Since 17 is less than 20 the estimate might be 70.
Estimate ------------- $70 Exact _....... ---------- $68
Stress the use and understanding of basic vocabulary and signs associated with multiplication and division as needed in the classroom.
Teach children to read and understand various types of mathematical sentences using the signs needed.

9X6=54

= 5 4 + 9 = 6
+ + (3 2) - (2 1) 0

5

3 =2

6 X (8 + 2) = 0

6X 4 =24

89

III. Fundamental Operations With Fractions

A. Common 1. Find 1/ 2, 1/3, 1/4, 1/5, 1/6, 1/8 of the whole object and of the group.

Review common fractions previously learned.
Discover parts of a whole by dividing and sharing paper, plyboard, cardboard, ribbon for badges, and string for yo-yos.
Talk about whole, half, quarter notes in music.
Extend understanding that a fraction represents a part of a group, compares one group with another, is part of the whole when the group is divided into smaller amounts.

2. Compare fractions with the whole object.and with other fractions; learn terms: numerator and denominator.

Learn to read and write fractions through sixteenths.

Find that

2(3'
the

n3(u,1m' e5r/a6l,

2( H of below

a whole object; the bar called

symbolize with a har between; learn denominator shows the number of

equal parts in the whole object or group, and the numeral above the bar

called numerator shows the number being considered.

Learn terms numerator and denominator.

Show that as the denominator increases, the size of the fractional part decreases, if the numerator remains the same.

For practice, to extend this concept, give pupils circles, squares, tangles with parts shaded and let pupils name the fractional part.

.l..

I

40

2"

...
1,. 3
90

3. Fractions written in different form.

Show that when the numerator and denominator are the same the fraction is equal to the whole.

= = 3/ 3 = 1; 4/4 1; and 6/ 6 1

Use mathematical sentences to compare fractions.

+ > + 1/3

2/ 3

1/4

2/4

< < 2/ 8 +3/8 <5/ 7

1/4

0

3/4

Arrange in order of largest to smallest:

1/2,2/3,1/8,3/4,1/6,7/8

in a group of 42 children, '/ 6 were below age 10. How many were less than 10 years old?

Use flannel board, cardboard, pie tins, paper plates, fraction board to show relationship to whole.

Note: fraction board made with wooden frame into which wooden or cardboard strips can be slid. Each fraction family can be a different color.

WHOLE

1/2

I

1/2

1/3

I

1/3

I

1/ 3

1/ 4

I 1/4

I 1/4

I 1/4

I I I 1/5

1/5 I 1/5

1/5

1/5

I I I I I 1/6

1/6

1/ 6

1/6

1/6

1/6

I I I I I 1/11 1 1/8 1 1/8 1/ R 1/ 8 1/ 8 1/ 8 1/ 8

= Show that fractions may have different number names as: 41 6 = 2/ 3 ; 214 = 1/2;
2/ 6 1/S'

4. Introduce like fractions
and begin addition and subtraction.

III]
I
.3

Learn that fractions are like fractions when the size of their equal parts is the same or when they have the same denominator.

Show that in adding like fractions the numerators are added and the denominators remain the same.

+ 218 + + 111 + 5 / 9

3/8 = 5/8

1/1
0

=

21 1
8/ 11

=

41 1

91

.----------------

B. Decimal
1. Continue using point in U. S. money.

Show that in subtracting like fractions, subtract the numerators and the the denominator remains the same.

o - = 5/6 - 4/ 6
= 3/ 8

l/S 4/8

Stress the idea that the decimal point is used to indicate which numbers are whole dollars and which are parts or cents. We say "and" to separate dollars and cents.

Let children solve problems involving the use of U. S. money expressions.

= Use place value pockets, flannel board, and hundred board to show that 1
whole 10 tenths, with ten markers placed in the tenths place. Show that
~ of ten tenths = 5 tenths as we put five markers in the tenths place and

write it .5.

= = 1/2

5/10

50/100

= = 1/2
10/ 10

.5 .50
= 1.00

100/ 100 = 1.00

Make a chart to show the comparison of common and decimal fractions of money:

Common

1 quarter
= 3 quarters = 1 half-dollar

1/4 dollar 3/4 dollar 1/2 dollar

1 nickel

1/20 dollar

1 dime

1/10 dollar

1 cent

= 1/100 dollar

9 cents

9/ 100 dollar

Decimal

or $ .25

or

.75

or

.50

or

.05

or

.10

or

.01

or

.09

2. Use fundamental operations in dealing with money.

Emphasize that the numbers to the right of the ones place are the decimal parts of the whole as divided into tenths, hundredths, etc.
= 100 pennies $1.00 or a whole
1<1 is 1/100 or $.01.
Show regrouping in adding and subtracting dollars and cents:

Ul
tIl.g
~~eS ~ ;a ~ $46.73 +31.29
$78.02

$57.41 -24.56
$32.85

Give practice in solving problems in which the process is adding or subtract ing money. Help pupils see that in adding and subtracting money, the compUtation is performed as if they were dealing with whole numbers. The onlY thing different is to put in the point and put the dollar sign for the cents sign. 172 cents is the same as $1.00 and 724 and may be written $1.72.
Give practice in changing horizontal to vertical:
$16.00 + $9.75 + $.89 + $184.15 + $49.60

92

IV. Relations
A. Equivalence
B. Inequality C. Order of decimals

These relations have been stressed throughout tne gwae unaer varIOUS topics.

The Commutative and Associative Laws of addition and multiplication offer excellent opportunity to develop this understanding as:
a+b=b+a 3+4=4+3
= a b = b . a
34 43
(a + b) + c = a + (b + c) (3 + 4) + 5 = 3 + (4 + 5)
7 +5=3+ 9
(a X b) X c = a X (b X c) (2 X 3) X 4 = 2 X (3 X 4)
6 X4=2X 12
Through the use of the number line children should gain skill in making comparisons as:

398 ".. 297
398 > 297 < 297 398
Help children to see where the ones place is in a number; to understand that the first place to the left of the decimal is ones place, the second place is the tens place, and the third place to the left is the hundreds place; but the first place to the right of the decimal is the tenths place and the second place to the right is the hundredths place.

"'0"
i=:
ctl
:':"s ....0c.:.l

"'0"
Q) J-t
"0
:i=:s:
..c:l

.'iQ.=".:.)

'Q")
i=:
0

3333

:S

"0

Q)

..'."..c:l

J-t "0

.i=: ....Q)

:i=:s:
..c:l

33

Help children to understand that to find out how much larger or smaller one number is than another, one uses subtraction. Then help them to see that to find out how many times as large one number is as another, one uses division. Comparing two numbers by division means the same as finding the ratio of two numbers, the ratio of 15 inches to 3 inches is:

3 I 15 or 5
Rate involves comparing one measure with another. 4 packages of chewing gum cost 15'1. We state the rate as 4 pkg. for 15'1. The rate is symbolized by the ratio 4/15, This symbol does not represent a fraction but represents a rate.

Proportions are equivalent ratios: If 15</ will buy 4 pkgs. of gum, how many packages will 30., buy?

15
4

=

30

4

nor 15 =

n

4

30 or 15 =

30 nor

n30

= --1:5r-

This proportion expresses the problem.
Per cent can be shown using frame work of proportion when solutions to per cent problems are found by using equations of ratios. 30 of the 45 pupils enrolled are present. What percent are present? Since 100 represents all pupils enrolled we write it as:
= 30 n
'45 100

93

v. ~easurernent

A. Standard units of measure.
1. Review 'and extend understandings of dozen, inches, feet, yards, pints, quarts, gallons, pecks, bushels, ounces, pounds, calendar, weeks, months, years.

Review and extend understandings of measurement. Review all tables of measure. Stress that the type of unit used depends on the thing to be measured.
Let children make displays in the arithmetic center of the room of various measuring instruments.
Estimate and measure different objects; lengths; heights; areas, using foot ruler, yardstick, and tape measure as: children's heights, bulletin board for display arrangement, book shelf, tables, windows, floor space for games or relays, desks for group situations, and playground areas such as baseball diamond:
= Home base to pitcher 60 feet. = First base to home = 90 feet.
Second base to first 90 feet.
Third base to second = 90 feet.
Children can weigh themselves, make comparisons, and find averages. They can bring in objects of different weights, weigh them in the classroom,. and get experience in changing ounces to pounds and pounds to ounces. How many ounces in % pound?
How much will Jh lb. of candy cost at $.65 a pound? If a bag of peanuts weighs 24 ounces, what is its weight in pounds?
Make a list of things their parents buy or sell by the peck, bushel, or quart. Bring in and demonstrate with peck and bushel containers.
Give experience in solving problems involving dozen, make comparisons of things bought by pounds and by dozens; collect and use various sizes of containers to show relationships.
Use water with a little coloring in it to show that:
= 2 cups 1 pint
2 pints = 1 quart
4 quarts = 1 gallon.
Make a chart of things we buy in 4 pint, Jh pint, pint, quart, gallon.
Make up problems as: How many Jh pint glasses of milk can be served from 1 quart? If lh pint of orange juice costs 8 what will one pint cost?
Make cookies and candy using recipes involving measurements. ~stimate distances: (a) from home to school, (b) longest distance walked in one day, (c) traveled at one time, (d) to nearest town.
Read odometers.
Make up own tables of measures.
Learn and use the basic terms of measurements and their abbreviations 81 needed.
Review and continue the study of the calendar: days in week, days in month. some 28, 30, 31; leap year, special days, birthdays.
Spell and write days, months, holidays.

2. Read and write hours and minutes; a. m., p. m., noon, midnight.

Children can read and write time: 11:05 o'clock, 3:45 o'clock. 2:30 o'clocJE.
Review different ways to express time as: 8:30 is half-past eight and 3. is three forty-five or a quarter of 4.

94

3. 'Build foundation for modular arithmetic.

Write: 80 minutes as 1 1/ 3 hour. A. M. is 12:00 midnight to 12:00 noon. P. M. is 12:00 nOon to 12:00 midnight.
Study time schedules for school day; plane; train; bus; radio network; etc.
Use circular number line to help children understand the addition and subtraction of hours.

12

= 11:00 + 3 hours 2:00

9:00 + 6 hours = 3:00

= 2:00 - 3 hours 11:00

4. Read thermometers; read and write temperature.

6
Bring in and share clippings of temperature readings over the nation and make comparisons with their own area. Review that temperature is measured in degrees (0); water freezes at 32; boils at 212. Discuss the effects of temperature of the kiln in firing clay. Make and read models of thermometers. Register the temperature of the body, of an oven,and of the air inside and out. (Use cardboard and red and white elastic to make a thermometer.) Give experience in reading and recording various temperatures. Discuss thermometers used by doctors.

100
90 80
70 60
50 40 30
20
10 0
10 ~o

5. Change measures from one unit to another; like units can be added and subtracted.
I. Count and make change.

Build understanding of and give practice in changing units of measure

and in adding and subtracting like units:

= 2 lbs. 24 OZS. 3 lh lbs.
5 ft. 16 in. = 6 I/S ft. or 76 in.
= 400 lbs. 0 ton. = 1/8 lb. 0 OZS.
27 in. = 0 yds.

1/.. doz.eggs = 0 eggs.

7 yda. = 0 ft.

3 ft. 6 in.

4 lbs. 9 OZS.

+2 ft. 5 in.

- 2 lbs. 7 OZS.

5 ft. 11 in.

2 Ibs. 2 ozs.

Provide opportunity for children to make their own change from lunch, stamp, and milk money; count dimes, quarters, nickels, and pennies to see the relationship to the dollar; discuss earnings, allowances, .savings, and the spending of money.

95

Make up and solve problems using these experiences with money.
Example: Bill bought a bag of candy for 27tJ.. He gave the clerk a half-dollar. How should the change be counted? Say 27, 28, 29, 30, 40, 50. What coins did Bill get? 3 pennies, 2 dimes.
Example: Count change from one dollar for: 72, 27, 98tJ, 464, 33.
Count change from a quarter: 19, 13, 23, 8.
Count change from a half-dollar: 36, 144-, 31, 27, 439'.
Make chart to show how coins may be written, their equivalent value, and relationships:

B. Metric-discover metric system as found on rulers and yardsticks

1 $ .01 penny

5

.05 nickel-S pennies

10

.10 dime-2 nickels--10 pennies

25

.25 quarter-S nickels--25 pennies

50

.50 halfdollar-2 quarters--5 dimes--10 nickels--50 pennies

100</ 1.00 dollar-2 half-dollars-4 quarters--10 dimes--20 nickels -100 pennies

Acquaint children with the metric system by having available rulers and yardsticks for them to examine.

VI. Geometry
A. Review previously learned geometric concepts

Recognize and be able to identify geometric forms.
Know characteristics: rectangle has four square comers and four straight lines; square is a rectangle with four equal sides.
Collect objects as cans, boxes, and cartons to illustrate geometric forms.
Make and use models of geometric figures from paper, cardboard, wood, clay, papier mache, and wire.
Use pegboard and string or rubber bands to demonstrate figures and to compare the perimeters of two different figures having the same area.

D

:6
D


List and measure things in the classroom that have geometric shapes: windows, doors, table and desk tops, globes, books, bulletin board, and balls.
Find the perimeters and areas of squares and rectangles.
96

B. Learn about parallel lines and right angles.

Observe lines that are parallel and angles that are right angles. Help children to know that parallel lines never meet when extended but stay the same distance apart. Squares and rectangles are figures which have parallel lines and right angles.

_ _....~:...

~ parallel lines

L("-- right angles

Use pictures of playground, roads, and ball fields to see parallel lines and right angles. Use figures collected and made to name parallel lines, number of sides, and right angles.

VII. Charts, Graphs
A. Learn to make and interpret simple graphs and charts.
B. Extend ability to draw maps.
C. Begin scale drawing.

Review concepts developed in earlier grades.
Let children make and keep simple graphs and charts ob temperature readings, daily or weekly grades, saving stamps purchased, scores for teams, and daily attendance.
Give experiences in drawing maps and reading directions, elevations, and distances from maps, charts, and graphs.
Make a map of the neighborhood and estimate distances to various places from school.
Use road maps for practice in measuring distances; use strips of paper to compare distances on map to scale.
Develop understanding that any small distance can stand for any number of miles or feet. Maps of the same area can be drawn to different scales.
Children can make simple scales in connection with maps.
Take two maps, compare size, and practice reading the scale used in both; then measure distances between places.
start scale drawing with small areas as desk or table top. Decide on scale to use. Then make scale drawings of rooms in their homes, schoolroom, ball field and playground areas, using '14, :Jf.l, or 1 inch equal to a foot, depending on the size of the paper used.

VIII. Problem Solving
Solve various type problems. 1. Compare two different
groups, each of known size.
2. Add a group of unknown size to a group of known size to produce a new group whose size is also known.

There were 695 pupils in our school last year. There are 721 pupils this year. In which year were more pupils enrolled? How many more? How many less did we have last year?
A = 721, B = 695, A > B, B < A
Ann's father had some potatoes in his warehouse. He bought 243 bu. from John's father. If he now has 596 bU., how many were in his warehouse?
n+243=596

';I. Add a group of known size
to a group of unknown size to produce a new group
whose size is also known.

There were 379 boys in camp. Another group of boys joined them, making 509 boys in camp. How many boys joined them?
379+n=509

97

4. Separate into equal groups a group of objects whose number is known.
5. Find the number of objects in each sub-group when the total number of objects and the number of sub-groups are known.
6. Find a group of unknown size which is taken from a group of known size when the remainder is known.
7. Find how many items there were at first when the number sold and the number left are known.
B. Extend understanding of the steps of problem solving.

Mary had 96 apples. If she put 8 into each of several boxes, how many boxes did she use?
OO+8=n
Bill had 108 marbles. If he divides them equally among 12 boys, how many marbles will each boy have?
108 + n = 12

Mr. Brown owned 972 horses. He sold some of them, leaving him 317. How many horses did he sell?
= 972 - n 317

Mother baked some cookies to sell. She sold 278 cookies. If she had 159 left, how many cookies did she bake?
= n - 278 159

Emphasize careful reading and thinking about a problem before choosing the process to use.

Realize that there is more than one approach to a solution, and use various techniques involving adding, subtracting, multiplying, and dividing.

Help children develop the ability to recognize the situation in a problem to see if they need to find a total, or a remainder, to obtain the number of units in a given group, to divide a whole into groups, or to compare one thing with another.

Help them to interpret the situation and change it into symbols.
= Give practice in using the signs: to indicate equivalence; :/= is not equiva-
lent; < less than; > greater than; :1>. not greater than; <t: not less than and
equations to relate known facts to known processes, as:

8<0<11

4+ 0 (16 X

+ 0)

6++7

A
=

= 14 15

= <D+6)X3=24
0+ A 9

Allow for estimating outcomes and checking results.

Provide opportunity for children to develop problem solving skills through the use of:

Illustrations, diagrams, and drawings. Making problems from experiences. Making charts. Selecting the correct process or processes. Using dramatization. Answering specific questions. Giving solutions mentally. Centering a series of problems around one theme. Writing the number question for a problem. first using a word form, then a number sentence. Estimating answer previous to computation. Comparing answer with estimates. Deciding what facts are missing. Reading and discussing problems. Telling problems in own words. Finishing incomplete problems. Finding facts that are important to solutions. Using concrete materials and devices. Making analogous problems. Using data from bulletin board to make problems.

98

FIFTH GRADE

UNDERSTANDINGS TO BE DEVELOPED IN FIFTH GRADE
The fifth grade pupil's understanding of the ten-ness of the number system is essential. During the development of the program the pupil gains an understanding of cardinal and ordinal numbers, acquires a more usable knowledge of Roman Numerals and is able to compare their composition with that of the Arabic Numerals, becomes more familiar with sets and the language associated with them, and learns how numbers are used with systems of numeration other than the base 10 to which he has become accustomed. He becomes aware of the importance of regrouping in various instances, and recognizes that it is essential to be able to use the four fundamental processes - addition, subtraction, multiplication, and division with more ease and understanding. An intensive study of fractions which demonstrates an appreciation of their meanings, function, and essential use is stressed at this level. The program includes ratio and its use in expressing rate and comparisons. The pupil acquires competence in using various measures and begins to realize there is another system of measure - the metric system. By the end of the year the fifth grader has developed skills in comprehension of geometric forms, recognizes various curves and solids, understands angles and rays, knows how to find area, and knows the meaning of volume. He is capable of using graphs, charts, and scales and of finding solutions to many problems. His ability to use with meaning the language of mathematics has been extended.
101

CONTENT

TEACHING SUGGESTIONS

I. Numeration
A. Decimal notation 1. Understand number!' to 9 places; read, write, and use.

Review place value as taught in previous grades, use charts such as:

Thousands

Ones

"'0"

(l)

r-.

"0

l::
..:c::l

til
..l(.:l.:),

'"Q)
l:: 0

I ! 5

6

1

"'0"

r(-l.)

"0 l::
..:c::l

'"s::
..(.l.),

rn
(l)
s::
0

I I 2

9

6

Ask such questions as:
In the number 561,296 How many tens are there in the number? 56,129
How many hundreds in the number? 561
How many thousands in the number? 5612
Emphasize the building of our system

= 10 = 100
= 1,000 = 10,000

10 X 1 10 X 10 10 X 100 or 10 X 10 X 10 10 X 1000 or 10 X 10 X 10 X 10

Show how to place commas between periods in large numbers.

Explain that large numbers are read by beginning at the left and reading toward the right:

In 256,375,824

The 2 means 2 hundred million 200,000,000

The 5 means 5 ten millions 50,000,000

The 6 means 6 millions 6,000,000

The 3 means 3 hundred-thousands 300,000

The 7 means 7 ten-thousands 70,000

The 5 means 5 thousands 5,000

The 8 means 8 hundreds 800

102
'I
1",

2. Continue counting forward and backwards by 2's, 3's, 5's, 10's.
3. Extend concept of ordinals through hundredths.
" Understand, read and write decimals; use in easy addition and subtraction.

The 2 means 2 tens 20
The 4 means 4 onei 4
200,000,000 50,000,000 6,000,000
300,000 70,000 5,000 800 20 4
256,375,824
Make counting meaningful and interesting by employing the unusual. For example, use such exercises as:
Count by 2's to 120 starting from 42.
Count by 3's to 150 starting from 51.
Count backwards by 5's to 25 starting from 95.
Count backwards by 3's to 27 starting from 93.
What number comes fifth from 2097?
What number is between 395 and 397?
Such exercises can be scattered throughout the day as mental arithmetic to keep children thinking even after desks have been cleared for a break, or for lunch.

To distinguish between cardinals and ordinals such illustrations as the following are helpful:
o0 0 0 0 0
123 456
One may say this is the the third block. Point to it. One may also say "3 comes after 2, 5 comes after 4," showing order.
In order to show what the numeral represents indicate that 3 means 3 objects.
In the decimal system of notation a number is ten times larger than the symbol which is immediately to the right of it. The ones place is the center of the system.

hundreds

tens

ones

tenths hundredths

11.---_11.....--~_I~._=_I_1

In the number 88.88, each 8 is ten times larger than the one to the right of it. The number to the right of a decimal point is less than a whole number.

Decimals must be related to common fractions in order to be fully understood. Show children that
= .50 2 (use dollars first) .5 = lh

.95

-

95 100 -

19 X 5

19

5

20 X 5 =2QX S

19
2QX 1

19 _ _ 0 20

The preceding example shows that a decimal fraction is changed to a common
= fraction by using the Fundamental Theorem of Arithmetic such as 5/5 = 1;
10/10 1.

103

Contrast the ease of adding amounts of money written as decimals to that of adding the same amounts as common fractions:
$ .50 .75 .10 .25 .35
. $1.95

B. Roman numerals
1. Extend concept of Roman numerals beyond LXXX; understand meaning of L, C, D, M and their combinations.

Review symbols for Roman numerals
1=1
V=5
X= 10
= L = 50
C 100
D = 500 M = 1000
Explain that a 1 placed in front of another of the symbols subtracts one from it:
IV=4,IX=9
In like manner, a smaller symbol placed in front of a symbol for a larger number indicates subtraction:
= XL 40, X (10) is subtracted from L (50)
Also, if the symbol for a smaller number is placed after another symbol, it indicates addition.
= = LX 60 ex 110
Practice exercises for increasing concepts of Roman numerals.

Place on board: This month is VII. This year is MCMLXII. The lesson is on page LXXIIL Please find page; write month and year on your paper.

Find chapters in books. Find chapters in the Bible in Psalms. Read old clocks with Roman Numeral faces.

C. Other systems of numeration.

Read dates on buildings.
Read and write dates of importance.
To help children understand numeration in other systems use comparative illustrations as:

104

1. Continue study of Base 5. 2. Introduce Base 8.

BASE TEN
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

BASE FIVE
1 2 3 4 lOu... 11 u ... 12f T 13uT 14u ... 2O f ... 21 u ... 22f ... 23f ... 24fl ... 30u ...

