A ninth grade remedial mathematics course

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A NINTH GRADE REMEDIAL MATHEMATICS CCURSE

A Project Presented to the Faculty of the School of Education The University of Southern California

In Partial Fulfillment of the Requirements for the Degree Master of Science in Education

hy Chas. F. Baldwin August 195D

UMI Number: EP46175

All rights reserved IN FO R M A TIO N TO ALL U SER S The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion.

UMI Dissertation PtMshing

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T h is project report, w ritten under the direction of the candidate’s adviser and ap p ro ved by him , has been presented to and accepted by the F a c u lty of the School of E d ucatio n in p a r tia l f u lf illm e n t of the requirements f o r the degree

of M a s t e r of

Science in Education.

Date.

" ' D ? ...................................

A d v is e r

Dean

ii TABLE OF CONTENTS CHAPTER

PAGE PART I. FUNDAMENTALS HOW TO DO BASIC MATHEMATICS

1.

ADDITION:

HOW TO FIND THE SUM OF A GROUP

OF NUMBERS 1*

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

PROCEDURES s How to increase your speed in adding numbers

2*

ARRANGEMENTS

2

How to line up the figures

in order to find the correct sum • * • • « • 5.

2,

CHECKING RESULTSs

How to be sure your

answer is correct

• • • • • • . • • • • . .

SUBTRACTIONS

ARRANGEMENTS

INSTRUCTIONS:

. . . • • • • • • • •

CHECKING RESULTS:

MULTIPLICATIONS TWO NUMBERS 1.

• • « . . • • • • • • •

How to keep from making

HOW TO FIND THE PRODUCT OF .......................

PROCEDURES:

11

How to line up the figures

in order to find the correct product • • • • 2.

7

8

. . . . .

ARRANGEMENT:

6

How to subtract one number

mistakes • g.

6

How to align the numbers

from another number 3.

3

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

for better efficiency 2.

2

HOW TO FIND THE DIFFERENCE

BETWEEN NUMBERS 1.

2

How to multiply two numbers

11

iii CHAPTER

PAGE correctly . . . . . . .

3. CHECKING RESULTS :

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

How to prove your

answer is correct • • • • • • • • • • , . • 4.

11

12

DIVISION: HOW TO FIND THE QUOTIENT OF TWO QUOTIENTS........... 1.

ARRANGEMENT:

16

How to place the numbers

to find correct quotient 2.

PROCEDURES:

16

How to figure out the

number of times one number will go ...........

into another number 3. CHECKING RESULTS:

How to prove your

quotient is correct • • . • • • • • • • • • 5.

17

FRACTIONS: HOW TO ADD. SUBTRACT. MULTIPLY .........

AND DIVIDE MIXED NUMBERS 1.

ADDING FRACTIONS:

21

How to find the sum

of two or more mixed numbers 2. SUBTRACTING FRACTIONS:

. . . • • • •

3. MULTIPLYING FRACTIONS:

21

How to find the

difference between mixed numbers

• • • . •

22

How to compute

the product of two mixed numbers 6.

16

• • • • •

DECIMALS: HOW TO MAKE FRACTIONS EASIER . . . . 1. ADDING .AND SUBTRACTING DECIMALS:

How

to find the sum and difference of pro­ blems that require the use of a decimal

22 26

lv CHAPTER

PAGE point . . . . . . . . ..........

2.

MULTIPLICATION OP DECIMALS:

. . . . .

26

How to place

the decimal point when finding the quo­ tient of fractional numbers . . . . . . . . 3.

27

DIVISION OP DECIMALS: How to place the decimal point when finding the quotient of a fractional number



27

PART II. PRACTICAL MATHEMATICS HOW TO SOLVE EVERY-DAY ARITHMETIC PROBLEMS 7.

PERSONAL MATHEMATICS:

HOW TO SOLVE COMMON

HOME PROBLEMS........ ................ .. . * ,1. FINANCING: 2.

How to budgetyour money

BARGAINING:

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

RECIPES:

PAVING: How

FLOORING:

............ • • • •

35

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

BUSINESS: 1.

36

How to measure a room for

a rug • 8.

34

to reduce the cost of

making a driveway 6*

. •

How to decrease and increase

recipes for cooking 5*

33

ERRORS: How to prevent mistakes in change making and sales charges * . . .



32

How to shop for best food

prices 3.

...

32

36

HOW TO MANAGE YOUR M O N E Y ....