BASE EIGHT
1 2 3 4 5 6 7 10...... 11 p 12 13 14 .. 15., ..
16.,." 17.....t

The numeral 14f.... means (1 X 5) + 4 and is read one, four base five (or
one five and four).
The numeral 17., means (1 X 8) + 7 and is read one, seven base eight (or
one eight and seven).

Base ten 364

Base five 124

300 3 (100) 3 (10XlO) 3 (10)'

60 6 (10) 6 (10) 6 (10) J

4 4 (1) 4 (n 4 (10) 0

100 1(55)
1(25) 1(5)2

20 2(5) 2(5) 2(5)1

4 4(1) 4(1) 4(5)0

In the preceding chart 4(1) is written 4(10)0. 100 is another name for 1.
Give children better understanding by transforming numbers from one base to another:
82 Base ten changed to Base five. Find powers of 5 as far as is necessary to determine the divisor of 82. This is found to be 25.

Powers/of 5
50 = 1
= 51 5
52 = 25
= 53 125

~82T_3_
75

5

71

5

_1_12~

read from top to bottom

2 -0-

v

Then 82 in base 10 = 312f1...

82 Base ten changed to Base eight. Find powers of 8 as far as is necessary to determine the divisor of 82. This is 64.
105

Powers of 8
80 = 1
= 81 8
82 = 64
83 = 512

~82_1_1_
64

read from top to bottom

,_1_~~

2 -0-

v

Then 82 in base ten = 122 0 1

D. Extend understanding of modular arithmetic.

Review mod 12 as used in clock arithmetic in Grade Four under section V of measurement.
Modular Arithmetic is obtained from a "clocklike" number system. It is sometimes called a finite number system because a limited set of numbers is involved. In computing with hour numbers unusual results are often obtained. H it is 8 o'clock and 7 hours are added the result is 3 o'clock. The sum of 7 and 8 with clock numbers is 3. In any modular system the name numeral does not appear.

Use Mod 4:

o
2+2=0 1+2=3
2 +3 =1

~2
H the hand is at 2 and moves for two spaces it is at O. H it is at 1 and moves 2 spaces, then it is at 3. H it is at 2 and moves for 3 spaces, it stops at 1. Do not count the numeral with which you start.
3 + 4 = 2 Mod 5 Begin at 3 and move four spaces clockwise.
2+3=OMod5 Begin at 2 and move three spaces clockwise.

106

Make an addition table for Mod 5

+01234
00

12

2

4

I- -

-3

-

-1
l-i- I-

4

3

Give practice in adding and subtracting in many mod systems.

II. Fundamental Operations

A. Addition and subtraction

1. Review basic facts in addition and subtraction.
2. Review regrouping process in subtraction.

Check addition:

One method is casting out nines. When the sum of the excesses of nines in the addend equals the excesses of nines in the sum of the addends, the addition is correct. An error of a multiple of nine in the sum is not discovered in this method.
237 ~ excess of 9's = 3 1
= = 425 ~ excess of 9's = 2 ~ excess of 9's in
196 ~ excess of 9's 7 J this sum 3
= = 858 ~ excess of 9's 21 excess of 9's 3
To determine the excess of nines in 237: Add the digits: 2 + 3 + 7 = 12,
12 is 3 more than 9. Cast out the nine and keep the 3 to use in the check. Similarly, to cast nines out of 425, add:
= 4 + 2 + 5 11 or 2 more than 9 and
1 + 9 + 6 = 16 or 7 more than 9.

The sum of the excesses, 3, 2, 7, is 12, which is 3 more than 9. Therefore after all nines have been cast out of the addends the result is 3. Now cast out nines from the sum of digits in 858.
= = = 858 ~ 8 + 5 + 8 21; 2 + 1 3 or 21 - 18 3.

The result is 3 as it was in the addend; therefore the problem is correct.

Check subtraction similarly:

476 excess of 9's = 81 -123 excess of 9's = 6 f

difference is 2

353 excess of 9's = 2

= 751 excess of 9's
= -249 excess of 9's
= 502 excess of 9's

4} 13 6 --6
7 +-+7

Review a problem in subtraction to show regrouping processes

3124 3 thousand 1 hundred 2 tens 4 ones

- 786 -

7 hundreds 8 tens 6 ones

regroup:
10 hundreds
2 thousands +1 hundred 2 tens 4 ones
or 2 thousands 11 hundreds 2 tens 4 ones

107

3. Review and extend understanding of sets.

regroup again:

2

thousands

10

hundreds

+120

tens tens

4

ones

or 2 thousands 10 hundreds 12 tens

regroup another time:

2 thousands or 2 thousands now subtract
2 thousands or 2338

10 hundreds 10 hundreds 7 hundreds
3 hundreds

10 ones 11 tens 4 ones 11 tens 14 ones 8 tens 6 ones
3 tens 8 ones

Review sets: What is a set?
A set is a collection of numbers, objects, or ideas as:
a set of books a set of dishes a set of girls in a class a set of natural numbers from 1 to 8 inclusive
Any person or thing in a set is called a member (or element) of the set. Sometimes there are no members in a set. It is said to be an empty or null set A capital letter is used to denote a set:
A = {boys on the team}
If the elements of Set A and Set Bare:
Set A = {2, 3, 4, 6} Set B = {1, 2, 3, 4, 5}
then the union of Set A and Set B is all elements of both sets
= A U B {t, 2, 3, 4, 5, 6}
The intersection of Set A and Set B is the elements common to both sets.
A n B = {2, 3, 4}
If a set is entirely contained within another it is called a subset.
Set A = {1, 2, 3} Set B = {1, 2, 3, 4, 5}
Then Set A is a subset of Set B.
Sometimes two sets are equivalent:
Set A = {girls in the 5th grade}
Set B = {girls in 5th grade who have completed 4th grade}
Then Set A = Set B
If Set C = {boys in grade 5}
Then Set B ,,;. Set C.
Have children work problems using sets such as:
If U = {O, 5, 10, 15, 20} what is the solution set for the following:
= 2 + x = 1 2 3n 15 x+2=2
Find solution sets for the following, using the natural numbers: 3+n<12
14 - n > 9
5-x,,;. 0

108

4. Continue column addition.

Work these:
= A {I, 2, 3, 4, 5} = B {O, 2, 4, 6}
C = {I, 3, 5}
1. An B

D ={}
E = {5}

2. A U B

3. C n E 4. D n E

= The set R {4, 7, 11, 14}. Set W represents the elements of set R of which
3 is a factor.

Set W = { } or fl.

What is the set of w?ole numbers which will make each of the following a true mathematical sentence?
12 > (2 X D) < (1,.2 X 0 14)

If it is agreed to let the number in the 0 be 2, what is the set of numbers

which will satisfy the A in these mathematical sentences?

= (a) A + 0 7

= + (b) A 0

31,.2

(c) A = 0 + 2

Review examples such as:
5 21 213 7462 7 62 187 1243 6 75 652 8065 3 83 217 9032 4 27 900 1105 2 19 805 3

57629 80513
90762 31904

806592 897641

5. Extend subtraction to include 6 place numbers.

Review examples such as:

876541 342030

905621 194302

900,000 187,652

6. Use number line extensively

The number line is most important to developing mathematical concepts. Try many uses of it daily. Some examples are:

6+9=15

,-------'-------,

o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2+3=5
,.---A--.,
o 1 2 3 4 5 6 7 8 9 roll ~~ M

To Bridge: 45 +7

Explain that in adding 7 to 45 it is easy to see that 5 added to 45 would be 50 and the other 2 would make 52. Thus the bridge is made from one decade to another.

40

45

50

55

I

-I - - - - > -I- > . .

I

+5

+2

109

B. Multiplication and division
1. Review basic facts in multiplication

= To subtract: ., - 8 f
,---A--., r,_.A---.,.
. o 1 2 3 4 5 6 .,
'------ ~----'

Try new ways of multiplying

Scratch Inverted pyramid Lattice Russian peasant (or doubling)

27 I
32 f
8t
IS
64
8 I6 I4

3 tens X 2 tens = 6 hundreds (in hundreds place)
= 3 tens X 7 210; 2 hundreds and 1 ten.
Add the 2 to the 6 hundreds, thus scratching the 6,
= making 8 in the hundreds place and 1 in the tens place.
2 X 2 tens 40; add the 4 tens to the 1 ten, placing the sum 5 under the 1, which is now scratched out.

2 X 7 = 14; the 1 ten is added to the 5 tens, making the 6 and scratching
out the 5. The 4 is placed in the ones column.

INVERTED PYRAMID

64 70 X 60 = 4200

79

9X 4= 36

= 4236 60X 9= 540
54 70 X 4 280 28

5056

LATTICE

236 X 125 = 29,500

RUSSIAN PEASANT
175 X 5!J-
350 29
7QQ 14
1400 7 2800 3 5600 1
10150

Double first number, halve second all the way until you read 1 on the right column (disregard fractions). Strike through lines having even num bers in the halving column. Add numbers in the doubling column.

Teach these generalizations:

To multiply by 10 annex zero.

To multiply by 5 annex zero, divide by 2.

To multiply by 11 mmuullttiippllyy bbyy 10I} then add.

110

243 X 1 243 X 10
2673

To multiply by 9
mmuullttiippllyy bbyy 110}
45 9

then subtract.

450 45 405
To square a number 23 X 23

23"
= 20 by itself 400 = 20 X 3, doubled 120 = 3 X itself 9
400 120
9 529
Teach the commutative and associative properties of ~ultiplication:
= 3 X 4 4 X 3 Commutative Property

(3 X 4) X 2 = 3 X (4 X 2) Associative Property

Teach some ways of checking multiplication and division:

25 X6 150
Add six 25's: 25 25 25 25 25 25 150

Divide 150 by 6. Divide 150 by 25.
Subtract 25 from 150 six times. 150 -25-1
125 -25-2 100 -25-3
'75
-25 -- 4
50
-25 -- 5
25
-25-6 -0-

2. Review basic facts in division

Review basic facts by use of games or devices to allay boredom such as: Relay races Timed exercises - pupils keep own charts Circle of numbers with divisor inside circle Ladder of quotients to be found Ponds to cross without getting wet-basic fact islands to help pupils across.

3. Continue multiplying by three place factors.

Review and expand use of three-place factors. 236 Stress correct placing of partial products. Show that the product is 152 the sum of several products:

200 X100
20000

30

6

X100 X100

3000 600

200
X50 10000

30

6

50 50

1500 300

111

-----------------

200

30

6

X2 X2 X2

400 60 12

20000 3000
600 10000
1500 300 400
60 12
35,872

Extend same principles into 4-place multipliers.

4. Use zeros in various positions Review the generalization that any number multiplied by zero is zero. Show

in both factors.

several examples using zero in the various places of the two factors.

1310 1206 1065 8962 2957 125 259 236 150 203

Explain that the two numbers which are multiplied to give another number called the product are called "factors."

5. Begin dividing by two-place divisors, including dividing by 10.

The easiest two-figure divisors to use are the ones that end in O.

45
10)453 40
'53
50
3'

Divide 453 into 10 groups, 4 hundreds cannot be divided into 10 groups but 4 hundreds can be regrouped into 40 tens.
Add the 40 tens to the 5 tens. 45 tens can be divided into 10 groups 4 to a group.

4 tens 5 ones
10)4 hundreds 5 tens 3 ones
(40 tens + 5 tens)
45 tens 3 ones 40 tens
5 tens + 3 ones (50 ones + 3 ones)
53 ones 50 ones
3 ones remaining
Therefore the first quotient figure is 4 in the tens place, 4 tens in each of 10 groups is 40 tens, which leave 5 tens; 5 tens can be regrouped into 50 ones and combined with 3 ones, making 53 ones; 53 ones divided into 10 groups gives 5 ones to a group leaving 3 ones as a remainder.

6. Divide by other numbers ending in zero.

Other divisors ending in 0 are used similarly:
30) 6 hundred
60 tens +4 tens 64 tens
2 tens 30) 64 tens
60 tens 4 tens
40 ones +2 ones 42 ones
112

30) 642

7. Divide by numbers ending in I, 2, 3 or 4 rounding downward.
8. Divide numbers giving twoplace quotients ending in 5, 6, 7, 8, 9 rounding upward.
9. Use tests for divisibility.

lone i..0)42 ones
30 ones 12 ones remaining

Divide 642 into 30 groups: 6 hundreds will not divide into 30 groups. 6 hundreds can be regrouped into 60 tens and combined with the 4 tens to become 64 tens. 64 tens will divide into 30 groups with 2 tens in each group, leaving 4 tens over. The four tens can be regrouped and combined to make 42 ones.

42 ones divided into 30 groups gives lone in each group and 12 ones remain ing.

21 R 12

30) 642

2 tens and lone will be in each of the 30 groups, with 12 ones

60

remaining.

42

30

12

When the second figure of a two-figure divisor ends in 1, 2, 3 or 4, it is necessary to round downward:

43) 372

Based on the understanding previously developed, the actual

process of grouping into 43 groups needs only to be discussed

as to meaning. To comtinue the explanation, show the 43 can

be estimated as 40 groups, and the process will be similar to the process of

those examples with divisors ending in O.

28) 572

When the second figure of a two-figure divisor ends in 5, 6, 7, 8, 9 it is necessary to round upward.

Consider 28 as an estimated 30 group divisor and proceed as in other examples with divisors ending in 30.

Extend tests for divisibility.

To test for divisibility by 2, 5, and 10:

If the ones place is divisible by 2, the whole number can be divided by 2. Any number can be divided by 2 if its ones place has 0, 2, 4, 6, 8.

Any number ending in 5 or 0 can be divided by 5.

Any number ending in 0 can be divided by 10.

To test for divisibility by 4: any number can be divided by 4 if the number indicated by the last 2 digits can be divided by 4. (Children should already know that 100 and 1000 can be divided by 4.)

To test for divisibility by 8: any number of thousands can be divided by 8; therefore, if the last 3 digits of any number can be divided by 8, the number is divisible by 8.

To test for divisibility by 3 or 9 show children how to discover the sum of the digits represents the remainder when a number is divided by 9.

324 = 300 + 20 + 4

When 1000, 100, 10, or 1 is divided by 9 the remainder is 1, then when 300 is divided by 9 the remainder is 3.

When 20 is divided by 9, the remainder is 2.

o
+ + = When 4 is divided by 9, the remainder is 4. That is 9)4 with 4 remaining.
3 2 4 9; which is divisible by 9.

Then the number is divisible by 9. Any number is divisible by 3 or 9 if the sum of its digits can be divided by 3 or 9.

113

10. Begin to use short division.

Teach short division after thorough understanding of the meaning of the

process is accomplished:

2 ) 48

Since 48 is the same as 40 + 8, the division process is as

2 ) 40 + 8 shown. 40 or 4 tens divided by 2 is 20 or 2 tens. 8 divided 20 + 4 by 2 is 4. There are two 2's in 4; write 2 under4; there are

four 2's is 8; write 4 under 8.
2 .> 4 tens + 8 ones 2 tens + 4 ones

2) 48
24

2 ) 90
2 ) 80 + 10 40 + 5

There are four 2's in 8 tens and 1 remammg; write 4 under 8. There are 10 ones. There are five 2's in 10 ones.

2 ) 8 tens 10 ones

4

5

2 ) 90 45

11. Practice estimating answer.

Use such practice exercises as:

21 ) 590 47 ) 5100 58 ) 621

estimate 30 because 21 is near 20 and 59 is near 60
estimate 100 + estimate 10 +

12. Check results of computa tion.

Check results of computqtions in division.
Check division by multiplying the factors and adding the remainder. Check by repeating the process.
Check by adding partial products plus remainder. Lead children to discover this for themselves.

Example:

9

9

3 I 28

9

27

9

-1

1

28

Check by subtracting 9 from 28 to find if there will be 3 nines with one remaining.
28 - 9 ~ 1 nine
19 - 9 ~ 2nines
10 - 9 ~ 3 nines --1 ~ 1 remainder

Check by casting out nines. This check is: the excess of nines is the product of the excesses in the divisor, and the quotient added to the exceSs in the remainder equals the excess of the dividend. Lead pupils into the discovery of the partial product plus the remainder.

114

Example: Method A 61
14 / 857 84
17
14
3"
Excess of 9's in divisor is 5 Excess of 9's in quotient is 7
Excess of 9's in 5 X 7 = 35 is 8
Excess of 9's in remainder is 3
+ Excess of 9's in 8 3 is 2
Excess of 9's in dividend is also 2.

Method B Add partial products 840 14
3 . 857

Check results of multiplication by reversing order of factors:

52

27

27

52

1404

1404

or by dividing the product by one of the factors to determine the other factor.

27 52 / 1404

52
27 i1404

Check multiplication by casting out nines. The check is: The excess of nines
in the product equals the excess of nines in the product of the excesses in both factors.

342
56 2052 1710
19152

Excess of 9's in 342 is 0 Excess of 9's in 56 is 2
Excess of 9's in product of 2XO=OisO
Excess of 9's in product is O.

As in addition, an error of a multiple of nine in the result is not discovered by casting out nines.

13. Review finding averages.

Charts are good to' use to further develop the concept of averaging. Find the average temperature for the month of February:

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

First Week
40
41 36 35 30 28 28

Second Week
30 32 32 34 82 30
30

Third Week
28
16 16 20 26 30 30

Fourth Week
32 34 36 36 35 34 33

14. Understand factor, prime, and multiples.

A factor is one of two or more numbers whose product is a given number.
Example: 7 and 2 are factors of 14 because 7 X 2 = 14.
A prime number is a. whole number that can be divided only by itself and one. Examples: 1, 2, 3, 5, 7, 11, 13, 17, .. When a counting number is written as a product of primes it is said to be factored completely.

115

To find prime factors of a whole number divide successively by the smallest prime:

2~ 2 X 2 X 3 X 3 are the prime factors of 36

2~

3~
31 3 -1

8=2X2X2 12 = 2 X 3 X 2

A multiple of a whc1e number is the product of the number and any natural number. 14 is a multiple of 7 because 7 X 2 equals 14.

It is often helpful for the chiI'd to be able to factor a number. When writing the simplest name for a fraction as 15/21 it is necessary to find a common factor or divisor for both 15 and 21.

If one factors them:

= 1 5 = 3 X 5
21 3 X 7
The child can readily see that both numerator and denominator can be divided by 3 without changing the value. (Fundamental Theorem of Fractions).

= = 15+3=5
21 + 3 7 or 15/21 5/7

When adding unlike fractions it is necessary to factor in order to find the least common multiple or denominator. Factor both denominators in the following:

+ 2/
3

3=

3/8
3X1

8=4X2

A common multiple must contain the factors of each, or it must contain 3, 4, 2, or 24.

+ 2 / 0

5/8

3X4X2xl=24

6=2X3

8=2X4

A common multiple must contain a 3, 4, and 2. Only one 2 is used since there is only one 2 in each number.

III. Fundamental Operations With Fractions

A. Common fractions

1. Understand fractional parts of a whole object.
a. Equal parts of a whole object.

Review and expand meanings of fractions. First meaning: a fraction is one or more of the equal parts of an object.

1

1

2'

4

116

3

2

8"

"3

In pure mathematics fractions are defined as points on the number line between whole numbers.

b. Comparison of fractional parts

o 1/2 1 3/2 2
Fractions are used to show parts. A fraction has two terms, the numerator and the denominator.
1 numerator
2"" denominator
The denominator shows the number of parts into which an object has been divided, and the numerator sholVs the number of parts being considered.
In the first illustration the 2 in :Ih shows the drawing is divided into p~ the 1 shows that one part has been shaded. Both the numerator and the denominator are called terms of the fraction numeral.

Second meaning: a fraction shows comparison.

1/2

1/4

1/8

1/3

1/6

1/5

~~~~~~

These figures show that the more equal parts into which an object is dividell the smaller each part is. Thus 1/4 is smaller than 1/2 and 1/8 is smaller thn 1/4, Also 1/6 is smaller than 1/s.
Sometimes more than one part of an object is considered.

3/4

2/3

7/8

The first circle has been divided into 4 parts, and 3 parts of the circle are shaded, so 3/4 of the circle is shaded; in the second, 2/s of the circle is shaded; and in the third one 7/8 is shaded.

Co Division (numerator by denominator)

A fraction may show comparison of 2 lines. The first line is % as long as the second line.
Third meaning: a fraction means numerator divided by denominator: 3 pies shared by four boys

Each boy gets 4. of each pie. In the end he will have % of a pie.
= 3 + 4 %
117

2. Further develop equal fractions

One fraction numeral may be the same as another fraction numeral.

1/2

2/4

4/8

3. Continue to simplify fractions.

----

1/3

2/6

3/6

~~~

= = 1/2 of an object is the same as 2/4 and 4/ 8 is also the same. 1/ 3 2/6, 3/6 1/2"

When two fractions are equal it is always better to write the fraction in its simplest terms:
3/6, 4/8, 6/12 all equal 1/2,

Therefore, when the final result of an operation shows one of these it should be stated as 1h. For example:

4. Find fractional part of a group

What part of the circle is shaded? The circle has been divided into 4 parts,

and 2
2/4 =

of the parts are shaded; therefore 2/4 of the circle is 1/2, so in simplest terms 1/2 of the circle is shaded.

shaded.

However,

The fundamental theorem of fractions can be generalized:

1 the numerator and the denominator are both divided (or multiplied) by

the same non-zero number, it does not change the value of the fraction:

= = 3/6 -+- 3/3 = = 4/ 8 -+- 4/4 = = 6/12 -+- 6/6

1/2 or
1/2 or 1/2 or

If2 X 3/3
1/2 X 4/4 1/2 X 6/6

3/ 6 4/ 8 6/12

Fractions show number of equal parts into which a group has been divided.

~DDD

2 parts of 8 have been shaded.

~DDD

2/ 8 -+- 2/ 2

1/4 of the squares have been shaded.

[!] [!] ~ [!] 0 D

~ ~ ~ [!] D D

~ [!] ~ [!J 0 0
= 1~ \lut of 18 have dots in the centers. 12/ 18 -+- 6/ 6 2/ 3 2/3 of the blocks have dots in the centers. 118

5. Add like fractions

Fractions that have the same denominator are called like fractions. Like fractions can be added.

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

1f4 has been dotted.
4 of the square has been shaded with lines.

How much of the whole square has been shaded in some manner?
+ = = 1/4 1/4 2/4 ; 2/4 1/2
1/2 of the square has been shaded in some manner.
1II:.Till ."0.-.:'

+ = 2/8

1/8

3/8

+ = 3/ 8

1/ 8

4/8

6. Change rational numbers with numerators larger than denominators (improper fractions) to equivalents of whole numbers plus fractions

DDDDD
EB D

Put all the fourths together There will be 1 and 4 or 14.

Use the number line to show how to make changes.

1

2

7. Add whole numbers and If Jane has 2lh sets of spoons and Mary has Ih of a set of spoons, how many

fractions

do they have together?

119

0 0 0 0 0 0 o0 ce~ ~ 0 0 aB

000 0 0 000
ft D ft 6 U I A ~

~I 0 0 0
A~ !

Vz Set

1 Set 1 Set

I I ~ ~ ~ ~ lfz Set

There are 3 sets.

Sometimes unlike fractions must be added and regrouping is necessary.

2 1/ 4 Can be shown pictorially and

.. +15/8 the enclosure shows the sum. ,-------------------------------------

lEB EB I

I

I

I

i

1

+

1

8. Subtract like fractions

1

+ 5/8

--------------------------------------~

Change the 1/4 to 2/ 8 &0 that the parts will be alike and there will be 3 and 7/ 8 objects.

0 0 B 21h

00 0

Suppose 1h of a circle were taken from 21h circles, what would be The result would be 2. If V4 were taken away the result would be 2%.
OOD

120

9. Subtract mixed fractions

To take away 1/4 the half would have to be changed to 2/4 and then 1/4 subtracted leaving 2 1/ 4,
It takes % of a yard of ribbon to make a bow. How much would be left in a piece 11,6 yards long?

10. Write remainder as fractions when sensible

1 yard

1.1h yard

11,6 - % = %

The whole piece of ribbon must be measured in fourths in order to take three fourths from it.

When fractional parts are important they should be retained:

Two boys had 3 watermelons. If each boy ate 2/ 3 of a melon, how much was left?

Amount left over was one melon and 2/3 of another.
A piece of paper was used to cut out valentines. Each valentine was 5 inches long. The paper was 26 inches long.

B. Decimal fractions
1. Interpret decimals as an extension of our number system.

Obviously the scrap of paper left was too small to be of value. Why bother with the fraction?

In decimal fractions all denominators are powers of ten; therefore it is easy to write them as a continuation of the place value children are accustomed to using.

'"'"
.C..I.l

s::
'="..c::

.se'.l:>n,:

'Cs":I:l
0

425

..c'.":,:

..c'.":,:
s:: ..C.I,l

'"C...I.l '"=s::
..c::

15

Children are familiar with odometers and can readily understand how the instrument works. Show a sample of one in class:

Have several read;ngs made. Explain the tenths place.
To visualize the comparison of wholes and decimals a line may be used; or a chart as shown:
1

121

-eeu

'(3

D:o II

100

10

1

J,..o
10

),00

2. Express tenths in relation to common fractions

Since tenths are one-place decimal fractions, a comparison with common fractions is easier with their use.

.1, 1/10

= .2, 2/ 10 1/ u
= .4, 4/ 10 2/ 5

= .5, 5/10 = 1/2

.6,

= 6/ 10

3/(;

.8, 8/ 10 4/ r;

/

(Because of the fundamental theorem of fractions)

3. Express hundredths in relation to common fractions and to money

Hundredths are also easy for children since 100 is the denominator.

= 50/ 100

1/2

30/ 100 ;::::: 3/ 1\)

= 25/ 100

1/4

= 75/ 100

3/4

compare with 50 cents ($.50) compare with 30 cents ($.30) compare with 25 cents ($.25) compare with 75 cents ($.75)

4. Understand decimal mixed A mixed decimal is a whole number and a decimal. 1.75 means 1 and 75/ 100, fractions

If a boy has 7.45 boxes of BB's, he has just about 7lh boxes because 7.50 is 7lh; and .45 is almost .50.

2.35 is about a third of one more than 2. 6.70 is almost three-fourths of one greater than 6.

A decimal number line is good for children to expand their concept.

o

1

2

.1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9

Mixed decimals can be shown by means of such a line.

5. Add tenths and hundredths or relate to adding common fractions

Decimals can be added as can common fractions. As in other numbers keep place under place-<lnes under ones, tenths under tenths, etc. A good thiJII
to remember is that, if the decimal is under the decimal, the numeralS "n1
all fall in place.

122

6. Subtract tenths and hun dredths

Bill bought gasoline three days in one week. He bought 5.9 gallons on Monday, 8.2 gallons on Wednesday, and 10.4 gallons on Saturday. How much did he buy all week?

= 5.9
= 8.2
= 10.4 = 24.5

5 9/ 10 8 2/ 10 10 4/ 10
23 15/ 10

24 5/ 10

The 15 tenths were regrouped to become one and 5 tenths. The one was addea to the ones column.

On a trip the odometer read 61274.8 at the beginning and 61706.3 at the end. How long W1:S the trip?

13
6 10 5 (H) +

61~!~(~

~

612748

-~-43--1--:-5

Regroup 1 hundred and add 10 to tens, regroup a one to become 10ths and add to the 3 tentbs.

Pocket chart may be used to show regrouping using decimals.

Example: 356.21
-124.79

In the diagram below regroup: take one of the 6 ones, make 10 tenths; add t6 the 2 tenths; take a tenth, add 10 to the hundredth.

Relations

No. PKGs 3 pkgs.
hundreds
Remove 1
leayes 2

5 pkgs. tens
Remove 2
leaves 3

6 pkgs ones
Remove 4
leaves 1

.- --., ';' .......... ...

_1:-._--..l

2

1"---'

tenths of hun-

pkg.

dredth of pkg.

Remove 7
leaves 4

Remove 9
leaves 2

Review the commutative and associative laws in showing equivalence
1 + 2 =2 + 1 = 3 + (2 + 4) (3 + 2) + 4
Review various names for the same number as a means of showing equivalence.
123

R

B. Inequality C. Order of decimals D. Ratio
1. Introduce rate pairs
E. Proportions

3 + 2 is another name for 5
6 X 4 is another name for 24 27 - 12 is another name for 15 10 pennies is the same as 1 dime
Many things are equivalent; more are not. Children must learn the vocabulary and symbolism of inequality.
A number line is helpful in teaching inequality.

o 1 2 3 4 5 6 7 8 9 10

On the line "6" is greater than "4" or 6>4 3 is greater than 2, or 3>2 3 is less than 7, or 3<7

Give children examples with instruction to insert the proper sign and prove exercises such as:

3X6-2X12
18 < 24

(3 X 8)+ 2 - (4 X 5)- 3
26 > 17

To prepare for decimals to follow, one must establish a thorough under standing of the relations between ones, tens, hundreds, and thousands. Also the ability to read dollars and cents is necessary as an introduction to decimal fractions.

.o...

XI-<

0

X.... .0... oX

..oX......

0
00....

ten thousands 1

Place value chart

.o...
X0
01-< ....
.... oX
.oX... .00...
thousands
1

o.....

s

.oX...

.X...

hundreds

tens ones

1

1

1

= 11,111

Ratio is easily understood by children in the form of rate-pairs: If a boy can bUy 2 marbles for a nickel, how many marbles would the boy get for 25c?
N=nickel O=marble NNNNN
00 00 00 00 00 5 nickels (25) will bUy 10 marbles.

A number line will show the same information:

o

5

10

15

20

2

4

6

8

For every nickel there are 2 marbles. The ratios are:

5

1/4, 15/6, 2/ 8, 25/10

Tmarbles

Seen as fractions their ratios are equivalent to each other. Equivalent ratios are known as proportions.
Children can find unknown:

124

If 5 buys 2 marbles, 25 will buy ? marbles?
5/2 _ 25/?
The relation between 5 and 25 in equivalent fractions is the same as tne relation between 2 and ?

F. Per Cent
Use ratios to introduce per cent

10 marbles

If ratios are compared with parts of 100, they introduce per cents.

Suppose Sam were selling magazines. There were 25 houses on his street. He sold magazines to 15 families out of the 25. His ratio of magazine sales would be 15/25, How many would he sell at the same rate if he stopped at 100 houses?

= 15/25 ?/100 = 15 X 4 60

= 25 X 4 100 then

60 out of 100 or 60% (per cent). "Per cent" means "out of 100" or per 100.

v. Measurement
A. Standard units of measure (English)
1. Review concept. and skills previously learned about measure

= = Show relations of measures such as 12 in. 1 ft.; 36 in. or 3 ft. 1 yd., etc.
Have children do as much actual measurement as possible. Have visual aids showing as many measures as are feasible. Tape measures are very helpful.
Stress estimating measures before actual measuring is done.
Use measures as need arises in the classroom.
Review the following:
Inches, feet, yards, rods, miles Pints, quarts, gallons Pints, quarts, pecks, bushels Ounces, pounds, tons Seconds, minutes, hours, days (A. M. and P. M.) Days, weeks, months, years, leap years Decades, scores, centuries (A.D., B.C.) Time zones Freezing point, zero, above and below zero, boiling point Pennies, nickels, dimes, quarters, half dollars

2. Change measures to larger and smaller units

Study tables of equivalents, substitute equivalents when necessary.
Compare measures that are comparable.
If a curtain is 54 in. long, how much cloth would be necessary to make 4 curtains?
54 in. 4 216 in. change to yards
= 36 in. 1 yd. = 216 -;- 36 6 yards
Before being rescued a man was afloat for 75 hours after his plane went into the ocean. How many days was he afloat?
= 24 hours 1 day = 75 -;- 24 3 days and 3 hours

125

-----,---,-----------

3. Develop skill in using fractional parts of measures
4. Learn appropriate measures for various commodities
5. Learn meaning of square measure

How many nickels in 6 quarters? Such problems give children the practice they need in changing measures.

Recipes furnish excellent examples of practice in taking fractional parts of liquid and dry measures.

A recipe for candy called for:

1 pt. milk % pt. sugar, etc.

Mary had only 1 cup of milk. How much sugar should she use? Why?

A pattern called for 2 yards of ribbon, 2% yards of lace to edge it.

Susan decided that if she left off the trim on the pockets it would take only 2/ 3 of the necessary ribbon and lace. How much ribbon and lace would she need?

= 2/ 3 of 2 yards 4/ 3 or 11/3 yards = = 2/ 3 of 2 1/2 2/ 3 of 90 in. 60 in. or 1 yard, 24 inches
or 1 yd. 2 ft. or 1 2/3 yards.

Add and subtract measures such as:

1 ft. (12+) 18 in. 8 yds. ~ ft. $ in. 2 yds. 1 ft. 9 in. 4

6 yds..

9 in.

What measure would you use to tell about such things as:
your trip (miles) the gas you used (gallons) the milk and eggs for breakfast (quart & dozen) your new dress (yards) your jet flight (hours)

Show that square measure is the kind of unit used when one is measuring things in which the length and width are considered such as carpet for a floor, surface of wall to be painted, a bulletin board, and volley ball court surface.
The unit of one square inch would be used if one was measuring the surface of a sheet of construction paper that would be used to construct a geometric figure or make a graph. The unit of one square foot or one square yard would be used if one were measuring the surface. of a room for a carpet.

1 square unit

A rectangular figure {) inches long and 3 inches wide may be considered as a row made up of 5 square inches taken 3 times or a row of 3 square inches taken 5 times.
5 inches

_~I~' _ _ 3 inches

1. sq. 1. sq. m., m.

1 -- 1 I

126

--------------

B. Metric system of measures 1. Know meaning
2. Compare with English system

A rectangular coordinate graph chart is an excellent aid to use in teaching this concept.
Bring a meter stick to class. Compare it with yard stick. Notice that the divisions on the meter are in tens.
Tntroduce terms and meaning: meter, decimeter, centimeter, millimeter. Find measures of various objects both in yards, feet, and inches and in meters, decimeters, and centimeters.

VI. Geometry
A. Find perimeters of polygons, circumference of circles

Review definition of these as simple closed curves.

parrallelogram

CJ

triangle equilateral
isosceles rectangle square

6
6
~
I
D

equal sides 2 sides equal

quadrilateral (figures formed by union of 4 line segments)

circle

o

polygon (any figure formed by union of line segments) Introduce meaning of term perimeter, "the total length of a closed curve." Explain meaning of circumference. Generalize formula for perimeters:
= square - p 48 = triangle - P a + b + c = rectangle - P 2L + 2W
Have children construct figures with line segments of given lengths:
Draw a triangle using the given lengths.

B. Find area of squares, rec-
tangles, triangles, and similar Polygons

Find perimeters by actual measurements. Teach use of compass and straight edge. The area of a square is easy to see by inspection.
EE 2 2X2 4 2 127

--------------
Generalize simple formula: Area of square is found by multiplying a side by a side. Expand idea to include rectangle.

C. Recognize parallel and perpendicular lines

6
= 3 X 6 18. Count squares. Generalize formula for area of a rectangle.
A =lw
= Show how line divides rectangle into 2 triangles, so area of triangle 1h that
= of rectangle. A 1h lw.
Having learned to recognize figures with parallel and perpendicular lines, children readily learn the terms:

parallel lines
Examples: opposite sides of a square, opposite sides of a rectangle, like railroad tracks, like floor and ceiling, etc.

D. Recognize three dimensional (solid) figures; learn names

perpendicular lines
Examples: adjoining sides of a square or rectangle, wall and floor, picture frame length and width, table top and legs (card table).
Cube, rectangular solid or prism, triangular prism, pyramid, sphere, cylinder, cone
A cube is already known because of the familiar ice cube or baby blocks. Its name needs to become part of the child's vocabulary, and he needs to be able to draw one.

A cube has 6 faces and 12 edges.

A rectangular three dimensional figure is also well known to children. They deal with sticks of butter, bars of candy, boxes of cereal, and the like. They need to call such figures by name and to recognize them .
This figure has 3 pairs of similiar faces: 2 ends, top and bottom, and 2 sides.
A triangular prism is perhaps the least familiar to children, but not difficult. Examples: wedges, some building blocks Two faces of the prism are triangular, the other faces are rectangles.
2 triangular faces 3 rectangular faces
128

Line segments shared by two faces are called edge. Points of intersection of 2 line segments are called vertice. This figure has 5 faces, 9 edges, and 6 vertices. Pyramid: a sort of tent

This pyramid has 5 faces, 5 vertices, and 8 edges. Cylinder
C~)_ _--')
We buy cans of food in cylinders; other examples: lipstick, stick candy spaghetti strips. A cylinder is formed by rolling a sheet (quadrilateral) over and joining sides. Figure formed at ends is a circle. Cone: Familiar paper drinking cups, clown hats, horns are good examples. Sphere - familiar object is a ball

E. Become familiar with sets of points and accompanying terms

A line is a set of points.
A "---------" B A line extends in both directions. Two points determine a line. Three points not on a line determine a plane.
Example of planes: sheet of paper, card, table top, window glass

Two lines intersect in a point.

Two planes intersect in a line.

120

Part of a line is called a segment.

.----"--.

AB

_ _ _0

0 _ _-

Part of a line and the end point make up a ray.

F. Develop meaning of terms for use with circles

A

B

_ _0 _ - - 0

An angle is formed by the union of two rays which are not on a line and which have a common end point.
c

~

~

ACandAB

Ray AC and ray AB meet at end point A, which is the vertex of the angle.

B
Space is the set of all points. A circle is a special closed curve made up of points that are all the same distance from a point called the center. The line segment from the center to a point on the circle is called a radius.

VII. Graphs, Charts, Scales

Pictographs and bar graphs are introduced at this level.

A. Read, interpret and make simple picture, bar, line, and circle graphs.

Each star represents a home run in the season. Compare home runs made by the boys.
Sam***
Joe * * * * * * Ellis * * * * *

Who was the "most valuable player?" Represent this by bar: by circle:

SAM

JOE

ELLIS

o 1 2 3 4 5 6 7 8 9 10

Line graphs are used to show rate pairs.

t

Candy bars sell at 3 for 1041. How much would 9 bars cost? Lines cross on 304

----- 15

12

9

6
3 v---

~

~ ~

o

10

20

30 40

130

Much information can be pictured in such graphs. Take advantage of grades, runs, scores, averages of various kinds, measures and the like to portray by graphs.
These exercises give children a good background for future work with graphs and scales.
Make a scale drawing of:
= the baseball field 1 in. 20 ft.
the classroom 1 in. = 10 ft.

B. Read and understand maps and globes
C. Read and interpret simple tables

Correlate scale drawing with understanding maps in geography. The ability to read tables is most important. These can be found in books in new~papers, in ~agazines, in various pamphlets, and in reports. Find them; a~k chIldren to brIng them to class. Have exercises on reading information gIven.

VIII. Problem Solving

A. Emphasize steps in solving problems

Help children form the habit of following certain steps in obtaining solutions.
1. Determine what is known. Children must be able to read and state what is understood from the reading.
2. Decide on process to use. Should the child add, subtract, multiply, or divide? Are there two or more steps to follow? If so, what is the process of each step?
3. Estimate results. This is important so that a child may recognize an absurd or impossible result.
4. Write a number sentence. This is important so that the child will be en. abled to "work" the problem.
5. Determine the result. Compare with estimated result.

B. Solve multiple-step problems

It is necessary for those who learn mathematics to become adept at taking one step at a time and knowing what each step is.

Mary had 58 in the bank. She bought 2 apples for a nickel each, some bananas for 154 and a bar of candy for a dime. How much did she have left?

Step 1 Step 2 Step 3

What did the apples cost?
= 2 X 5 10
What did all the purchases cost?
10 + 15 + 10 = 35
What was left?
58-35=23

Multiple-step problems are aids to reasoning.

C. Solve problems of various types

Problems that involve addition and subtraction situations
John had 3 books. Sue had 5. How many did they have together?
3+5= 0 John had 3 books. He and Mary together had 8 books. How many did Mary have?
3 +0 =8
Mary had 5 books. She and John together had 8 books. How many did John have?

131

D. Use symbols and the language of sets in solving mathematical sentences

0+5 = 8
There were 12 cookies in the box for John and Mary. If John ate 4, how many were left for Mary?
12-4=0
There were 8 cookies in the box. If mother left 12, how many had John eaten?
12-0=8
John told Mary he had eaten 4 cookies. He had left 8 for her. How many were there at first?
= 0 - 4 8
Problems involving separating a group into equal parts: The number of the group is known; the size of the group is not known.
There are twelve boys in the cub pack. If the pack received a basket of peaches how many will each boy receive if there are 75 peaches?
The number of groups is not known, the size of the group is known:
How many bags of candy will mother be able to send to the bazaar if she made 72 pieces and she puts 6 pieces in a bag?
Problems using ratio to symbolize rate and comparison situations.
At 2 for 15 how many bars of chocolate can be bought for 75?
= 2/ 15 n/75
Susie is lh as tall as her daddy. If Susie measures 3 feet, how tall is her daddy?
liz = 3/n
Problems involving finding averages.
Alice kept her scores in spelling for the week:

Monday

88

Tuesday

93

Wednesday

88

Thursday

86

Friday

90

What was her average score for the week?

Problems that children think up are excellent for keeping interest alive and for drill.

Build problems from experience using mathematical sentences such as:

What is the set of numbers {M, R} such that M = R + 4

M+R=20 The pupil might volunteer such sentences as:

John had four more marbles than Tom. Together they had 20. How many did each have? Yesterday we had 12 silhouettes of the class made. Today we made more in order to have one for each of our 32 children. How many did we make today? Find the solution set. 12 + N = 32
Practice putting the activities of the class, whether from reading or from actual experience into the language with which they should become familiar.
A set of scores made today in our ball game by the boys was {5, 4, 3, 6, 9}.

The girls' set of scores was {2, 7, 8, 6, I}.

132

E. Establish mathematical proof

In solving equations children will be called Upon to give reasons for processes
= chosen or for various combinations made. It will be good training to use state-
ments of proof. Why is 2/4 1/2 ?
= 2/4 1/2 because of the fundamental theorem of fractions, or the law of 1,
because the divisor 2/2 is 1.

3X2=2X3

Why are these products equal? CXlmmutative Law

= 2 3 X 8 = 2 0 X 8 + 3 X 8

184

160 + 24

Why is the first product equal to the sum of the other two? Distributive Law

Use these and other "reasons" for saying why certain steps are taken in solving problems.

133

I(
,

SIXTH GRADE

UNDERSTANDINGS TO BE DEVELOPED IN SIXTH GRADE
The sixth grade pupil expands his grasp of the number system to enable him to read, write, and interpret numerals through billions and to know both cardinal and ordinal uses of them. He extends his ability to work with fractions to include decimals, large and small, in all operations and is able to change one fractional form to another. He understands the importance of rounding off integers to any place.
The pupil demonstrates mastery of the fundamental processes in using large numbers and is able to estimate answers before performing operations. He uses laws or properties of numbers with understanding _ commutative, associative, and distributive. The idea of sets and their terminology are ex. tended and numbers are compared in various bases. The use of ratio is extended to include per cent. Ideas about measurement of various kinds are broadened. A deeper appreciation of the importance of accuracy in solving problems is developed. The pupil correlates more effectively his understanding of arithmetic with that of geography in studying latitude, longitude, meridians, and degrees. Knowledge of graphs, charts, and scales to interpret harder problems in expanded. Concepts of geometry are extended to include working in three dimensions.
137

CONTENT

TEACHING SUGGESTIONS

I. Numeration
A. Decimal notations 1. Read, write, and use num bel's through billions.
2. Extend the concept and use of ordinal and cardinal numbers.
3. Extend the base-ten nume ration system through the decimal system of notation for fractions.
B. Roman numerals 1. Read and write Roman numerals with facility.
2. Make comparisons with Arabic system.
C. Other systems of numeration

Review place value through millions. Explain billions as an extension showing:
All places follow the ten relationship. Each place is ten times its predecessor (moving left from decimal point).
Use display to review:
275 is 2(100) + 7(10) +5(1)
2(10X10) + 7 (10) + 5 (1) = + 2(102) + 7(10') 5(10)
Use a base other than ten to further develop meaning:
2134 in base 5 is:
+ + 2(53) 1(52) 3(51) + 4(5)
This serves as an introduction to exponents.
Have children practice reading large numbers so that they can determine by inspection the positional value of any digit to at least 10 places.
Use larger numbers in sequences and ask children to tell what the place of certain numbers is. For example, in the series 2751, 2752, 2753, tell which the middle number would be in an orderly arrangement.
Extend place value concept to decimal fractions in millionths. Compare mil lionths with millions showing ones as the center of the decimal system.
Show how to round off to nearest hundred, thousand, etc.; also to nearest unit, tenth, hundredth, etc.
Show that the numerator of a decimal fraction is the numeral itself and that the denominator is indicated by the place value position of the last digit of the numeral.
Example: . 48_48 _48 . - 100 - 10'
In each decimal fraction the denominator is a power of ten.
If 3 digits follow the decimal point, the denominator is 10" or 1000.

Review methods of reading and writing Roman numerals taught in earlier grades.
Extend to snow that the bar over a letter multiplies the value of the letter by 1000.
= n D X 1000 = 500,000 = = M M X 1000 1,000,000

Teach the history of Roman numerals and compare with that of Arabic numerals. (See James and James, Mathematical Dictionary)

To increase appreciation of Arabic numerals show examples of using Roman numerals in cumbersome situations.

Review method of writing in base 5.

Base 10
10 20 30 50

Base 5
20 ? ? ?

138

To change from base 10 to base 5 divide by successive powers of the base. Example:
42 = 1 of base square (or one 25)
3 of base (or three 5's) 2 ones ~ 42 .~ base 2 (or 25)

25 ,~..I17 ~ base (or 5's)

15
""""2" ones

Thus 42 base 10 = 132 in base 5. 132t1n means:

(25)

(5)

base'

base

ones

1

3

2

To change from base 5 back to base 10: 132un = 42 in base 10.

base'

base

one

1

3

2

1X25=25 3X 5=15
2 X 1= 2
42

To change from base 10 to base 8:

base'

base

64's

8's

ones 1's

42 in base 10 has no 64's.
8 I 42 I 5 -+ base or 8's
-40-

2

-+ ones or l's

So 42 in base 10 = 52 in base 8.

To reverse the process:
= 52.,gh'
5X8=40
= 2 X 1 2
42 in base 10.

= To change from base 10 to base 4: 42 222rou,

-ba- se' 16's

-

-

I'
I

base 4's

ones 1's

16 i 42 ~
32
4TWT2 - 8'-
-2-

2 X 16 + 2 X 4 + 2 X 1 = 42

Base 5
1 2 3 4 10 11

Base 8 1 2 3 4 5 6
139

Base 4
1 2 3 10 11 12

Base 10
1 2 3 4 5 6

D. Review and extend the idea of sets and set language.
E. Extend understanding of modular arithmetic.

Many sixth grade children will not have had any experience with sets. Those who have will enjoy using the terminology. Refer to Section IT A 3 in the fifth grade guide for this work.
Build Mod 12 and Mod 7 systems by using months of the year and days of the week. In developing Mod 7 think of the days of the week as:
Sunday number 1, Monday number 2, Tuesday number 3, Wednesday number 4, Thursday number 5, Friday number 6, and Saturday number O.
There is no number larger than 7. Therefore in this system the sum of 4 and 5 is 2.
In building Mod 12 the months of the year may be used in the same manner.
Use Mod 5 dial to find the sum:
3+2= 2+4= 4+4= 3+4=
Use Mod 5 addition table to find:
4+1= 3+4= 3-1= 2-4=
= 3 + (2 + 4)
= + + (3 - 1) (2 4)
(2 - 3) - 3 =
Make multiplication tables to further illustrate different modular systems.

II. Fundamental Operations With Whole Numbers

A. Addition and subtraction 1. Compute with high degree of accuracy and speed.
2. Calculate mentally sums and differences.

Reinforce the meaning of: Addition as the basic operation; subtraction as the inverse of addition; multiplication as repeated addition; division as repeated subtraction and the inverse of multiplication.
Give timed test to increase speed.
Give drill to insure accuracy.
Rapid addition can be achieved by much repetition. One way of diversifying the process is to add across and down.

-'>

23

23 2 tens and 3 ones add 4 tens 63

" 6 tens and 3 ones add 2 ones 65

~
42

6 tens and 5 ones add 1 ten 75

" 7 tens and 5 ones add 5 ones 80

~
15 8 tens add 6 tens

140

" 14 tens add lone

141

~
61

14 tens and lone add 1 ten

151

",. 15 tens and 1 one add 9 ones 160

~
19

140

Check by adding up, then down.

Check by combining numbers to make ten, then adding:

I10 ~~1~5t 10 ~fJ 10 l60

Rapid differences are not hard to achieve but estimating answers helps a child to determine feasibility of answers.

8574 estimate 9000 - 3156 estimate 3000
5418 result less than 6000 because 8574 is considerably less than 9000.

7952 estimate or round to 8000

- 2897 round to

3000

5055 answer will be about 5000

In determining exact answers mental regrouping must take place.
14 10} 12 8 +4 10} 7 fl ~ 2+ 2 8 9 7 One can't take 7 from 2. Regroup a 10 into ones, and add 5 0 5 5 10 ones to 2 = 12 ones (leaving 4 tens).
Subtract 7 from 12 = 5. One can't take 9 tens from 4 tens - regroup 1 hundred into 10 tens; add 10 tens and 4 tens = 14 tens (leaving 8 hundreds). Subtract 9 from 14 = 5; subtract 8 from 8 = 0; subtract 2 from 7 = 5.

3. Estimate results of operations.

The following is an example of estimating by "rounding off" forward and backward by decades.

24 44 17 63
19

20 backward 40 backward 20 forward 60 backward
20 forward

In subtraction
76592 1984

approximately 76600 - 2000

4. Extend understanding of associative, commutative, and closure properties of addition.

Extend understandings.

Review generalizations already understood.

Commutative Law

4+3=3+4

aadd~d~inngg

up =
down

14~~ )

grouping numbers and getting sum 5

Associative Law
+ 6 + 7 = (3 + 3) + 7 = 2 + (3 7)
7 + 8 = (7 + 7) + 1 = 7 + (7 + 1)

141

Doubles are important in number relations too.

= 5 + 5 = 10
4 +4 8

3 7

+ +

3 7

=
=

6
14

This will enable a child to build on doubles, since

8 + 8 = 16 then 8 + 9 = 17

Closure in addition is easily understood: that any whole number added to another whole number gives a whole number.

The additive identity element should be understood:

Zero added to any number equals the number.

Use these properties in "proving" facts.

5. Review concept of sets studied in previous grades.

A set is a collection of objects, numerals, ideas as: a set of boys on the field; a set of chairs in a circle; the set of even numbers on the clock. Sets are designated by capital letters and enclosed in braces, as:
A = {2, 4, 6, 8, 10, 12}
Set A is the even numerals on the clock face.
The union of sets has elements of one set combined with the other:
SetA={2,3,6}
= SetB={I,2,4,5}
A U B {I, 2, 3, 4, 5, 6}
The intersection of 2 sets is that part of one that is found in the other:
SetA={2,3,4} Set B = { 1, 2, 3 } AnB={2,3}
A subset is a set that may be contained within a set:
Set A = {2, 3, 4} Set B = {2, 3, 4, 5, 6}
thus B contains A or A c B (A is a subset of B)
An empty or null set has no elements. Set W = {2,3} Set Y = {7, 6} Set Z = Wny = {} or f)

6. Relations of sets.

Sets Hand J are said to be in one.to-one correspondence if each element of set H can be matched with one element of set J, and each element of set J can be matched with one element of set H.
If Set H = {7, 8, 3, 4, } and Set J = {a, b, c, d}
then set H and set J are said to be in one-to-one correspondence. Two sets that are in one-to-one correspondence (that have the same number of elements) are said to be equivalent.
Two sets that contain the same elements are said to be equal sets. The arrangement of the elements does not effect the equality of the sets.
Set L = {SIR' .5, 3} Set M = {3, 1/2, I} Set L = Set M
Use other examples such as:
1. What is the set of even numbers> 14 and < 32?

142

,$

2. What is the set of counting numbers less than 32 which are multiples of 6?

3.

What is matical

the set of sentence?

numx b+ersx

which
< 10

will

make

the

following

a

true

mathe-

4.

What is matical

the set of sentence?

Nodd+nu2m<ber1s4 which

will

satisfy

the

following

mathe-

5. Is Set R a subset of Set W?
= Set W {3, 5, 7, 9, 11} = Set R {3, 8, 9}

6. Is Set A in one-to-one correspondence with Set B?
= Set A {14, 17, 19, 13}
= Set B {17, 14, 13}
7. What is the set of odd numbers for which the following statement is true?
3 N < 12

B. Multiplication and division.

1. Master multiplying and dividing by powers of ten

To multiply a number by any power of ten move the digits to the left the same number of places as there are zeros in the multiplier, and annex the same number of zeros as:
= 10 X B 80
100 X 8 800
= 1000 X 8 8000
To divide by powers of ten, take off zeros in the dividend to compare with those in the divisor. Division which does not result in an integral quotient should be delayed until decimals are studied.

2. Divide by a three-digit di- Review principles of dividing by a two-digit divisor. Provide practice in

visor (factor)

rounding off and estimating.

286 ,I 5951

How many 3 hundreds in 60 hundreds? The quotient will be about 20. The quotient will begin in the tens place.

Follow 3 steps:

Estimate how many figures there are in the quotient.

Use trial factor to find first figure in the quotient.

Use information found to place first quotient figure and proceed as in all previously learned division.

3. Extend use of tests for divisibility.

Review tests for divisibility and extend tests to include 7 and 11.

A number is divisible by 7 if twice the last digit subtracted from the remain-

ing number is either a multiple of 7 or a zero. This rule is practiced for

numbers with no more than 4 digits.

.

A number is divisible by 11 if the sum of the odd-numbered digits minus the sum of the even-numbered digits is a multiple of 11 or zero.

4. Check by casting out nines.

Check by casting out nines. In casting out nines the check is:
The excess of nines in the product of the excesses in the divisor and the quotient added to the excess in the remainder equals the excess in the dividend.
143

123
586 / 72365
586
1376 1172 ~ 1758
2ii7

Excess of 9's in quotient is 6
= Excess of 9's in divisor 1
lX6=6 Excess in remainder = 8
8 + 6 = 14 excess = 5
= Excess in dividend 5

Multiplication and division may be checked by Mod 9.

5. Estimate quotients mentally; calculate some products and quotients mentally, and correct estimated quotient figure.

Estimating a quotient figure too small:

4
35 ,i1ij3 140
43 (1st trial)

6 35 / 183
210
(2nd trial)

5
35 / 183 175 -8
(3rd trial)

First trial will not work because the remainder is larger than the divisor (or could be as large).

Second trial will be wrong because the product of the quotient and the divisor is larger than the dividend.

The third trial is correct because' the product of the quotient and thedivisor is not larger than the dividend, and the remainder (if any) is smaller than the divisor.

Much of the testing is done without actually writing, but by thinking through the process.

6. Interpret the concept of averages, use of averages as an approximation.

The term average really is a word which is understood as the ''mean.'' It is sometimes used to indicate the middle number in an ordered arrangement of numbers and called the "median." Sometimes it is the number most often repeated in a set of numbers and is then known as the ''mode.''
Examples: (mean) John worked 7 hours 1st day 8 hours 2nd day 12 hours 3rd day 10 hours 4th day 8 hours 5th day
Total 45; 45 + 5 = 9 average
How long was his average working day? If just the numbers were used; as: find "mean" of the numbers 7, 8, 12, 10, 8, the same result would be used.
Example: (median) (mode)
Find the median in the following set of numbers: 19, 15, 17, 19, 16, 15, 17, 21, 19, 18,21,20.

15 15 16 17 17 18 - 18 and 19 are the middle numbers and are called the "median."
19 19 - 19 is repeated most often, and is therefore called the "mode."
19 20 21 21

144

Example: average of approximation
If an average daily cost is known, one would be able to estimate total cost of trip.
Suppose friends kept records of their trip and spent $6 a day (average) for all their expenses. How would they know the amount to save for their next trip if they planned to be gone 2 weeks?
$6 X 14 days = $84

'1. Understand more fully the Commutative, Associative, and Distributive Laws.

The Commutative Law

Addition and multiplication are both commutative.

Example:

3+8=8+3 3X8=8X3

Knowing this principle cuts approximately in half the basic addition and multiplication facts a child must learn. The Associative Law Addition and multiplication are both associative; that is, addition or multiplication of several factors may be performed in any chosen order as:
4+5+3=~+~+3=4+~+~ 4X5X3=~X~X3=4X~X~
The Distributive Law 6 X (2 + 4) = (6X2) + (6X4)
This can be used in understanding multiplication in several places:
5 X 432 = 5 X 400 + 5X30 + 5 X 2

III. Fundamental Operations With Fractions

A. Common fractions

1. Addition and subtraction of fractions.
a. Reinforce the concepts already learned.

Review terms of a fraction:

Numerator tells how many parts there are of an object. Denominator tells number of parts originally in the whole object.

In %0, object was divided into 4 parts, 3 of which are used.

There are many names for the same value; they are called equivalent fractioD80

= = =. 1/2 = 3/6 = 4/s = 6/12

3/4 6/s 9/12 . .

2/3 = 4/6 = 6/D

15

3x5

35

3

3

"20= 4zI5 =4"X'5 = .. X 1

Fractions are usually written in the simplest form, that in which the numerator and denominator have no common factors (except the numeral 1).
The term "simplest form" instead of "lowest term" is used so that children will not form the concept that the fraction has less value. Lowest sometimes has the connotation of least or smallest.
Fractions belong to the set of numbers called rational numbers, taking their place on the number line between the whole numbers.

145

b. Add and subtract like fractions
c. Add and subtract unlike but related fractions.

whole numbers

o

1

2

3

1/2 2/2 3/ 2 4/ 2
rational numbers

5/2 6/2 7/2

Review the three meanings of fractions:

As one or more of the equal parts of an object, 3/ s (3 out of 5 equal parts).

As a means of division, 3/ 8 (3 cakes shared equally by 8 persons).

As a means of comparing numbers, the ratio idea: 2/ 5 (2 balls out of 5 are red).

In reviewing fractions use a number line to show relationships:

o

1

I I 1 I 1/12 2/ 12 3/ 12 4/ 12 5/ 12 6/ 12 7/ 12 8/ 12 9/ 12 10/12 11/ 12 12/12

1/6

2/ 6

3/ 6

4/ 6

5/6

6/6

1/4 1

2/4

I 3/4

4/4

l/a

2/ a

3/a

1/2

2/ 2

Like fractions have denominators in the same terms: S, %, 7/ S' %; these like fractions all have 8 as denominator.
To add like fractions, add the numerators:
S+%+7/S +%
1 eighth
= 3 eighth (review commutative law)
7 eighth lfs + % 0/8 + lfs etc.
5 eighth
16 eighths or 16/8 or 2
To subtract like fractions, subtract the numerators.
11/ 16 - 5/ 16 11 sixteenths
-5 sixteenths
= 6 sixteenths 6/16 or 3/8
2:14 - %

Take one and regroup into fourths; add to the one-fourth:

= 1 5/ 4 - 3/4

12/4

12/4 = 11/2

To add unlike fractions, those with denominators having common factors are more easily added:
Y4+%+%+%.
In order to add, one must use a common denominator, the fourths must be changed to eighths;
= = 1/ 4 2/ 8, 3/ 4 6/ S
+ + + 2/ 8 3/ 8 5/ 8 6/ s = 16/S = 2
The terms of a fraction can both be multiplied (or divided) by the Sa1Jle number without changing the value of the fraction.

146

i$i

d. Add and subtract unlike and unrelated fractions.

To subtract related fractions without regrouping:

= 7/16 - 3/8
= 7/ 16 - 6/ 16 1/16 = = 2 11/ 12 - 1 5/ 12 1 6/ 12

1 1/ 2

= with regrouping: 7 1/ 4 - 2 3/ 8

+ = (6 4/ 4
= = 6 5/ 4 -

1/4 ) -
2 3/ 8

2 3/ 8 6 10/ 8 -

2 3/ 8

4 7/ 8

To add fractions that are unrelated-that is, having denominators that do not have common factors:

+ 1/4

2/5

One must find a common denominator. A number which has factors that are
the same as the factors of the denominators will serve as a common denominator.

Review writing prime factors

2. Multiplication and division of fractions.

= 12
15

=

3 3

X X

2 5

X

2}

20=2X2X5

4 has factors (other than 1 and itself) 2, 2

5 has no factors other than 1 and itself

Therefore 20 is the least common denominator for 5 and 4

+ = + = 1/4

2/ 5

5/ 20

8/20

13/20

+ 3 / 8

7/18

These cannot be added unless they have a common denominator. To obtain this, the denominators must be factored.

The factors of 8 are 2, 2, 2 and of 18 are 2, 3, 3.

So the common factors are 2, 2, 2, 3, 3 making the common denominator 72.

= 3/ 8
= 7 / 18

27/72 28/72

therefore

+ = + = 3/8

7/ 18

27/72

28/72

55/72

To multiply a fraction by a whole number, associate the process with addition.

a. Multiply a fraction by a Each member of the family had half a cantaloupe for lunch. If there were

whole number

4 people in the family, how many cantaloupes did they eat?

b. Multiply whole number by a fraction

= = 4 halves :If.! + :If.! + :If.! + :If.! 4/2 2

= = 4 X 1/2

4/2

2

Generalize: To multiply a fraction by a whole number, multiply the numerator of the fraction by the whole number.

To multiply a whole number by a fraction, use the word "of" in the statement to introduce the concept. Compare with finding fractional part of a whole object: 1/2 X 16 is the same as 1/2 of 16.

147

II
o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 11 16

c. Multiply a fractional num-
ber by a number represented by a whole number and a fraction.

Since the understanding of multiplying a whole number by a whole number and a whole number by a fraction has been established, the principles can be used in such examples:

4 X 5J,2 =
4 X (5 + J,2) = + (4 X 5) (4 X J,2) = + 2 0 + 2 = 2 2
5J,2 and (5 ',2) are 2 names for the same number.
or
since 5J,2 = 11/2
4 X 5~ =
= = 4 X 11/2 44/2 22
Thia is equally well understood in examples IIUCh u:
+ = 6~ X 7=
(6 ~) X 7
= 6 X 7 + ' , 2 x 7 =
42 + 3',2 45',2
= = = = Since 6',2 IS/ 2, then 6J,2 X 7 IS/ 2 X ., ft/2 4l5~

d. Multiply a fnctIoa b1 a
fraction.

To multiply J,2 X :If. visualize:
~ ~1---------I-----1

Divide length into 4 parts. Divide width into 2 parts. Count number of parts in whole figure-a. Then 1h X :If., which is one measure, is lIiI of the figure. Another way to visualize the problem:
I
Take 1h of the object

Now take :If. of the ',2
The part so marked shows :JAa of the original rectangle. TherefGl'l
',2 X :If. = lIiI. 148

e. Reclprocala

Generalize: The numerator of the product of two fractions is the product of
the numerators, and the denominator of the product is the product of the denominators.
= = 2/a X 5/8 10/24 5/ 12

Apply properties of numbers to fractions (rational numbers).

= 1/2 X 5/7 5/7 X 1/2 (Commutative Property)

+ + = + + + + (1/2 1/3)
1/5 X (4

1/4 1/2 (1/ 3 1/4) (Associative Property)
1/2) = 1/5 X 4 1/5 X 1/2 (Distributive Property)

Remind pupils that the Closure Property also holds true for rational numbers

(fractions) because in addition or multiplication (as shown):

+ = 1/8

3/8

4/ 8

+ = 5/ 8 3/8 a/ 8 (or 1)

= 1/ 5 X 3/4 3/20
= 1/2 X4/ 2 2/2 (or 1)

The sum of any two rational numbers is a rational number.

The product of any two rational numbers is a rational number. (Whole numbers are rationals since they can be written in the form 8/8)'

Reciprocals: The reciprocal of a rational number is the number by which it is multiplied to give one.

= 1/2 X 2 = 1 2 is the reciprocal of 1/2

2 X 1/2 2/3 X 3/2

1
=

1/2 is the reciprocal of 2 1 2/a is the reciprocal of

3/2

3/2 is the reciprocal of 2/s

These will be helpful in division of fractions.

f. Divide a whole number by a fraction.

This is measurement division-Examples: How many strips of paper % yard long can be cut out from 3 yards?

1 yd.

2 yd.

3 yd.

4 4 4 .t. 4 4 4 4 4 4 4 lf4 4 lf4 4 4
~ '-v----' ~ ~

How many %. yard measures are contained iIi 3 yards?
r-----'\
= = = 3 + %. 12/4 + 3/4 4/ 1 4

The whole number must be replaced by the equivalent rational number having the same denominator as the divisor.

g. Divide a fraction by a whole number.

What part of a strip of fudge candy would each of 4 boys get if they were given lh a bar together?

4 of lh = s of whole
The shaded part shows the lh bar candy they shared. The lh must be separated into 4 parts for the boys.
lh + 4 is the same as saying each boy gets 4 of the lh. lh -7- 4 then = lh X 4 = s.
149

h. Divide a fraction by a fraction; use reciprocals.

To divide in such examples as 2h -+- 'I.,
think: How many '14 's are in 2h 1 Compare with the idea that 6 -+- 2 means how many 2's are in 6.

h....

I--r--I--r--I-,----"I i

The illustration shows that there

are two '14 's in lh and it shows

that subtracting 4 two different

times uses up the lh.

'-~_.. V'f

= Therefore 2h -+- '14 2

Some children will be able to follow the complex fraction illustration:

lh numerator '14 denominator

In studying about reciprocals the pupils have already learned that a number

multiplied by its reciprocal gives 1 and that multiplying both terms of a

fraction by the same number does not change its value. They can follow

the proof then as shown:

= = 1/2 4/ 1

2

2

1/4 4/1 "]

H you multiply the denominator by 4/1 you can multiply the numerator by 4/1 so that the value of the fraction is not changed.

To change '14 to 1 multiply by its reciprocal.

After several proofJ

= 2/ 3 -+- 3/4
2/3 X 4/3 8/g

= 5/S -+- S/.,
5/S X 7/3

= 3 5 / 24

1 11/ 24

point out that the "invert the divisor and multiply" rule is just a short-cut which is developed from the reciprocal method. Stress that correct usage involves multiplying by the reciprocal of the divisor. The divisor and its reciprocal repr.esent different numbers.

The common denominator method will also be helpful in explaining the division process.

6 -+- 3/s =
= 48/ S -+- 3/ S =
48 -+- 3 16

i. Divide with mixed fractions

Dividing with numbers containing both whole numbers and fractions presents little difficulty once the mixed fraction is changed to fractional form.

Example:

2 1/ 2 -+- 3/4 2 1/ 2 is the same as 5/ 2

5/ 2 -+- 3/4, follow through procedure already explained.

7/ S -+- 1 1/ 2 11/2 is same as 3/ 2

7/ S -+- 3/ 2, follow through procedure already explained.

4 2/ 3 -+- 1 1/ 4

= 4 2/ 3

14/3

= 1 1/ 4

5/4

14/3 -+- 5/4, follow through procedure already explained.

The method of changing the terms of the problem may need some explanatioD for the children.

Why does 21/ 2 = 5/21

150

Use the number line to show.

o

1

2

There are two lh's in 1; four lh's in 2; and another lh. There are five lh's in
2lh; 2lh = 5/2,

B. Decimal fractions

1. Recognize and use decimals through hundredths as a new form of fractions.

Compare decimal and common fractions in various ways. The number line is a good illustration.

1

I .1 .2 .3 .4 .5 .6 .7 .8 .9

Ll 1.2 1.3

1/10

2/10 l/s

3/10

4./10 5/10 2/"
1/2

6/ 10 3/ 5

7/ 10

8/10 i/10
4./ 5

H each section of the number line were divided into 100 parts one part would be .01 or 1/100'

Review method of reading, that the last number gives the decimal its name as determined in a place-value chart (See Grade 5 Decimals).

.4, 4 tenths .25, 25 hundredths

2. Read, write, and use decimal fractions through thousandths; learn special terms.

Extend the concept: .714 = 714 thousandths.
Have children read such exercises as:
.5, .012, .47, .09.
Point out the use of the word "and" for the decimal point in reading whole numbers and decimals.

7.02 is read "seven end two hundredths." Practice :reading fractions of all types:
.7, 8.51, 119.09, 652.6, 9.25.

Use in examples showing mileage where odometers are read. Have children make odometers with paper strips, weaving through a long strip cut with windows to show numbers.

3. Express decimals as common fractions and common fractions as decimals.

"Decimal" is a word that comes from a Latin word deci meaning ten. Denomi nators of decimal fractions are powers of ten.
100 = 10' = 10 X 10
= = 10 10' 10 X 1
.1 = 1/10 = = 1/101 1/10 x 1 .01 = = 1/ 100 1/1 0" = 1/10 )[10 .001 = = 1/1000 1/ 10" = 1/IOxl0 XI0
To find what common fraction a decimal fraction represents, place the indicated numerator over the indicated denominator and write in simplest form:

.25 numerator is 25 Jr./100 = 1/4.
denominator is 100

151

To express a common fraction as a decimal fraction:
4 means lout of 4 partl> lf4 is the same as how many out of 10, 100, or 1000.
4 = 1/100
= = Think: 4 X 1 100. The other factor is 25. Then 1 X 25 25.

Both terms of a fraction can be multiplied by the same number without changing its value.
lh = 60/100 4 also means 1 + 4

4/1
4/1.0 .25
4 /1.00

since 1 is not divisible by 4 it must be considered 10 tenths.
or 100 hundredths Divide: get .25

Several examples of this type will help the children generalize that the same fraction can be found by dividing the numerator by the denominator, annexing zeros as needed.
2 no.5
, / 1.00

Some decimal fractions will not be found 10 easily.

.3333 3 / 1.OO

l/S = .33331/.

Such fractions are called repeating fractions because the division is never even. Some fractional equivalents should be committed to memory since they are used so much. However, a child shuuld always have the security of knowing how to find the equivalent of any fraction, common or decimal.

Table of commonly used equivalents:

= 1/2 .5

= 1/4
1/8

=

.25 .33

1/3

= 1/10 .10

= 8/4
2/ 3

=

.75 .66

2/8

4. Add and subtract decimals with accuracy and speed.

The only concept with which children are not thoroughly familiar in addiDI and subtracting decimals is the use of the point itself. Simple explanatioDi that whole numbers, tenths, hundredths, etc. are added to their own kindwhole number to whole number, tenths to tenths, hundredths to hundredthl. should be taught.
1.2 + ,05 + 3.76 + .807 or
1.2
.05 3.76
.807 5.817

152

The vertical form can be simplified by annexing zeros and having all. places filled:

1.200 .050
3.760 .807

= 1.2 1.200 = .05 .050
3.76 = 3.760

5.817

Notice that the decimal points will come under other decimal points.

In subtracting, follow regrouping procedures already learned. Decimal point will come under decimal point because of the place value of each of the numbers, as in addition.

Check addition and subtraction of decimals as in addition and subtraction of whole numbers.

5. Multiply a decimal fraction by a whole number.

First estimate product:
= 2.9 2.9 is almost 3; 3 X 5 15. The product will be between 14 and 15.
5 If 1, 4, and 5 are the digits in the product, the point must come be14.5 tween 4 and 5 to make the product more than 14 but less than 15.
Generalize then, after several examples, that there are as many decimal places in the product as there are in the multiplicand.

6. Multiply a whole number by a decimal fraction.

Notice that the preceding generalization holds true when the decimal fraction is in either factor.
22 .9 19.8

7. Multiply a decimal fraction by a decimal fraction.

Estimate product:

7.8 2.1
78
156
1638 16.38

7.8 is almost 8 2.1 is a little over 2
= 8 X 2 16

Therefore the point will be between 6 and 3. The correct proQuct w111 be 16.38 since this is nearer 16 than 1.638, .1638, 163.8, or 1638.

Draw conclusions through generalizations that there are as many decimal places in the product as there are in both the factors.

8. Verify results in multiplication; apply principles of multiplication to decimals.

Also notice that
= = 7.8 X 2.1 2.1 X 7.8 16.38.
One serves as a check on the other.
Show that the commutative property of multiplication holds true with decimal fractions as it does with whole numbers.
The associative property also applies to multiplication of decimal fractions.

153

- . - '. . . . . ._ _2

_

9. Divide a decimal by a whole number.

A

B

Ones

Tenths

m- 0000

c

- Ones
D

Tenths
0000 0000000000

Tenths
m..
E

-- Hundredths Tenths
Inooauo
F

Hundredths
000000 ,....000000000

Tenths
cal;!
1.

Ones IO~ Hundredths

-

00 ooooooooco

10. Divide a decimal by a decimal.

What number do the cards represent in A? These cards are regrouped in B. What number is shown in B?
Divide the cards in B into two equal parts. How many cards will there be in each group? What number do the cards in each group represent? Use the diagram to show that the following solution is correct:
.7 2 I 1.4
Write in words the number the cards represent in C or D.
Divide the cards in D into two equal groups. What number do the cards in each group represent? Use the diagram to show that the following solution is correct:
.08 2 I .16
What place does the zero hold in the quotient of the preceding problem? What is shown by the cards in E? in F? Divide the cards in F into two equal groups. What number do the cards in each group represent? Use the diagram to show that the solution given is correct:
.15
2 I---:i: == 2 I .30
Estimation .51 2.75 2.75 is not quite 3, and .5 is lh.
How many halves in 3? The quotient will be not quite 6, since the dividend was not quite 3.
Generalize that there are as many decimal places in the quotient as the difference between the decimal places in the dividend and the divisor.
2.75 (2 places) .5 (1 place)
difference (2 - 1) == 1 place, quotient will have 1 place.
Use another form also to show the division
.5 I 2.75 .5 X 10 2.75 X 10 (Fundamental theorem of fractions)

154
J

-----------------------

5.5
5 /--zr:5
25
25
25

Some children may already be familiar with the caret. They should be shown

that in using the caret one does not move a decimal point. 27.5 -:- 5 may
be written ~ and since.~ is another name for 1, 27.5)( 10 = 275 or 275 -:- 5

.5

10

.5 X 10 -5-

or.5 / 27.5

11. Divide a whole number by a decimal fraction.
12. Use short method of dividing by powers of ten.

.6 / 435
.6 / 435.
.6 / 435.0 725
.6/\,/ 435.0",

This kind of problem follows the pattern just examined.
= The child should understand that, if no decimal is
written, it comes at the end of the numeral: 435 435. Also zeros can be annexed without changing the value of the fraction. Use caret to show multiplication by 10 and divide.

Generalize through several examples that to divide by powers of ten one moves the decimal point to the left as many places as there are zeros in the divisor.
75
10 / 750 750 -:- 10 = 75.0
70
-00 875 -:- 10 = 8'7.5, etc.
50

13. Round off decimal fraction. In order to facilitate the use of fractions, explain rounding off to desired places.

Round to nearest tenth:

.67~ .7 .43~ .4 .28~ .3
.172~.2

Round to nearest hundredth:

.007~.01
.2674.27 .311 ~ .31

14. Extend understanding of four fundamentals.

Provide experiences to gain a better understanding of the fundamentals.

Complete the following with "=," ">," or "<."

8 X 3/4 --

14 -:- 2

7.4 - 4

6.8 X 1/2

+ 2/ 3 -:- 2/6 - - 1 + 2 5/ 6 X 2 ------__ 28

4X X 1/7

3/4
2 1/ 2

IV. Relations
A. Equivalence

Throughout the mathematics program the idea of equivalence is stressed. In working with numbers such symbolism as the following is most helpful.

= + 5 0 7 1/ 2 = o + 3 9/2
+ 8 2/ 3 4 1/ 2 = 0

o - 10 = 12.03
15.2 - 4 = 0
= 19 - 0 11 1/ 2

155

2 X 0 = 14
[]X.5=20
7 X 3.4 = 0

50+0=10 0+4=16 18+9=0

The open sentences in the expressions above become closed sentences when the variable is replaced by the replacement set. If the values in that set make
a true mathematical sentence, the set is called the solution set. Fractions have many equivalents (See section ill).

Money has many equivalents. (A nickel is equivalent to 5 pennies.) Decimals have equivalent common fractions.

Per cents have decimal and common fraction equivalents.

One idea that should be emphasized is that there are many equivalents for a number:

+ 8, 5 3, 4 X 2, v'64, 16/2, 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1, 2 (5-1),

2 X 12, and 2 X (10 + 2) are

-3-

3

all names for eight.

B. Inequality.
C. Order of decimals D. Ratio
1. Distinguish between the use of the ratio symbol and the use of the fraction.

Teachers need to be aware of the importance of teaching inequality as well as equivalence.
Symbols to use and understand the meaning of: <, >, =F 5 < 9 five is less than 9
4 > 2 four is greater than 2
4 + 0 > 4 if 0 =/= t), is positive n < 3; 5> n; 1 > n. (Solutions to come from counting numbers.)
These open sentences will become true statements when replaced by their solution sets:
n < 3 solution set {1, 2}
5 > n solution set {1, 2, 3, 4} 1 > n solution set { } A solution set may have many members, one member, or no members.
A particularly important relationship that children should understand is that of "tenness" in olir number system. As they study decimals they should become familiar with the fact that decimal places are just extensions to the right of the decimal point with any place still being ten times larger than the place to its right. (See section ill B.) Ratios may be written in fractional form, but the colon is used also in ratio symbolism:
1/5 or 1:5, both mean "one per five" Explain that ratios can be used as fractions but that all fractions are not ratios.
Use graphs to picture the ratio relations: Gum costs 10c for 3 packages.

~~-----

3 6 9 12 15 18

156

Let horizontal line show multiples of 3; let vertical line show multiples of 10; by looking at the oblique line, one can easily see that 12 packages cost 4Oc.

2 Use ratio in solving problems Ratio is used to show rate or comparison.

that involve the idea of rate

or comparison.

A boy noticed a sign in the store: Gum, 3 packages - 10ct

He wondered how many packages he could buy at the same rate for 5OIf.

The ratio:

Then multiply both terms of the fraction by 5 to make an equivalent fraction

3/ 10 =15/ liO
So the boy could get 15 packages for 50ct.

D. Proportion

Notice that the truth set as a replacement for the variable n in sentence 3/10 = n/50 can be found from the "cross products."
= 3/10 n/ 50
10 X n = 3 X 50
= 10n 150 = n 15
Generalize then that two ratios are equal if their cross products are equal. The generalization will prove helpful when denominators will not divide evenly as they did in the above illustration.

For example:

Find the number of kites that can be bought for 50." at the rate of 3 for 15cf.

3/15 = n/50

= 15n 150

n = 10

Introduce the term proportion as being two equal ratios. Show that proportions may be written in two forms:

2:5 = 8:20
=or
2/5 8/20,

E. Per Cent

In many books diagonal lines will be used for the fractional form:
= 2/5 8/20,
Review meaning of per cent and symbol "%" "per hundred" as

1. Extend concept of per centrecognize symbol

25% = 25 out of 100

25% of lOOn =

12 means
300

25/ 100

=

n/12

n=3

2. Write fractions as per cents

Show how fractions may be written as per cents.

= lh 50%
lf4 = 25%

= lh of 100 50 = lf4 of 100 25

3. Write ver cents as fractions See charts in texts for complete listing:

25% means 25/ 100
If both terms of a fraction are divided by the same number, the value of the fraction is not changed. (Fundamental Theorem of Fractions.)

25+25 1
100 + 25 =4'

157



4. Write decimal fractions as Decimals are based on tens, and 100 is related to 10; the change then, from

per cents

decimals to per cents, is very easy.

5. Write per cents as decimal fractions

.5 is the same as .50. Zeros annexed after a decimal do not change its value because:

5/10 X 10/10 = 50/ 100 = 50%

Generalize the process of changing decimal fractions to per cents; move decimal two places to right and use symbol "%."

= = .25

~/ 100

25%

= = .62

62/100

62%

= = .08
1.12 =

8/100
112/100

=

8% 112%

The inverse relation is not difficult for children to understand.

= = 25% = ~/100 = .25
62% 62/ 100 .62

V. Measurement

A. Standard units of measure
(English)
1. Review concepts and skills previously learned.
2. Introduce the idea of cubic measure.

Review measures:

linear liquid
dry
time time zones

calendar temperature weight money square

Show that cubic measure is the kind of unit used when one is measuring things in which the length, width, and depth are considered, such as the capacity of a box, the amount of air in a room, or the capacity of a refrigerator. One cubic inch would be used as the unit if one were measuring the size of a loaf cake pan. One cubic foot or one cubic yard would be used as the unit if one were measuring the capacity of a railway freight car.

f(11
tal 1

1 cubic unit

In the first layer of the following figure there are 2 rows of 6 cubic units each which is 12 cubic units. There are 3 such layers. Then 12 cubic units taken 3 times will be 36 cubic units.

= 6 cubic units X 2 = 12 cubic units.
12 cubic units X 3 36 cubic units several illustrations, particularly those using wooden cubes in various arrangements, will help develop the concept of 3 dimensional measurement 3. Add and substract measures. Measures of all kinds must be combined at times. Apply principle of regrouping.
Find sum of:
158

4. Apply principle of regrouping in multiplying and cliriding measures.

3 gals. 3 qts. 1 pt.
I 1 gal. 2 qts. 1 pt.
4 gals. 5 qts. 2 pts. 1 qt. ~
1 gal ~ 6 qts.
5 gals. 2 qts.

How much cloth must be bought to make 3 skirts if 1 skirt has 27 inches in it?

27 3
81 inches

36 inches = 1 yd.
2
36 / 81 72
9""

2 yards 9 inches
= 9/86 1/.
21/. yards

A board. is 12 ft. 6 in. long. How many pieces 2 ft. 3 in. long can be cut from It?
Change 12 ft. 6 in. to 150 inches.
Change 2 ft. 3 in. to 27 inches.
Divide: 5 boards can be cut
27 / 150 135

5. Solve problems using all processes with measures.

Problems such as those already shown indicate the type to be used in all kinds of measures. It will be necessary to remember that all four processes, addition, subtraction, multiplication, and division should be used with linear, dry, liquid, weight, time, temperature, and other measures. Arrange to have problems calling for changing from one, denomination to another.

6. Approximate measures of different kinds.

Concrete examples are necessary to help develop the skill of approximation.
Actual quarts, pints, pounds, pecks, and the like with many associations give the child confidence in "guessing."
Such things as these help: How much does this bag of sand weigh? About as much as 2 sticks of butter? A bag of coffee? How much water is in this jar? About as much as the :Ih pint of milk we drink each day?
How hot is the water we heated? Can we put a hand in it? How hot is the liquid when it bubbles?

7. Learn to solve readily and accurately problems dealing with money.

Plan excursions; use committees for cost of trip, cost of food, and cost of lodging.
Plan construction of various types. How much will material cost? Compare cost with substitute material. Share cost with class members. Keep accounts of various class projects; for example:
party costs, Jr. Red Cross contributions, individual records of allowance, receipts and expenditures.

159

Have projects for class members to help with family grocery purchasing. Keep individual weekly records. Use ads in the newspapers to decide where to buy an agreed upon list of goods as,
sporting goods, clothing, food, boats, and toys.

B. Metric system 1. Understand and use the metric system.
2. Learn relations between grams, liters, meters, and kilograms, kiloliters, and kilometers.

Review concept of meters, decimeters, and centimeters. Introduce grams and liters. Show actual measures; have scales and measuring devices for comparison with English system such as: A meter is a little more than 39 inches. A centimeter is equal to .4 inches. A liter is a little less than a quart. Kilogram is about 2.2 pounds.
Explain that the term kilo means one thousand. Call attention to various measures as:
kilometer, kilogram, and kiloliter.
Many movies with foreign settings state distances in kilometers instead of miles. One kilometer is approximately 0/8 mile. Commercial made metric charts may be used to an advantage.

VI. Geometry

A. Interpret latitude and longitude; know meaning of meridi-
an.

Teach children the meaning of the term degree. Show the relation of the earth to a sphere and that meridians mark off 15 spaces on the globe.
Correlate longitude and latitude problems in geography with the work in arithmetic.

B. Find perimeters and areas of polygons

Review section on perimeters and areas from fifth grade. Expand to enable children to recognize various polygons in a complex figure.

3
In this figure the perimeter is formed by adding all the numbers around the figure.

Explain the definition of perimeter. The perimeter of the figure here is 15.
To find the area: The area of the rectangle is 2X3=6 The area of the triangle is
2X 3= 3
2
Therefore the area of the polygon is 6 + 3 or 9.
Other examples can be used to give practice in finding figures within figures.
mo

c. Find volume of rectangular
solid.

An introduction to volume in the fifth grade will have Provd d

finding volume.

1 e readiDesa for

Review:

Use actual wooden illustrate 2 X 2 X

c5ub==es20t0

~~

2
Show other combinations:
3X3X4=36 2X4x3=24
Generalize: To find the volume of a rectangular solid multiply the length by the width by the height. A formula could be written:
= V lwh
Show that a cube is a special case.

3X3X3=27

3

Formula:

V=e3

D. Realize that cubes, cones, cylinders, spheres, and the like are solids.
E. Understand angles.

3
The children learned to recognize these figures in the fifth grade and to call them by name. They were introduced to the concept that these figures enclose space, but this concept can be expanded by having children actually make the figures.
Patterns can be found in most any book on paper sculpture. (Review Section from 5th Grade.)
An angle is formed by 2 rays, which have a common end point.

acute

right

obtuse

Angles are measured by degrees. There are 360 in a circle. As the rays spread apart the number of degrees between them increases. The length of the ray does not determine the size of the angle.

Have children measure angles using protractors. Learn names of angles: acute, less than 90; right, 90; and obtuse, more than than 90
Use terminology in the classroom throughout the day.

161

F. Understand meaning of line segment.

Find:
angle at which to hold a pencil angles made by walls angles made by arms during exercises
This concept was introduced in Grade 5 but must be constantly reviewed. A line is a set of points extending in opposite directions.

AB


A line segment is the set of points between two points on the line; AB is a line segment.

Is RS a subset of line segment AB?

A
_ _ _0

R

S

0_--0

B
0 _ _-

Is circle A a subset of circle B?

What point is an element of the set of points of each line?

A

E

D

C.

The following represents (union, intersection, subset, none)

G. Know meaning of congrueney.

Children can soon decide if one figure is congruent to another even without knowing the term. They divide a paper into 2 parts or 4 parts. They note arches that have curves corresponding to other curves. They are accustomed to gloves and some other apparel exactly matching.
Show them how geometrical figures are congruent also.
Line segments
.A---.B .C---.D

Triangles Other Polygons

6
/7

6
/7

Angles

~~

162

H. Introduce Law of Pythagoras.

If a triangle has a right angle the Law of Pythagoras holds true.
The square upon the longest side equals the sum of squares upon the other two sides.

VII. Graphs, Charts, and Scales

A. Maintain and extend skill in reading, interpreting and drawing graphs.

Review picture graphs, line and circle graphs. Give more data to be graphed in each form. Have rather complete graphed material bought from sources outside the classroom to be read in class.
Examine carefully graphs in geography, science, or other books. Stress the importance of reading carefully graphed material. For example, point out what changing a scale does to the mental image created. Show how an advertiser can use actual facts and yet create a picture to the reader that is not accurate.

B. Continue making scale drawings.

Make floor plans of houses to scale. Make models of furniture. Draw the city park or recreation center to scale. Have children make notebooks of their own scale drawings. Emphasize correctness.

C. Chart various experiences of childnln,

Children like to keep records of their own experiences.
They can keep up with spelling grades, weights and heights, scores in games, and other stich activities by graphing or charting these. By making their own, they will be better able to interpret others.

D. Chart statistical data.

Chart the increase in population. the growth in crop production, the rainfall during the month, temperature ranges, political campaigns, and voting records. Such data as these furnish excellent sources of material for charting and graphing.

E. Read maps to various scales.

Maps come in various sizes and types. Children have at their finger-tips a wealth of knowledge just from becoming familiar with maps. The arithmetic centered around maps is bound only by the knowledge and initiative of the teacher.

VIII.

Problem Solving It is generally agreed that arithmetic is best taught through problem solving
situations. There should be problems in each area already discussed in the guide.

A. Understand steps for problem Teach the children the steps to follow in working through a problem, as

solving.

follows:

Read and understand relationships.
Write in a mathematical sentence. Decide the operation or operations to be used.

163

Follow through on the operation(s). Determine results.
(Compare these steps with steps in solving problems in science or other fields.)
Teach children to read mathematics. This is a responsibility of every teacher in mathematics and must not be termed solely the reading teacher's job. Every discipline has its own terminology and word arrangement that must be understood in order for a person to become adept in reading it.

B. Solve problems of various types.
C. Solve problems involving numbers expressed as fractions.

Find the smaller quantity when the larger quantity and the excess of the larger quantity or the deficiency of the smaller quantity are known.
John had 12 marbles which was 3 more than Frank (or Frank had 3 less). How many did Frank have?
Find the quantity when the smaller quantity and the deficiency of the smaller quantity or the excess of the larger quantity are known.
Mary baked 4 dozen cookies which was 2 dozen less than Clara baked (or Clara baked 2 dozen more). How many cookies did Clara bake?
Find the ratio of one quantity to another (situations involving "times as many as" or "fraction as many as").
Sam has 30 cents and Mary has 10 cents. Sam has how many times the number of pennies that Mary has?
George picked 20 apples today, but yesterday he picked 120. His yield today is what part of yesterday's yield?
Find a second quantity when the first quantity and its ratio to the second quantity are known.
Sam has 3 times as many pennies as Sue and he has 30c. How much money has Sue?
George picked 1f4 as many apples today as he did yesterday. If he picked 120 yesterday, how many did he pick today?
Find a quantity when the other quantity and its ratio to the first quantity are known.
Sam has 3 times as many pennies as Sue. If Sue has 10, how many does Sam have?
George picked 30 apples today. If this is Y& as many as he picked yesterday, how many did he pick yesterday?

Mary had to make a doll outfit. She bought the following material:

lh yard cloth 4 yard lace 1 spool thread
6 buttons

(79</ a yard)
(10" a yard) (3 spools 10,,~ (2 for 10,,)

How much did the outfit cost?

Such examples furnish many uses of fractions in problem solving.

164

------------------------..,

abbreviate accurate addend addition altitude amount angle approximate Arabic Numerals area average
bar graph base billion boiling point
casting out centigram centimeter century circle graph circumference clockwise column combine common denominator common factor compare complex fraction counter clockwise cube cylinder
decade deeimal fraction decimal point decimal system degree denominator diagonal difference

UPPER ELEMENTARY LEVEL VOCABULARY

digit dimension direction distance divide
element end point equals equilateral equivalent estimate even numbers exactly example excess expense exponent
factors Farenheit figures fractions
geometric gram graph gross
hemisphere horizontal hundred billion hundredth
identity integer interseetion inverse isosceles
kilogram kiloliter kilometer

latitude length line graph liquid liter longitude
mean median meridian meter metric midnight milligram millimeter millions millionth minus mode multiple multiplication
negative noon numeral numerator
oblique odometer
parenthesis per cent perimeter period perpendicular pictograph polygon power prime prism problem product

property of number pyramid
quadrilateral quinary
radius ratio rational number ray rectangle rectangular solid reciprocal remainder repeating decimal right angle rounding off
scale solid speedometer square square measure speed symbols
temperature ten billion terms thermometer thousand thousandth time zone ton triangle trapezoid
uneven union
vertical volume

185

BOOKS FOR CHILDREN
Andrews, F. Emerson. Numbers Please. 1961. Little Brown & Co. Behn, Harry. AU Kinds of Time. 1950. Harcourt Brace Co. Bendick, Jeanne. How Much and How Many. 1947 J. B. Lippincott Co. Bendick, Jeanne and Levine, Marcia. Take A Number. 1961. McGraw-Hill Co. Brades, Louis. Mathematies Can Be Fun. 1955. J. Weston Walch Pub. Bradley, Duane. Time For You. 1960. J. B. Lippincott Co. Bragdon, L. J. Tell Me The Time Please. 1936. J. B. Lippincott Co. Brindze, Ruth. The Story of Our Calendar. 1949. Vanguard Press Clark, Garel. Let's Start Cooking. 1951. W. R. Scott Fenton, Carroll L. and Fenton, Mildred A. Worlds In The Sky. 1950. John Day Co. Flynn, Harry. Tick, Tock, A Story Of Time. 1938. D. C. Heath Co. Freeman, Ira. Fun With Figures, Easy Experiments For Young People. 1946.
Random House. Friends, Newton. Numbers: Fun And Fact. 1954. Scribner & Sons. Hogben, Lancelot. The Wonderful World of Mathematics. 1955. Doubleday & Co. Huffman, Peggy. Miss B's First Cookbook. 1950. Charles E. Merrell Karpinske. History Of Mathematics. Rand McNally. Lauber, Patricia. The Story Of Numbers. 1961. Random House. Leaf, Munro. Arithmetic Can Be Fun. 1949. J. B. Lippincott. Leeming, Joseph. From Barter To Banking. Appleton Century. Leeming, Joseph. Fun With Puzzles. 1946. J. B. Lippincott Leeming, Joseph. More Fun With Puzzles. 1947. J. B. Lippincott. Marshak, Ilin. What Time Is It? 1932. J. B. Lippincott Co. Meyer, Jerome S. Fun With Mathematics. 1952. World Book Co. Moore, Lillian. The Important Pockets Of Paul. 1954. McKay. Mott-Smith, Geoffrey. Mathematical Puzzles For Beginners And Enthusiasts. 1954.
Dover Publishing Co. Norman, Gertrude. The First Book Of Music. 1954. Watts Co. Sanford, Vera. A Short History Of Mathematics. 1930. Houghton-Mifflin Co. Schlein, Miriam. It's About Time. 1955. Scott-Foresman Co. Schlein, Miriam. Shapes. 1952. W. R. Scott. Schloat, G. W., Jr. Adventures Of A Letter. 1949. Charles Scribner Sons. Shackle, G. L. S. Mathematics At The Fireside. 1952. Cambridge Universal Press. smith, David E. Number Stories Of Long Ago. 1951. National Council of Teachers.
Of Mathematics Smith, David E. The Wonderful Wonders Of One-Two-Three. 1937. Warde & McFarlane Tannenbaum and Stillman. Understanding Maps. 1957. Whittlesey. Townsend, Herbert. Our Wonderful Earth. 1950. Allyn and Bacon Co. Werner, Elsa Jane. The Golden Geography Book. 1952. Simon Co. Wilcox and Burke. What Is Money? 1959. Steck Publishing Co. zarchy, Harry. Let's Make Something. 1941. Knopf. zarchy, Harry. Wheel Of Time. 1957. Crowell Ziner and Thompson. The True Book Of Time. 1956. Children's Press.

MATERIALS

FILMS

FILMSTRIPS

About Money (Cbildren'. Productiona)

Adding Fractions (Eye Gate)

Addition Is Easy (Coronet Films)

Addition Combinations (Society For Visual Education)

BolTOwing In Subtraction (Teachin.g Films Custodian) Adventures With Number Series (Popular SCience)

Decimals Are Easy (Coronet Films)

Amazing Arithmetic: Common Fractions (Eye Gate)

Division Is Easy (Coronet Films)

Arithmetic Experiences Series (Curriculum Filma, Inc.)

Extending Mathematical Ideas (Ginn and Co.)

Arithmetical Experiences Series (Eye Gate House, Inc.)

How To Add Fractions (Hunt)

Arithmetic Series (Young America Films)

How To Change Fractions (Hunt)

Basic Numbers Speed - i - 0 - strip series: MUltiplication

How To Divide Fractions (Hunt)

And Division (Society For Visual Education)

How To Subtract Fractions (Hunt)

Business Methods For Young People <V. F. C.)

Introduction To Fractions (Hunt)

Common_Denominators (Eye Gate)

Let's Measure: Ounces, Pounds, Tons (Coronet Films) Decimals And Percentage (Curriculum Films)

Maps Are Fun (Coronet Films)

Division Combinations (Society For Visual Education)

Meaning of Long Division (Encyclopaedia Britannica Films)

Dividing With Fractions (Filmstrip House) Fractions Series (Curriculum Films)

Measurement (Coronet Films)

Fraction Series (Society For Visual Education)

Multiplying Fractions (Knowledge Builders)

History of Measures Series (Young America Films)

New Horizons In Arithmetic (Ginn and Co.)

Introduction To Fractions (Jam Handy Organization)

Other Number Bases (Webster Co.)

Learning New Numbers: Fractions (Filmstrip House)

Parts of Long Division (Young America. Films)

Light On Mathematics (Jam Handy)

Simple Fractions (KnOWledge Builders)

Man And Measure Series (Filmstrip House)

Subtraction Is Easy (Coronet Films)

Measuring Time And Things (Dukane)

The Language Of Graphs (Coronet Films)

Multiplication Combinations (Society For Visual Educato tion)

The Meaning Of Percentage (Young America Films) Multiplication And Division (Young America Films)

The Number System (Encyclopaedia Britannica Films) Multiply Fractions: Numerator And Denominator (photo And Sound Production)

Using The Banks (Encyclopaedia Britannica Films)

Multiplying By Fractions (Eye Gate)

Weights And Measures (Encyclopaedia Britannica Films) Multiplying Fractions By Fractions (Curriculum Films)

Weights And Measures (Encyclopedia Britannica Films)Number Recognition (Society For Visual Education)

What Are Decimals? (Encyclopaedia Britannica Films) Subtraction Combinations (Society For Visual Education)

What Are Fractions? (Encyclopaedia Britannica Films) The New Elementary Mathematics Filmstrips in Color.

What Is Money? (Coronet Films)

Directed by Dr. Bernhard H. Gunlach (Colonial Films)

Your Thrift Habits (Coronet Films)

Units And Fractional Parts (photo And Sound Productions)

Using And Understanding Numbers: Decimal and Measurement (Society For Visual Education)

167

Abacus Area board Balls Blocks Bottle caps Bottles Boxes Bulletin boards Cabinets Calendars Carpenter's ruler Cases Cans Cardboard Cartons Chalkboards Clocks Clothespins Coat hangers Coins Compass

AIDS AND EQUIPMENT

Craftsman's T-square Crayons Cups Date stamp Decimal board Decimal chart Disc Division kit Flannel board Flannel board cut outs Fraction board Fraction kit Fraction wheel Fruit Games Geometric models Geometric pictures Globes Graph paper Hundred board

Jars Odometer Protractor Meter stick Metric chart Multiplication kit Muffin tins Number line Paper plates Percentage board Place value board Place value pockets Place value charts Peg board Pegs Pipe cleaners Price tags Projectors Puzzles Recordings Rectangular graph

Road maps Rulers Scales Scissors Set of weights Shelves Slide-Hide calculator Speedometer Spoons Steel tape Stop watch String Sun dial Tables Tape measure Telephone book Thermometers Time tables Toothpicks Travel schedules Yardstick

168

Volume I

REFERENCES FOR TEAcHERs

A. Periodicals

Mathemetics Student Journal. National Council of Teachers of Mathematics, 201 Sixteenth Street, N. W., Washing-

ton 6, D. C. (for grades 712)

.

School Science and Mathematics. Central Association of Science and Mathematics Teachers, Oak Park, illinois.

The Arithmetic Teacher. National Council of Teachers of Mathematics, 1201 SiXteenth Street, N. W., WashingtoD

6,D. C.

The Mathematics Teacher. National Council of Teachers of Mathematics, 1201 Sixteenth Street N. W., Washington

6,D. C.

B. Books
Archer, Allene. Number Principles and Patterns. Ginn and Co., 1961. Arbuthnot, Mary Hill. Time for Poetry. Scott Foresman and Co., 1952. Brueckner, L. J. and Grossnickle, F. E. Making Arithmetic Meaningful. John C. Winston Co., 1953. Buckingham, Burdette R. Elementary Arithmetic: Its Meaning and Practice. Ginn and Co., 1953. Demay, Amy J. Guiding Beginners in Arithmetic. 'Row Peterson and Co., 1957. Devault, M. Vere. Improving Mathematics Programs. Charles E. Merrill Books, Inc., 1961. Flieglet, Louis A. Curriculum Planning for the Gifted. PrenticeHall, Inc., 1961. Heniz, Mamie W. Growing and Learning in the Kindergarten. John Knox Press, 1959. Hickerson, James A. Guiding Children's Arithmetic Experiences. PrenticeHall, Inc., 1952. Hollister, G. E. and Gunderson, Agnes. Teaching Arithmetic in Grades I and II. D. C. Heath and Co., 1854. James, G. and James, R. C. Mathematics Dictionary. D. Van Nostrand Co., Inc., 1959. Lambert, Haze] M. Teaching the Kindergarten Child. Harcourt Brace and Co., 1958. Larson, H. D. Enrichment Program for Arithmetic. Row Peterson and Company, 1958. Marks, John L., Smart, James K. and Sauble, Irene. Extending Mathematical Ideas. Ginn and Company, 1956. Marks, John L., Purdy, Richard and Kinney, Lucien B. Teaching Arithmetic for Understanding. McGraw-Hill BOok
Co.; 1958. McSwain, E. T. and Cooke, Ralph J. Understanding and Teaching Arithmetic In the Elementary School. Henry Holt
and Co., 1958. Merton, Elda L. Arithmetic Readiness Experiences in the Kindergarten. John C. Winston Co., Moore, Elenora H. Fives at School. G. P. Putnam's Sons, 1959. National Council of Teachers of Mathematics. Insights Into Modern Mathematics: Twenty-third Yearbook. The COUD-
cil,1957. National Council of Teachers of Mathematics. Instruction in Arithmetic: Twentyflfth Yearbook. The Council, 1960. National Council of Teachers of Mathematics. The Growth of Mathematical Ideas: Twenty.fourth Yearbook. The
Council, 1959. Russell, David. Children Learn To Read. Ginn and Company, 1949. Scott and Thompson. Rhymes for Fingers and Flannelboards. Webster Publishing Co., 1958. Spitzer, Herbert F. Practical Classroom Procedures for Enriching Arithmetic. Webster Publishing Co., 1956. Spitzer, Herbert F. The Teaching of Arithmetic. Houghton Mifflin Co., 1961. Swain, R. L. Understanding Arithmatic. Holt, Rinehart and Winston, 1960. Taylor, E. H. and Mills, C. N. Arthmetlc for Teacher-Training Classes~ Henry Holt and Company, 1955. Willis, Ed. D. Living in the Kindergarten. Follett Publishing Co., 1956. Wheat, Harry Grove. How to Teach Arithmetic. Row Peterson and Co., 1956. Woodward, Edith J. and McLennan, R.C. Elementary Corcepts of Sets. Holt, Rinehart and Winston, Inc., 1959 Young, Fredrick H. Digital Computers and Related Mathematics. Ginn and Co., 1961.

C. Articles, Pamphlets and Bulletins
Arithmetic Learning Sequence for Elementary Schoel. Curriculum Bulletin No. 17, Wisconsin Cooperative Educational Planning Program. Madison, Wisconsin, 1958:
Deans, Edwina. "Arithmetic in the Primary Grades." National Elementary Principal, October, 1959, PP 22-28. Deans, Edwina. "Independent Work in Arithmetic." The Arithmetic Teacher, February, 1961. pp 77-78. Dean, Richard A. ''Defining Basic Concepts of Mathematics." The Arithmetic Teacher, VD:8, March, 1960. Denny, Robert. How to Challenge the Gifted in Arithmetic. Des Moines: The Denny Press, 1959. Glenn, William H. and Johnson, Donovan A. Short Cuts in Computing. St. Louis: Webster Publishing Co., 1961.
ae. Ginn Games for Arithmetic, Grades 3'8. Bulletin, Atlanta: Ginn and Co., 1959.
Gunderson, A. C. and Gunderson Ethel. "Fraction Concepts Held by Young Children." The Arithmetic Teacher. ber, 1957.

169

Gunderson, Agnes C. "Thought-Patterns of Young Children in Learning Multiplication and Division." The Elemen tary School Journal. April, 1955.
Harting, M. L.; VanEugen, Henry; Knowles, Lois and Gibb, Glendine. Charting the Course for Arithmetic. Chicago: Scott Foresman, 1960.
Heard, Ida Mae, "Developing Concepts of Time and Temperature." The Arithmetic Teacher, March, 1961, pp 124-126. Hohn, Franz E. "Teaching Creativity in Mathematics" The Arithmetic Teacher, VIII: 5, March 196!. Klas, Walter L. "Problems Without Numbers." Arithmetic Teacher, 8: 19-20, January, 1961. Morton, Robert Lee. Helping Children Learn Arithmetic. Chicago: Silver Burdett, 1960. Morton, Robert Lee. Teaching Children Arithmetic. New York: Silver Burdett, 1953. Parker, Helen C. "Teaching Measurement in a Meaningful Way." Arithmetic Teacher, 4:194-198, April, 1960. Perrodin, Alex F. Arithmetic Aids Through the Grades. College of Education, University of Georgia, 1959. 171p. Smith, D. E. Numbers and Numerals; Number Stories of Long Ago; Wonderful Wonders of 1, 2, 3. National Council
of Teachers of Mathematics, Washington, D. C. Stern, Catherine. "New Experiments with Multiplication." The Arithmetic Teacher, 8: 381-388, December, 1960. The Elementary and Junior High School Mathematics Library by Clarence Ethel Hardgrove. National Council of
Teachers of Mathematics. 1201 Sixteenth Street,Washington 6, D. C. 1960. 18p. Thorpe, Cleata B. "These Problem Solving Perplexities." The Arithmetic Teacher. 4: 152156, April, 196!. Unkel, Ester R. "Children are Naturals at Solving Word Problems." The Arithmetic Teacher. April, 1961, pp 161-163. Urbanek, Joseph J. (ed) Mathematical Teaching Aids. Chicago Schools Journal Vol. 35: 3-6 supplement. Chicago Teach.
ers College, 1955. Van Eugen, Henry. "Rate Pairs, Fractions, and Rational Numbers." The Arithmetic Teacher. VII:8, December, 1960. What are Kindergartens For? Association for Childhood Education International, Washington, D. C., 1959.
D. Courses of Study
Arithmetic-Mathematics Teachers of New Mexico Arithmetic Curriculum Guide. New Mexico: State Department of Education.
Board of Education, City of Chicago, Arithmetic Teaching Techniques. Chicago, 1956. Dekalb County, Board of Education, Activities and Aids for Teaching Arithmetic. DeKalb County, Decatur, Georgia,
1959. Denver Public Schools Mathematics Study, Mathematics Program of the Denver Public Schools. Volumes: Kindergar-
ten and Grade One; Grade Two and Three; Grades Four and Five; Grade Six. Denver, Colorado: Department of General Curriculum Services. Elementary Mathematics Curriculum Committee, Elementary Mathematics Curriculum. Washington, D. C. Public Schools of the District of Columbia. Gunlach, D. H. Basic Mathematics for Elementary Teachers. Bowling Green, Ohio: Educational Research Council of Greater Cleveland. 1961. Hawley, Newton and Suppes, Patrick. Suppes Arithmetic Project. Sets and Numbers, Book I; Sets and Numbers, Book
n. Holden-Day, Inc., 1960.
Hance, Lyle S. (ed) Tentative Course of Study in Arithmetic. Atlanta: John C. Winston Publishing Company. Mathematics for the Elementary School (Grades 4, 5, 6). Yale University: School Mathematics Study Group, 196!. Springfield Massachusetts Public Schools, Arithmetic Curriculum Guide. Springfield, Mass., 1960. Studies in Mathematics Education. A Brief survey of improvement programs for school mathematics. Atlanta: Scott
Foresman and Company. 1960. 71 p. The Greater Cleveland Mathematics Project GCMP Teachers Guide Primary Part 2, Preliminary Edition. Ohio: The
Educational Research Council of Greater Cleveland, Ohio, 1960. Wisconsin Cooperative Educational Planning Program, Arithmetic Learning Sequence for Elementary School. Madison, Wisconsin, 1958.
E. Mathematics Tests
Myers, Sheldon S. Mathematics Tests Avaliable in the United States. National Council of Teachers of Mathematics, 1201 Sixteenth Street, N. W., Washington 6, D. C. 1959.
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GLOSSARY FOR TEACHERS

ABSCISSA. If the ordered pair of numbers (a. b) are the coordinates of a point of a graph, the number a is
the abscissa.

<\BSOLUTE ERROR. One -half the smallest marked interval on the scale being used.

AIaBISO=LU-aT.EOVnAtLhUeEn.uTmhbeear blsinoleutaebsvoalluutee

of the value

real number a is the distance

is denoted by Ial
of a point from

If a >
zero.

0

then

1al

== a and if a < 0,

ACCURACY. The accuracy of a measurement depends upon the relative error. It is directly related to the number of significant digits in the measured quantity.
+ ADDITIVE IDENTITY. The number I in any set of numbers that has the following property: I a = a for all + a in the set. The symbol for the identity is usually 0; in the complex numbers it is 0 Oi, and in some systems
bears no resemblance to zero.

ADDITIVE INVERSE. For any given number a in a set of numbers the inverse, usually designated by (-a) is that number which when added to a will give the additive identity as a result.

Example: Additive inverse of 5 is -5 and the additive inverse of -5 is 5.

ALGEBRAIC EXPRESSION. An algebraic expression may be a single numeral or a single variable; or it may consist of combinations of numbers and variables, together with symbols of operation and symbols of grouping.

ALGORlTHM. (ALGORISM) Any pattern of computational procedure.
AMPLITUDE. The amplitude of a trigonometric function is the greatest absolute value of the second coordinates of that function. For a complex number represented by polar coordinates the amplitude is the angle which is the second member of the pair.
ANGLE. The set of all points on two rays which have the same end-point. The end-point is called the vertex of the angle, and the two rays are called the sides of the angle.
ANGULAR VELOCITY. The amount of rotation per unit of time.
APPROXIMATE MEASURE. Any measure not found by counting.
APPROXIMATION. The method of finding any desired decimal representation of a number by placing it within successively smaller intervals.

ARC. If A and B are two points of a circle with P as center, the arc AB is the set of points in the interior of LAPB on the circle and on the angle.
AREA OF A SURFACE. Area measures the amount of surface.

ARGAND DIAGRAM. Two perpendicular axes, one which represents the real numbers, the other the imaginary numbers thus giving a frame of reference for graphing the complex numbers. These axes are called the real axis and the imaginary axis.

ARITHMETIC 1I-IEANS. The terms that should appear between two given terms so that all the terms will be an arithmetic sequence.

ARITHMETIC SEQUENCE (pROGRESSION). A sequence of numbers in which there is a common difference between any two successive numbers.

ARITHMETIC SERIES. The indicated sum of an arithmetic sequence.

= + + ASSOCIATIVE LAW. A basic mathematical concept that the order in which certain types of operations are per-
formed does not affect the result. The laws of addition and multiplication are stated as (a + b) + c a (b c) and (a X 1?) X c == a X (b X c).

AVERAGE. A measure of central tendency. See mean, median and mode.

AXIS OF SYMMETRY. A line is called an axis of symmetry for a curve if it separates the curve into two portions so that every point of one portion is a mirror image in the line of a corresponding point in the other portion.

BASIC TABLE. The name given to any operational table in a base or place-value arithmetic: as, basic addition tables, subtraction tables, multiplication tables, division tables, power tables, logarithmic tables, etc.

171
l --------

BASE. The first collection in a number series which is used as a special kind of one. It is used in combination with the smaller numbers to form the next number in the series. In the decimal system of numeration, eleven which is one m0re than the base of ten literally means "ten and one". Twenty means two tens or two of the br:se.
BASE TEN. A system of numeration or a place-value arithmetic using the value of ten as its base value.
BETWEENNESS. B is between A and C if A, B, and C are distinct points on the same line and AB + BC = AC.
BINARY OPERATION. An operation involving two numbers such as addition; similarly, a unary operation involves only one number as "the square of".
BINARY SYSTEM. A system of notation with base two. It requires only two symbols: 0 and 1.
CARTESIAN PRODUCT. The cartesian product of two sets A and B, written AXB and read "A cross B" is the set of all ordered Paris (x,y) such that x A v B.
CHECK. To verify the cOiTectness of an answer or solution. It is not to be confused with "prove".
CIRCLE. The set of points in a given plane each of which is at a given distance from a given point of the plane. The given point is called the center, and the given distance is called the radius.
CIRCULAR FUNCTION. A function which associates with each arc of a unit circle (as measured from a fixed point of the circle) a unique point. The sine function associates with the measure of an arc the ordinate of its companion point, and the cosine the abscissa of the point.
CIRCUMFERENCE. The. length of the closed curved line which is the circle.
CLOSED CURVE (SIMPLE). A path which starts at one point and comes back to this point without intersecting itself represents a simple closed curve.
CLOSURE. A set is said to have the property of closure for any given operation if the result of performing the operation on any two members of the set is a number which is also a member of the set.
COLLECTION. Elements or objects united from the viewpoint of a certain common property; as collection of pi~tures, collection of stamps, numbers, lines, persons, ideas.
COMMUTATIVE LAW. A basic mathematical concept that the order in which certain types of operations are per-
= formed does not affect the result. Addition is commutative; for example 2 + 4 4 + 2. Multiplication is com-
= mutative; for example. 2 X 4 4 X 2.
COMPASS (OR COMPASSES). A tool uSed to construct arcs and circles.
COMPLEX FRACTION. A fraction that has one or more fractions in its numerator or denominator.
= COMPLEX NUMBER. Any number of the form a+bi where a and b are real numbers and i' -1.
COMPOSITE NUMBER. A counting number which is divisible by a smaller counting numbers different from 1.
CONCRETE. Belonging to things; that which is directly experienced by our senses.
CONJUGATE COMPLEX NUMBERS. The conjugate of the complex number a + bi is the complex number a - bi
CONJUNCTION. A statement consisting of two statements connected by the word and. An example is x + y = 7 and x - y = 3. The solution set for a conjunction is the intersection of the solution sets of the separate state-
ments.
CONDITIONAL EQUATION. An equation that is true for only certain values of the variable. Example: x + 3 = 7.
CONIC, CONIC SECTION. The curves which can be obtained as plane sections of a right circular cone.
CONSISTENT SYSTEM. A system whose solution set contains at least one member.
CONSTANT. A particular member of a specified set.
COTERMINAL ANGLES. Two angles which have the same initial and terminal sides but whose measures in degrees ailfer by 360 or a multiple of 360.
COUNTABLE. In set theory, an infinite set is countable if it can be put into one-to-one correspondence with the natural numbers. COUNTING NUMBERS. {l, 2, 3, 4, ... }
CONVERGENT SEQUENCE. A sequence that has a limit.
DECADE. A specified set of ten. It refers to the subsets of the natural numbers known as ten, twenty, thirty, ete
In measurement of time it represents a period of ten years.
172

DECIMAL EXPANSION. A digit for every decimal place. DEDUCTIV}; REASONING. The process of using previously assumed or known statements to make an argument for new statements.
DEGREE. In angular measure, a standard unit that is 1/90 of the measure of a right angle. In arc measure, one of the 360 equal parts of a circle.
DEGREE OF A POLYNOMIAL. The general polynomial aox" + a.x" - I + ... + a,,_.x' + a" is said to be of degree n if ao"&o. DENOMINATOR. The lower term in a fraction. It names the number of equal parts into which a number is to be divided. DEPENDENT LINEAR EQUATIONS. Equations that have the same solution set. DDfFERENCE. The answer or result of a subtraction. Thus, 8 - 5 is referred to as a difference, not as a remainder. DIGIT. Any one of the ten symbols used in our numeration system; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. (From the Latin, "digitus", or "finger".) DllIEDRAL ANGLE. The set of all points of a line and two non-coplanar half-planes having the given line as a common edge. The line is called the edge of the dihedral angle. The side or face consists of the edge and either half-plane. DIRECT VARIATION. The' number y varies directly as the number x if y = kx where k is a constant. DISCRIMINANT. The discriminant of a quadratic equation ax" + bx + c = 0 is the number b" - 4ac. DISJUNCTION. A statement consisting of two statements connected by or. Example: x + y = 7 or x - y = 3. The solution set of disjunction is the union of the solution sets of the separate statements. DISTRIBUTIVE LAW. Links addition and multiplication.
Examples: 3X14 = 3(10+4) = (3Xl0) + (3X4) = 30 + 12 = 42. 4X3lh = 4(3+lh) = (4X3) + (4xlh) = 12+2 = 14. a(b+c) = ab+ac.
DIVERGENT SEQUENCE. A sequence that is not convergent. DIVISION. The inverse of multiplication. The process of finding how many times one quantity or number is contained in another. For any real numbers a and b, b =fa 0, "a divided by b" means a multiplied by the reciprocal of b. Also, a+b=c, if and only if a=boc. DOMAIN OF A VARIABLE. The set of all values of a variable is sometimes called its domain. DUODECIMAL SYSTEM. A system of notation with base twelve. It requires twelve symbols: 0, 1,2,3,4, 5, 6, 7, 8,9, T, E. ELEMENTS. In mathematics the individual objects included in a set are called the elements of the set. EMPTY SET. The set which has no elements. The symbol for this set is t/J or { }. ENDPOINT. The point on a line from which a ray extends is called the endpoint of the ray.
EQUALITY. The relation "is equal to" denoted by the symbol "=". EQUATION. A sentence (usually expressed in symbols) in which the verb is "is equal to". EQUIVALENT EQUATIONS. Equations that have the same solution set. EQUIVALENCE RELATION. Any relation which is reflexive, symmetric, and transitive.
Reflexive: a = a Symmetric: If- a = b then b = a. Transitive: If a = b and b = c, than a = c. EQUIVALENT FRACTIONS. Two fractions which represent the same number. EQUIVALENT SYSTEMS. Systems that have the same solution set. EQUILATERAL TRIANGLE. A triangle whose sides have equal length. ESTIMATE. A quick and frequently mental operation to ascertain the approximate value of an involved operation.
173

EXPONENT. In the expression an the number n is called an exponent. U n is a positive integer it indicates how many times a is used as a factor.
an=aXaX '" xa
'---.r----J
n factors Under other conditiOliS exponents can include zero, negative integers, rational and irrational numbers.
EXPONENTIAL EQUATION. An equation in which the independent variable appears as an exponent.
= EXPONENTIAL FUNCTION. A function defined by the exponential equation y aX where a>O.
EXTRANEOUS ROOTS. Those roots in the solution set of a derived equation which are not members of the solution set of the original equation.
EXTRAPOLATING. Estimating the value of a function greater than or less than the known values; making inferences from data beyond the point which is strictly justified by the data.
FACTOR. The integer m is a factor of the integer n if mq = n where q is an integer. The polynomial R (x) is a factor of the polynomial P(x) if R(x) Q(x) = P(x) where Q(x) is a polynomial. Factorization is the process of finding the factors.
FINITE SET. In set theory, a set which is not infinite.
FUNCTION. A relation in which no two of the ordered pairs have the same first member. Also, alternately; A function consists of: (1) a set A called the domain, (2) a set B called the range, (3) a table, rule, formula or graph which associates each member of A with exactly one member of B.
FRACTION. A symbol "alb" where a and b are numbers, with b not zero.
FUNDAMENTAL PROCESSES OF ARITHMETIC. ').'he basic operations: addition, SUbtraction, multiplication, and division.
FUNDAMENTAL THEOREM OF ARITHMETIC. Any positive integer greater than one may be factored into primes in essentially one way; the order of the primes may differ but the same primes will be present. Alternately, any integer except zero can be expressed as a unit times a product of its positive primes.
FUNDAMENTAL THEOREM OF FRACTIONS. U the numerator and denominator are both multiplied (or divided) by the same non-zero number, the result is another name for the fraction.
GEOMETRIC MEANS. The terms that should appear between two given terms so that all of the terms will form a geometric sequence.
GEOMETRIC SEQUENCE. A sequence in which the ratio of any term to its predecessor is the same for all terms.
GEOMETRIC SERIES. The indicated sum of a geometric sequence.
GRAPH. The visual representation of relations.
GREATEST INTERGER FUNCTION. Is define by the rule: f(x) is the greatest integer not greater than x. It is usually denoted by the equation f(x) = [x].
GREATER THAN. The number a is greater than the number b if a is to the right of b on the number line. Also, a is greater than b if a-b is positive.
GREATEST LOWER BOUND. A lower, bound a of a set S of real numbers is the greatest lower bound of S if no lower bound of S is greater than a.
HARMONIC MEAN. A number whose reciprocal is the arithmetic mean between the reciprocals of two given numbers.
HEMISPHERE. U a sphere is divided into two parts by a plane through its center, each half is called a hemisphere.
= IDENTITY, IDENTICAL EQUATION. A statement of equality, usually denoted by _ which is true for all values
of the variables. The values of the variable which have no meaning are excluded. Example: (x+y)" x"+2.xy+y".
INCONSISTENT SYSTEM. A system whose solution set is the empty set.
INDEPENDENT SYSTEM. A system of equations that are not dependent.
INDEX. The number used with a radical sign to indicate the root. (3j In this example the index is three.) If no number is used, the index is two.

174

j

INDUCTIVE REASONING. The process of reaching a probable conclusion by observing what happens in a number of particular cases.
INEQUALITY. The relation in which the verb is one of the foll?wing: is not equal to, is greater than, or is less
than, denoted by the symbols -F' >, <, respectively.
INFINITE SET. In set theory, a set which can be placed in one-to-one correspondence with a proper subset of itself.
INTEGER. Anyone of the set of numbers which consists of the natural numbers, their opposites and zero.
INTERCEPT. H the points whose coordinates are (a,O) and (O,b) are points on the graph of an equation, they are called intercepts. The point whose coordinates are (a,O) is the x-intercept, and the point whose coordinates are (O,b) is the y-intercept.
INTERPOLATION. The process of estimating a value of a function between two known values other than by the rule or the table of the function.
INTERSECTING LINES. Twv or more lines that pass through a .single point in space are called intersecting lines.
INTERSECTION OF SETS. If A and B are sets, the in tersection of A and B, denoted by A n B, is the set of all
elements which are members of both A and B.
INVERSE. The opposite in order or operation. Thus counting .backward is the inverse of counting forward; subtraction is the inverse of addition; division is the inverse of multiplication.
INVERSE FUNCTION. H f is a given function then its inverse is the function (provided f is one-to-one) formed by interchanging the range with domain. The symbol for inverse of f is -1.
INVERSE VARIATION. The number y is said to vary inversely as the number x if xy = k where k in a constant.
IRRATIONAL EQUATION. An equation containing the variable or variables under radical signs or with fractional exponents.
= JOINT VARIATION. A quantity varies jointly as two other quantities it the first is equal to the product of a
constant and the other two. Example: y varies jointly as x and w if y kxw.
LAWS OF ARITHMETIC. The fundamental structural properties which govern the fundamental operations, as:
= = the Associative Law of addition and multiplication stated as (a + b) + c a + (b +c) and (a X b) X c a X b Xc) = = the Commutative Law of addition and multiplication, stated as a + b b + a and a X b b X a; = the Distributive Law of multiplication over addition and subtraction, stated as aX(b+c) axb+axc and
ax(b - c) = axb - axc.
LEAST COMMON MULTIPLE. The least common multiple of two or more numbers is the common multiple which is a factor of all the other common multiples.
LEAST UPPER BOUND. An upper bound b of a set S of real numbers is the least upper bound of S if no upper bound of S is less than b.
LESS THAN. An arithmetical relation which indicates that one value is smaller than another. On the number line, the point standing for the smaller number always lies to the left of the point standing for the larger number.
Example: seven is less than ten, 7<10.
LINEAR EQUATION. An equation in standard form in which the sum of the exponents of the variable in any term equals one.
LINEAR MEASURE. A measure used to determine length. LOGARITHM. The exponent that satisfies the equation b'" = n is called the logarithm of n to the base b for any given positive number n.
LOWER BOUND. A number a is called a lower bound of set S of real numbers if a <x for every xeS.
MAGIC SQUARE. A square of numbers possessing the particular property that the sums in each row, column, and diagonal are the same.
MATRIX. A rectangular .array of numbers.
Example:
[ t ~~ :~ :~ J1 as bs Cs ds a4 b4 c4 d4
175

KEAN. In a frequency distribution, the sum of the n measures divided by n is called the mean. MEASUREMENT. A comparison of the capacity, length, etc., of a thing to be measured with the capacity, length, etc., of an agreed upon unit of measure. Non-standard units are used before standard units of measure are introduced. MEDIAN. In a frequency distribution, the measure that is in the middle of the range is called the median. In geometry, a median of a triangle is a line joining a vertex to the midpoint of the opposite side. MODE. In a frequency distribution, the interval in which the largest number of measures fall is called the mode.
Alternately, in a frequency distribution, the measure which appears most frequently in the group is called the
mode.
= MULTIPLE. If a and b are integers such that a bc where c is an integer, then a is said to be a multiple of b.
MULTIPLICATION, A short method of adding like groups or addends of equal size. It may be illustrated on a number line by counting forward by equal groups. MULTIPLICATIVE INVERSE. The multiplicative inverse of a non-zero number a is the number b such that
ab = 1. It is usually designated by 1/. or a-I.
MUTUALLY DISJOINT SETS. Tv.'r. sets having no elements in common. NATURAL NUMBERS. Any of the set of counting numbers. The set of natural numbers is an infinite set; it has a smallest member (1) but no largest.
= N-FACTORIAL. The expression "n!" is read "n factorial". n! n(n ~1) (n -2) ... 21.
NULL SET. A set containing no elements. It is sometimes called an empty set. The symbol for the null set is p or { }. NUMBER SYSTEM. A number system consists of a set of numbers, two operations defined on the set, the properties belonging to the set, and a definition for equivalence between any two members of the set. NUMERATOR. The upper term in a fraction. NUMBER GROUPING. This is basic to the decimal system of numeration in that it recognizes certain special (roups whose sizes are powers of 10. That is, ones (10"), tens (101), hundreds (10"), thousands (10"), etc. NUMBER LINE. A line on which there is a series of points that stand for numbers arranged in order. A complete number line is unlimited in length and has zero as the reference point. NUMERAL. A written symbol for a number.
Example: Several numerals for the same number are: 8, VIll, 7+1,10-2,18/ 2, OBTUSE ANGLE. If the degree measure of an angle is between 90 and 180 the angle is called an obtuse angle. ONE-TO-ONE CORRESPONDENCE. A pairing of the members of a set A with members of a second set B such that each member of A is paired with exactly one member of B and each member of B is paired with exactly one member of A. OPEN SENTENCE. An open sentence is a sentence involving one or more variables, and the question of whether
= it is true cannot be decided until definite values are given to the variables. Example: x+5 7
ORDERED M-TUPLE. A linear array of numbers (a., a., Ila, , a...) such that a. is the first number, a. is the
second number, Ila is the third number, ... , and a.. is the mth number. ORDERED PAIR. A pair of numbers (a, b) where a is the first member and b is the second member of thepaJr. ORDINATE. If an ordered pair of numbers (a, b) are coordinates of a point P, b is called the ordinate of P. PARALLEL LINES. Two straight lines in a plane that do not intersect however far extended. PARALLELOGRAM. A quadrilateral whose opposite sides are paralleL PARAMETER. An arbitrary constant or a variable in a mathematical expression, which distinquishes various specific cases.
176

PARTIAL PRODUCT. Is used in elementary arithmetic with regard to the written algorithm of multiplication Each digit in the multiplier produces one partial product. The final product is then the sum of the partial proi ucts.
PAR.TIAL QUOTIENT. In long division, any of the trial quotients that must be added to obtain the complete quotlent.
PERIMETER. The sum of the measures of the sides of a polygon. The measure of the outer boundary of a polygon.
PERIOD. The number of digits set off by a comma in an integer or the integral part of a mixed decimal In a repeating decimal the period is the sequence of digits that repeats.
PERIODIC DECIMAL. The decimal representation of a rational number in which a sequence of digits repeats.
= Example: 1/7 = .142857142857 ...
Sometimes given as 1/7 .142857 PERIODIC FUNCTION. A function from R to R, where R is the set of real numbers, is called periodic if and
only if, f (x) is not the same for all x and there is a real number p such that f(x+p) = f(x) for all x in the d0-
main of f. The smallest positive number p for which this holds is called the period of the function.
PIffiASE. A numerical phrase is any numeral given by an expression which involves other numerals along with the signs for operations.
PLACE HOLDER. Any symbol designed to hold a place in a number or mathematical expression is a place holder. Thus, the zero in 10 is essentially a place-holder, because it stands for the quantity "not any". The zero has the same function in numbers like 60, 100, and 40,285.
PLACE VALUE. 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.
PLANE ANGLE. Through any point on the edge of a dihedral angle pass a plane perpendicular to the edge intersecting each side in a ray. The angle formed by these rays is called the plane angle of the dihedral angle.
POLAR COORDINATES. An ordered pair used to represent a complex number. The first member of the pair is the number of units in the radius vector and the second member is the angle of rotation of the radius vector.
POLYGON. A simple closed curve which is the union of line segments is called a polygon. POLYNOMIAL. An algebraic expression of the form aoxn + a,xU- 1 + ... + aU-IX + au sometimes designated by the symbol P(x).
POLYNOMIAL EQUATION. A statement that P(x) = O.
POLYNOMIAL FUNCTION. A function defined by a polynomial equation or f: x-?P(x).
PRECISION.. The precision of a measurement is inversely related to the absolute error. Thus the smaller the absolute error the greater the precision.
PRIME NUMBER. A counting number other than one, which is divisible only by itself and one.
PRISM. If a polyhedron has two faces parallel and its other faces in the form of parallelograms, it is called a prism.
PURE IMAGINARY. A complex number a+bi in which a = 0 and b "" O.
PYRAMID. A pyramid is a polyhedron, one of whose faces is a polygoB of any number of sides and whose other faces are triangles having a common vertex
QUADRANTAL ANGLE. If the terminal side of an angle with center at the origin coincides with a coordinate axis, the angle is called a quadrantal angle.
QUADRILATERAL. A quadrilateral is a polygon formed by the union of 4 line segments.
QUINARY SYSTEM. A system of notation with the base 5. It requires only five symbols or digits: 0, 1, 2, 3, 4.
RADIAN MEASURE. Angular measure where the unit is an angle whose arc on a circle with center at vertex of angle is equal in length to the radius of the circle.
RADIUS. Any line segment with endpoint at the center of a circle and the other endpoint on the circle is called a radius of the circle.
RADIUS VECTOR. A line segment with one end fixed a t the origin on the cartesian plane and rotating from an initial position along the positive x-axis so that its free end point generates a circle.
177

RANGE (OF A FUNCTION). The set of all elements assigned to the elements of the domain by the rule of the function.
RATIONAL EXPRESSION. A rational expression is a quotient of two polynomials or in symbols P(x)/Q(x) where P(x) and Q(x) are polynomials.
RATIONAL NUMBER. H a and b are whole numbers with b not zero, the number represented by the fraction 1" is called a rational number.
... - RAY. Let A and B be points of a line.-Then ray AB is the set which is the union of the segment AB and the set
AB. of all points C for which it is true that B is between A and C. The point A is called the end-poiat of
RECIPROCAL. Mu~tiplicative inverse.
RECIPROCAL FUNCTION. Pairs of functions in the set of real numbers whose product is 1
Example: (Sin 1/ (Csc 1/ = 1.
REFERENCE TRIANGLE. For any angle on the cartesian plane with vertex at the origin, the triangle formed by the radius vector, its projection on the x-axis and a line drawn from the end of the radius vector perpendicular to the x-axis is called the reference triangle.
REFLECTION IN A LINE. A point P has a mirror image 1" in the line AB if P, 1", and AB all lie in the same plane with P and 1" on opposite sides of AB and if the perpendicular distances PO and plO to the point 0 in AB are equal.
REFLEXIVE PROPERTY. H a is any element of a set and if R is a relation on the set such that aRa for all a then R is reflexive.
REGROUPING. The changing of the combinations of units, as: (a) Changing smaller units in addition and multiplication, as
= 12 ones = 1 ten, 2 ones
16 inches 1 foot, 4 inches
(b) Changing larger units to smaller units in subtraction and division as
= 1 ten = 10 ones
1 year 12 months
RELATED ANGLE. For any angle on the cartesian plane, the related angle is the angle in the reference triangle formed by the radius vector and x-axis.
RELATION. A relation from set A to set B (where A and B may represent the same set) is any set of ordered pairs (a, b) such that a is a member of A and b is a member of B.
RELATIVE ERROR. Ratio of the absolute error to the measured value.
REPEATING DECIMAL. A decimal numeral which never ends and which repeats a sequence of digits. It is indicated in this manner: 0.333 ... or 0.142857.
RESOLUTION OF VECTORS. The process of finding the vertical and horizontal components.
RESTRICTED DOMAIN. Domain of a function or relation from which certain numbers are excluded for reasons such as (1) division by zero is not permitted, (2) need for the inverse of a function to be a function.
RIGIIT ANGLE. Any of the four angles obtained at the point of intersection of two perpendicular lines. The angle made by two perpendicular rays. Its measure is 90 degrees.
RIGIIT TRIANGLE. A triangle with one right angle is called a right triangle.
ROUNDING. Replacing digits with zero's to a certain designated place in a number with the last remaining digit being increased under certain specified conditions.
SCALAR. In physical science, a quantity having magnit ude but no direction. In a study of mathematical vectors, any real number is called a scalar.
SCALE. A system of marks in a given order and at fixed intervals. Scales are used on rulers, thermometers, and other measuring instruments and devices as an aid in measuring quantities.
SCIENTIFIC NOTATION. A notation generally used for very large or very small numbers in which each numeral is changed to the form a X 10" where a is a real number containing at most three significant digits such
that 1 < a < 10 and k is any integer.
118

~-

Example:
6,708,345 = 6.71 X 106 .000000052 = 5.2 X 10-8
SEGMENT. For any two points A and B, the set of points consisting of A and B and all points between A and B is the line segment determined by A and B. The segment is a geometrical figure while the distance is anum ber which tells how far A is from B.
SEQUENCE. An ordered arrangement of numbers.
SERIES. The indicated sum of a sequence.
SIGNIFICANT FIGURE. Any digit or any zero in a numeral not used for placement of the decimal point. Example:

703,000

.0056

5.00
SKEW LINES. Two lines which are not coplanar are said to be skew.
= SLOPE. The slope of a given segment (P1P2) is the number m such that m yx')')--Yx11 where P1 is the ordered
pair (Xl' Y1) and P2 is the ordered pair (x2, Y2)' SOLUTION SET. The truth set of an equation or a system of equations.
SPHERE. The set of all points in space each of which is at a given distance from a given point. The given point is called the center of the sphere and the given distance is called the radius.
SQUARE. Formed by four line segments of equal length which meet at right angles.
STANDARD DEVIATION. The square root of the arithmetic mean of the squares of the deviations from the mean.
STATISTIC. An estimate of a parameter obtained from a sample.

= = = SUBTRACTION. To subtract the real number b from the real number a, add the opposite (additive inverse) of
b to a. a -b a+(-b). Also, a -b c if and only if a c+b.

SUCCESSOR. The successor of the integer a is the integer a+1.

D

l : l : ' SUMMATION NOTATION. The symbol at. The symbol

the Greek letter "sigma," corresponds to the

k=m
first letter of the word "sum" and is used to indicate the summing process. The k and n represent the upper and lower indexes and incate that the summing begins with the kth term and includes the nth term.

Example:
5
= ~ at a2+a.+8<+8o.
k=2
00
l : When the summation includes infinitely many terms it is written at. In this case there is no last term aoo
k
because 00 is not a number. The symbol 00 is used simply to indicate that the summation is infinite.

SYMMETRIC PROPERTY. 1 a and b are any elements of a set and if R is a relation on the set such that aRb implies bRa, then the relation is said to have the symmetric property.

TERM. In a phrase which has the form of an indicated sum, A+B, A and B are called terms of the phrase.

TRANSITIVE PROPERTY. 1 a, b, and c are any elements of a set and if R is a relation on the set such that aRb and bRc imply aRc then the relation is said to have the transitive property.

TRIANGLE. 1 A, B, and C are three non-collinear poi nts in a given plane, the set of all points in the segments having A, B, C as their end-points is called a triangle.

UNEQUAL. Not equal, symbolized by .

179

UNION OF SETS. If A and B are two sets, the union of A and B is the set AUB contains all the elements and only those elements that are in A or in B.
= = = Example: A {2, 8, 3}, B {5, 2, 7, 6} then AUB {2, 8, 3, 5, 7, 6} UNIQUE. One and only one.
UPPER BOUND. A number b is called an upper bound of a set S of real numbers if b > x for every xeS.
VARIABLE. A letter used to denote anyone of a given set of numbers. Another name for variable is placeholder in an equation.
Example: x+5 = 7.
VECTOR. In physical science, a quantity having magnitude and direction. In mathematics a vector is a matrix
r of one row or one column as (al bl cl ) or al 1
I a2 1
l:: J
VERTEX. The point of intersection of two rays is called the vertex of an angle. VOLUME. The amount of space occupied by a solid or enclosed within it is called the volume.
180

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I I 111 I ELEMI E9NTARY

GENE9 RAL

10

I ALGEBRA

MATHEMATICS GEOMETRY

MEDIINATTEER ALGEBRA

ADVA12NCED ALGEBRA

I 11 or 12 Consumer Mathematics

NUMER~TION

HISTORY

SYMBOLS FOR NUMBERS AND COUNTING 1. Hindu-Arable 2. Roman 3. Others
RECOGNITION OF CARDINALITY OF SETS (Groups)

Cardinals through 12
Ordinals through fifth
Readfug and writing numbers 15
Sets (Groups) to 3
Vocabulary introduced by using objects More than and less than

BASES FOR NUMER AT ION I. Decimal Notation (Base 10} and Place value 'l. Notation and Place Value for Other Bases
VARIABLES

Reading and writing to 99 Counting by 2's, s:~, lO's to 99 Reading and writing number names through 10 Ordinals through tenth
Sets (Groups) to -10 One to one match ing Members of a set Vocabulary extend ed to understand ing of more than, greater than, less than, and others .
Place value of on~s and tens Reading and writing two digit numerals

Counting experiences extended

1- Counting through 300 by l's, 2's, 3's, 4's,

through 200 by l's lO's, 5's, 2's;

5's; through 1000 by lO's and lOO's;

through 20 by 4's; tbrough 30 by 3's; ' backwards from 100 by l's, 2's, 5's,

through 1000 by lOO's

lO's

Ordinals through fifteenth

Ordinals through thirtieth

Retahdrionugghanxdnwriting Roman numerals

Reading number names through 100 Writing number names through fifty

Reading and writing through XXX

Recognizing, combining, separating, and rearranging through 18
Symbols for sets used informally Sets with objects demonstrated Vocabulary of sets extended

Abstract symbols to indicate members of a set
Symbol for empty or null set Use of braces ~ ~

Reading and writing 3 digit numerals Understanding of place value concept of
ones and tens strengthened Understanding of place value through
hundreds Decimal point with money expressions

Reading and w1iting 4 digit numerals (perhaps 5 and 6 digit)
Understanding use of 10 as the base of our system of numeration
Zero in l's, lO's and lOO's place Regrouping numbers through 100

Counting by 6's, 7's, S's, 9's, and back Ordina~ through hundredth

ward from 100 to 1

Reading and wr ing beyond LXXX

Ordinals through fiftieth

Understanding L, C, D, M, and their

Reading and writing through LXXX and combinations

recognizing D, C, M

Counting and op rating in base 8

Base 5 introduced

and base 5

Union of sets (U)
Intersection <nl
Subsets

Extend understa ding of sets

Reading and writing 9 digit numerals Understanding of place value through
1000's Making comparison of place value
using base 10 and base 5

Numerals to 9 pjaces Reading and writ ng decimals to hundredths Place value thro~gh millions Place value in tjase 5 and 8

Use of D. in mathematical sentences

Use of 0 and N in mathematical sentences

Use O, A, and N in mathematical
sentences

Use of 0 and N 11 equations; in problem solv'!lg

History of Roman numerals
Concept and use of ordinal and cardinal numbers extended
Reading and writing Roman numerals with facility
Comparison of Roman numerals with Arabic system
Extend operations in other bases Extend understanding of sets
Numerals to 12 places Base 5 and base 8 reviewed Place value in bases other than. 10 Reading and writing deo:-imals to
millionths Use mathematical sentences in
problem solving

Appreciation of decimal system through historical deV<llopment of other number systems Compare decimals and Roman systems of numeration Computation with bases through 9 Extend understanding of sets
Extend under standing of place value in the decimal system and other bases
Use in mathematical sent ences and formulas

Historical development of numeration, Hindu-Arabic, retaught and extended through research
Symbolism extended to include Babylonian and Egyptian Computation with bases through 12

Historical de-J Research to show

I vclopment of , how historical

algebra

background

helps pupils

discover what

constitutes the

modern num-

ber system

Appreciation of development in geometry from ancient to present time

Irrational numbers

Knowledge of Roman numerals and other symbols maintained
Bases 8, 12 and others reviewed

Introduce symbols used {'31'ticularly in geometry

Continuation of historical development of algebra
Pure imaginary number system, the symbol i Complex number

History of ef- Historical de

forts to solve equa-

velopments of taxation,

tions of de- investment

gree greater J and credit

than two

system

Polar form of complex numbers

Strengthen the concept of sets

Set notation including finite and infinite sets

Interpretation of sentences ex pressed in set notation

Set notation used in definitions and in theorems

Solution set of equation Xn=a

Intuitive consideration of idea that all infinite sets do not have same number of elements

Emphasize "base" and "position" by use of exponents

Logarithms to different bases and method of conversion

Main tain skills developed in our decimal notation

Use of variables extend ed Application
made in number sentences

Variable defined in set language
Domain of the variable

Usage reviewed and extended in equations

Quadratic va riables Equations
with three linear variables

Sine x and cos x as variables in real number system

EXPONENTS
SERIES AND SEQUENCES
PROBABILITY AND STATISTICS

Simple averages

Use in area formulas; in place value of the decimal and other number bases

Exponents used to emphasize role of base and position in the decimal system Zero exponents

~
Mean, median, and mode

I
Use of mean median, and mode Frequency distribution
I
I

Using bases other than ten

Positive, neg- 1 Review and exative integral tension exponents Zero exponents Multiplying and divid4Jg

li
!
I
I Histogram and frequency polygon
!

i
I
r
I

I

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:

--- - - - -

I

i
I
!

1 Negative, fractional, irra tion al exponents

Further inl'estigation with irrational number as exponent

Meanings, sum of series,
I sum of sequences, sum
Ii of infinite series, limit
Permutations Combinations

Number e as sum of infinite series

Characteristics of good investment and comput ing simple and compound interest
Computation ar;d interpretation of data. Percentile in terms of norm al curve

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OPERATIONS

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9 ELEMENTARY

IMATHE~TICS

ALGEBRA

GENERAL

ADDITION AND SUB TRACTION

Readiness for combining and separating groups

Combining and separating groups through ten with symbols

Mastery of 45 addi tion and subtraction facts Development of the other 36 basic facts Use of higher decade without grouping

ltastery of 81 basic combination facts;

facts involving zero

1Ad4daidndge;111,ds2,wiatnhdin3tedrnigailtzenruoms baenrds

with re-

groupmg

rubtracting 2 and 3 digit numbers with

internal zeros and regrouping

Adding 4 digit numbers, regrouping to Addition of 7 digit numbers

IO's, lOO's, and 1000's

Subtraction with regrouping

Adding 7, 1 digit numbers; 5 and 6,

2 and 3 digit numbers; and 5 four

digit numbers

Subtracting with regrouping

Commutation "'ith high degree of mast Whole numbers,

ery

decimal and

common frac-

tions

Use of

AlgFb~aic e~-

properties pr SS10nS

m solving

problems

Application of commutative; associative, and distributive properties

MULTIPLICATION AND Readiness DIVISION

Regrouping of 2's and 3's to form other groups

FRACTIONS (Common)

Introduce the Understanding
idea of one-half and writing .!.
of a whole 2 Understanding
and writing .!.. of a whole 4

PROPERTIES
I

Adding twos and doubles Separating twos and

biscovering and mastering basic multi plication and division facts with 2's, 3's, 4's, 5's, and 10's

Developing .multiplicatiqn and division Multiplication with 4 digit multipliers

facts with 6's, 7's, 8's, 9's Multiplying by 2 and 3 digit

multiplier

using zero in 2 digit divisors

different

places

Use of 4 digit multipliers; 3 digit dividers Tests for divisibility extended

Whole numbers, decimal and common frac

Use of Alg braic exproperties pr !ssions an solving

doubles

~ultiplying by 1 digit multipliers Dividing by 1 digit divisors

Dividing by 1 digit numbers without remainders

with

and

Tests for divisibility continued Rounding divisors to get first

quotient

Casting out 9's Distributive law

tions

problems

Understanding what is meant by aver figures

age Introduce factorin~

I

Meaning of ' and of 1

an object ;- ;"' 2 3' ,. 5 e a of a whole and of a grou

" " ' 1 ti~r Familiarity with: -{ & ~

lJse of symbols for fractions and use of lJIterms numerator and denominator

of a group and of an se of decimal point in reading and

object /

writing money expressions

Introduce use of deci

I I I I I I I mal point

Finding parts of whole and of groups Equivalent fractions

Simplifying fractions

Addition, subtraction, multiplication,

Use of fundamental operations in deal and division of fractions

ing with money Relationship between
decimal point

the cent

sign

and

Adding and subtracting tenths and hundredths with regrouping

Adding and subtracting like fractions Decimal mixed numbers

Estimation of results

Understanding of commutative, associative, and distributive properties

.

Multiplying and dividing fractions Short cuts of multiplying and dividing
by IO's, 100's, and 1000's Estimation and rounding off Use of reciprocals Distributive law Multiplying and dividing decimals
Use of proper terminology .

Fractions in simpies~ form Division of frac tions by. use of Principal of One; by common denominator method
Extend use of Laws of Arithmetic

Extend use of frac-

c~~lex frac

tiona

~t onal expres

S1 S

Properties of a

field

Field properties

Application of properties to solving probIems Practice in computation using fractiona
I Caosmsomciuattaivtiev,e,
and distribu-
I tive

FACTORING

f se of Factors

Use of Factors

Use of factors in dealing with fractions Use of factors in more complex frac- Factoring com-

Use o_f terms factor, prime, multiple

tions and in getting common denomi- posit numbers

nators

mto primes

H. C. F.

L. C. M., Nat ural numbers

H. C. F.

(primes)

andL. C.D. Co~mon monomi

use in

al factor

solving Dif erences of

problems squares

Perject squares

Qutatic tri-

no ials

Importance of factoring in solving problema

PROPERTIES OF

1 and 0

I

I

I

piscovering generalizations: n+o-n; Understanding of the generalizations in Reviewing generalizations

Extend use of property of 1

I I n--o=n; n-n=o; n xo=o; oxn=o 1 x n=n; n+ l=n

reference to 0 and 1

Use of property of 1 in simplifying fractions

n+O n; O+ n n nx l=n; lxn=n n-O=n; n+l=n

~t Emphasize pr~erties an use

If ab 0 then a=~or b= O;
(-a) =0; 1- a

axl a:.1.!-a a+O=a; axo-o

PROBLEM SOLVING

I

10 GEOMETRY

I I l1 INTERMEDIATE ALGEBRA

12 ADVANCED ALGEJ!RA

I C1o1nsourm1e2r
Methematlct

Addition and sub- Vector addition of

-

traction with com complex numbers

plex numbers

MultiplicatiQn and division of complex numbers, 12 =1
Rationalization of denominators
Rational expressions Complex algebraic fractions

Multiplication and division of complex numbers in polar form DeMoivre's theorem Powers and roots of comnlex numbers

I

I Properties of field, integral domain, group

Basic laws reviewed Secial cases of
actorization ineluding trinomials reducible to dif ference of squares

Factor theorem, synthetic division

I Use of property of . Idea that 1 and 0

I one in simplifying fractions

need not resemble

1 and 0



TYPES OF PROBLEMS Daily activity

One step problem Gradual introduction of abstract terms

LOGIC

CONCEPT OF PROOF

Readiness

Using concrete

with concrete materials

materials

Pictorial representstions
Development of prob!ems with word symbois and number symbois Relating all four processes in simple oral and written problems

Analyzing and solving one step prob lems using mathematical sentences
Estimation of answers Checking of computation

Analyzing and solving problems invol Using four fundamental operations

ving four fundamental processes

Using ratio to symbolize rate and com

Various types of problems

parison

Estimation of answers

Finding averages

Checking of computation

Using equations related to processes

taught

Readiness through step by step reasoninl

Finding quantities in var!ous situations Three basic per-

Expressing numbers as fractional num- centage prob

erals

Iems

Variables with solution set containing Life situation

one or more digits

problem

Processes with variable extended to in

elude: a X

-=-

b c

Extend step by step reasoning

Analyzing problem solving procedure Estimation of
answers

Extension Extension

Lin~~ equations Qualiratic equations Systems of linear 1 quations
i:r:.

Matching and other simple procedures used to show correctness of answers Comprehending inverse process

Extend procedures to show correctness of answers

Using more than one solution

Using Laws of Arithmetic as reason. Checking
I

Using Laws of Arithmetic Us~ more than one solution Ch ing

Checking reasonableness of solution set

Giving reasons for steps in problem solving

I
Direct and indirect proof

Extension Extension

Proof of statemenu concerning geometric fiiUI'es
Statement, eon verse, inverse, and contraposl tive Deductive and inductive reasoning Direct and indi~ rect proof used

A~plication of functlon to physical processes Worded problems involing one variable, two variables, quadratic variables

Solutions of simultaneous equations using set language for representing solutions
Practical problems usinft trigonometric unctions

Conjunction Disjunction

Direct and indirect proof of selected theorems

Algebra as a logical system with facts capable of being proved from a basic set of postulates Proofs emphasized throughout course

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I I I 9
I ELEMENTARY ALGEBRA

9 GENERAL MATHEMATICS

10 GEOMETRY

11 INTERMEDIATE
ALGEBRA

I

12 ADVANC ED ALG EBRA

I CO1N1 SoUr Mf2ER M~TH EMAT IC S

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used in under-

Commutative Jaw of addition

problem

tions

tions

Systems of linear equations

Proofs of theorems

standmg a collec- and multiplication

solving

Systems of 2

Systems of quadratic equations

tion of l'Oins

As&ociative law of addition

_

Use of many

equations with

Systems of linear equations with

Experiences with measurement

and multiplication Use of concrete objects to show

names for the same

2 unknowns

three variables

-1; = T2

number

~ UlUALITY

Readiness

Mathemati~
sentences

Opportunity to learn 5 is less than 6, and 4 is greater than 3 Use of number line to show addition

Demonstration with counters and number line that sume numbers are greater than or less than other numbers

Skill in making comparisons:

unequal, greater than, less

than Symbols:

< , > , -1-

Use of terms with proper symbols:
<>

Use of symbols in problem solving

Nature of ine ualities Number sent nces with inequali-
ties

Nature of inequalities Number sentences with in-
equalities

Linear inequalities Properties of inequalities

Revie\V and extension of in equalit ies

Exterior, remote interior angles of triangles
Longer side and sum of cther two sides

Quadratic inequalities Graphs of quadratic inequali-
ties

Axioms of Inequalities Proofs of theorems

fUNC TIONS VARIATION

'

I

Graphs of

Graphs of simple compari-

simple com- sons

pariSOilJ

Introduction of concept: Linear Quadratic

Linear, quadratic functions Exponental function Logarithmic function

Idea of inverse function Detailed study of circular polynomial, rational, exponental, logarithmic,
trigonometric fu nctions

Ratio and pr portion in number sentences

Direct Inverse J oint

Application in formulas and in mathematical sentences

Direct variation: Y-mx+b where b-0

IATIO AND PROPORTION

Identifying equal groups Understanding_ of
.!.. !. !.
2

Demonstration of 1 to 3 or 2 to 5 correspondence with objects
Use of problems of rate type in clarifying multiplication and division

Ratio used to express rate and comparison

Meaning of terms used in rates and comparisons

Use of colon between terms Finding missing terms
Solving problems of per cent

Ratio as the comparison of one integer wit another (except 0)
Rate as a ratio Scale drawinll

Used in solving per cents Scale drawings

Review and reteaching of ratio Properties of proportion

Sine, cosine, tangent Similarity of geometric figures Area of sectors

Trigonometric functions as ratios in r ight triangles

IERCENT

Readmess experiences through classroom activititoS related to 100%

Classroom experiences: 100%, 50%, 25%

Experiences extended

Use ratio to introduce per cent

Per cent compared with common and decimal fractions
Meaning of three cases of per cent
Use in problem solving

Per ce.nt by t~e of ratio and pro

portion

Use of and

per less

ctehnIns

greater 1

than

100

Rounding off per cent

Use of per ce ts in business

Extend use of per cents

Extend use of per cents

Paying cash . through loans and using carrying charge.

MEASUREMENT I~ NIT~ S O=F =ME=AS=UR=E ==Di~ne= =cvfh~e~= ,lod'fpo;= m,os;te,~ nmctyloaoa= lncrekdye,= ,x- =TR= heheolalcuil~ onfrg,ghna= tionitmuido= reno:n~ oteof- ~Idui= nergnest= isftaya. sinndgn= aereadd= nemdd~ eauss--== Ideuia= nnnrcdtiilt= fusiudcnea= ittirsoe= nloafta= ionmnde= sahusis= puesreo= bmfeetsw9 ntatenet7 dno- ~CU~ osuaennrv~ eooetfrhcs~ oesiotrnant= ninfdrua= oermdd~uonn~ iets u~onfi~ tmteo~ as-~C~tfMoornoe~ avmannero~ iosntnihgoe~ enourfn~it ~PM~ rdieinnaiv~ cniimispnig~ uloelntoio~ fpflciucr~ eabgtiicr~ oonmu~ peaianns~ dgure== ADdiomf~ vfveaeenarr~ estnauEgcr~ nieengsglb~ isoevwe5ismegyehes nttterimac~ cnodum~ nmetiana~ssgusr~ eand7TNMSqae~ auuanatsdiru~ ceralealr= stoimtoou= tefdaess= ucriee= n, celo~ng~ itu~ de 1~-========~=--Svlci'=in2eeno=tnif=nicu=mnb=oetr=a- ~~Luri=anmde=eaia,rn,=1dae=rgera=e,e,v=ol~- r

================rR=ad=ia=ns=a=s =un=its=f=or=m=ea=s-= ur ing ar c, angles Methods of conversion to degrees

,

==E=x-te=n=sio=n==0=(
use of metric measure

weight

coins

Adding and subtracting like

square meas-

Accuracy and precision

tion

units

ure

Use of meter sUcks

~TIMATION

Skill in esti Estimating using ob- Sums and differences by round- Rounding numbers to 1000s

mating

jects in classroom ing off

Rounding large num-

Estimating in linear,

Importance o precision

square, and cubic meas-

Continued

Estimation as compared with

Use of number line marked in

bers

ure

accuracy and

~

units of 100 to find approxi-

precision in

mate differences

measurement

____ ~'-------------~--------------~----------+--------------r----------------------r----------------------r----------r-----------------1----------~------------+---------------------t------------t------------1-----~, ------~--------------------~--------

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FUOW CHART

K

2

3

4

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6

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I I I ELEMlNTARY GEN~!k'_L ALGEBRA . MATHENI[ATICS

10 GEOMETRY

11 INTERMEDIATE
ALGEBRA

poiNTS, LINES AND PLANES

Recognition

of rectan-

gle, cle,

lsinqeu,arcei,r-

triangle

Explore square, triangle and rectangle

Straight line, square corner

IdentiJ~ lines and surfa es

Parallel lines and right angles

GEOMETRIC CONCEPTS

Using number lines to show re-

lations

Lines, points, plane, ra ment, and end point

ya1elfiinneedseg-

Perpendicular linea

Study of angles: right, acute, obtuse

Points, lines, planes, and their properties

Recollliz.ing 1, 2, 3, dimenalonal space

Points, lin~:S and planes a~ sets

Ruler, postulate, rays, segments, congruent segments; parallel lines; linea and planes; perpendicu lar lines and planes

One-to-one correspondence between ordered pairs and points on the xy plane. One-to-one correspondence between ordered triples and points in three-space

Graphs of functiona and relationa on Cartesian plane
Addition and subtraction of geometric vecton

GEOMETRIC FIGURES

Cube

CONSTRUCTION

ANALYTICAL GEOMETRY "

Cylinder, sphere

Recognizing triangle, circle, square, cone, cube, cylinder, and sphere

ldentp ing and reprod\ cing circle, squar~, triangle, recta gle, cube, sphex~. cone, cylinder

Making, collecting and using geometric figures Finding area and perimeter of square and rectangle

Defining closed curves, parallelo gram, triangle (equilateral and isosceles), rectangle, sqnare, quadrilateral, . circle, polygon
Studying area and perimeter Recognizing prism, pyramid and
rectangular solid

Using term: degree . in 1longitude1. latitude problem
Extenaing study of area Finding volume Using term "congruent" Law of Pythagoras

Extension of concepts of 6th

grade

Perimeters of the triangle, quad-

rilateral, pentagon

Areas of tangle.

tphaeratlrliealnogglrea,ms,qucairrce1lerec

Symmetry, similar triangles

Measurement of angles

Area. and volume Measurement of angles

Construction of simple figures by means of straight edge and protractor
Constructing figures to scale

Extension of simple construction of scale drawing
Three demensional figuies from drawings
Using Compass

Bisection of angles and lines

Perpendiculars

.

Angles equal to riven aDiles

Construction of inscribed polygons Perpendiculars to a line
at a point on the line
and perpendiculars to a line from a point not on the line

Application of measurement of geometric fi gures

Rays, angles, triangles, polygona, quadrilaterals, circles, spheres, prisms, cones, cylinders

Conic sections

Rmeveinewtal ocfofutnrduea- tiona Skill in e of the straikht edge, compass,
j and the protrac-
tor

Inscribed and circumscribed circles of triangles Inscribed and circumscribed polygons of circles Segments of line in proportion

Derivation of slope-intercept form Introduction of. 3-dl~ensional graph

3-dimensional geometry Equation of plane

Tangents to curves; areas under curves
Intensive study of straight line and circle with proofS of theor ems from plane geometry
Extension of study of conies, polar form of equations, paramet ric equations, rational forms

GRAPHS CHARTS

Readiness Readiness

GRAPHS AND CHARTS

Observing and manipulating

Reading thermometers Making simple graphs of daily experiences

Maankdiniictsuimreplgerapbhasr of cl ssroom experie ces Drawi g simple maps of school,
I neig~borhood and
class

Malting and interpreting simple graphs
Extending ability to draw maps and make and read scale drawings

Picture graphs Bar graphs Line and circle graphs Reading maps

Extension of construction of graphs and maps to include large numben and scaling

Construction of graphS: Histograms, bar, broken line, circle, pictograms
Scale drawings.

~eadiness
Observing and constructing simple charts

Making simple charts Using information on charts Developing awareness of the function of a chart

Making weather cha~, number chart of basic facts

Making and inter- Reading simple charts and tables Reading of charts and tables

preting charts ex-

extended to include inter-

tended

pretation and construction

from statistical data

Reading charts

Extend construction of graphs

1'lumber line Coordinate
~lane
Linear and quadratic equations

Interpretation and conatruction of graphs

Introduction of 3-d.imensional graph Linear equation Quadratic equations in form; x2+y2=r2

Graphs of linear equationa Quadratic equations Solution of systems

Frequency distribution chart and table

Esltaetimsteicns ~ Extension of grade 7 d 8
skills Inequalitiet in a plane

Table of trigonometric functions

Table of Logarithms

Plotting graph of function defined by algebraic equation
Study of graph of exponential and logarithmic funcllim
Wgrirtainpgh equation from points . on Advantage of quantity aDd quality bu1 ing. Home ownership vemu
- rent.