SAVINGS ACCOUNT:

How to use a book

39

V

CHAPTER

»

for saving money 2. CHECKING ACCOUNT:

.........

amounts to a bank 4• BANK FORMS: bank drafts 5* INSURANCE: 6* INVESTMENTS: money 9.

SHOP:

• • . . « • • • •

40

How to take larger • . « • • • • • • • • * •

40

How to fill out checks and •

40

How to provide for your future .

41

How to earn more for your

• • • • • • • • • • • • • • • • • • •

41

HOW TO SOLVE PROBLEMS IN YOUR HAND

WORK . . ................... 1. BOARD FEET:

44

How to find the cost of

lumber • • • • • • • • • • • • • • • • • • •

a.

39

How to place money

in a bank for current use 3. PREPARING DEPOSITS:

PAGE

ALLOWANCE FOR WAST:

How to estimate

enough wood to offset loss in Jointing • • • 3. HEIGHTS OF OBJECTS:

45

How to find out how

tall a tree is without climbing it • • • . « 4. PULLERS AND BELTS:

44

45

How to find the num­

ber of revolutions a machine Is turn­ ing



45

PREFACE This course syllabus was prepared during the summer session of 1949 under the supervision of C. C. Crawford, author of Functional Education*

Professor and fellow grad­

uate students have contributed toward the formation of this syllabus for teachers of high school remedial mathematics courses* Most of the current texts on remedial mathematics have failed to relate the problems used with actual life exper­ iences.

However, enough drill exercises have been includ­

ed so as not to Ignore practice in the basic fundmentals* In this course an attempt has been made to show how mathematics can be useful in everyday life* The wording and sentence structure is formed for slow students of the ninth grade*

Also, the problems included

in this syllabus are planned to meet the practical needs of students in this grade.

Teachers of remedial mathem­

atics should find this outline extremely valuable as a guide to be used in conjunction with a text or as a text within itself*

1

PART I. FUNDAMENTALS HOW TO DO BASIC MATHEMATICS The problems contained in the following section of this course outline are a review of the simple arithmetic processes*

All of the exercises contain the difficulties

that usually confuse the student that has been having trouble in mathematics.

Would you like to have a higher

school mark, and have a chance for a higher income in life? If so, study the next six chapters carefully in order to achieve these objectives*

t,

2 CHAPTER 1. ADDITION HOW TO FIND THE SOM OF A GROUP OF NUMBERS A.

MOTIVATIONi

Rewards that will come If you can add

quickly and accurately. 1*

ECONOMY OF TIME:

If you can add quickly valuable

minutes will be saved* 2,

SELF ASSURANCES

Adding figures correctly will

promote confidence in future mathematical problems* 3.

HAPPIER HOME:

You will be of help to your parents

in checking the accuracy of their bills* B.

DIRECTIONSs

Suggestions to follow to increase your

adding abilities* 1*

PROCEDURESs

How to increase your speed in adding

numbers• a*

Start at the bottom and add up the column of figures.

b.

Don't search the column for„-y»fusy combinations.

c.

Always place the "carry over" at the bottom of the next column.

d.

Don't peck out the numbers with your pencil. This reduces speed without increasing accuracy*

e*

Glance at the size of the numbers.

You may

know the answer immediately without having to add each column separately* 2.

ARRANGEMENTS

How to line up the figures in order

3 to find the correct sum. a.

Write all numbers the same size*

This per­

mits better alignment. b.

Place the columns directly underneath each other.

c.

Make sure that the right side of the columns forms a straight line.

d.

Remember that the left side does not neces­ sarily form a straight line.

3.

CHECKING RESULTS!

How to be sure your answer is

correct* a.

Add the columns the opposite direction (Prom top to bottom.)

b.

Glance at the figures and make an estimate of what the answer should be.

c.

Place commas after each group bers.

of three num­

This will help to approximate correct

answer. C.

PERFORMANCE!

Problems that will help you to increase

your speed and accuracy when adding. 1.

DRILL EXERCISES!

Work the following addition com­

binations as rapidly as possible. a.

9 7

b.

8 7

c.

9 8

d.

79 88

e.

768 979

f.

2.

645 978 452

g.

LIFE PROBLEMS:

791 850 420

h.

496 734 502

1.

7643 8724 9876

Solve the following practical

exercises* a*

Add your parents grocery bill to see if the grocers answer is correct*

b.

Keep a list of your daily expenditures.

Add

them as often as necessary* c. Find the total cost of all your school books* d* Find the total number of minutes you spend in class each day by adding all the classes to­ gether* e.

Find out how much money your parents spend for rent, utilities, and groceries each month*

f . Determine by adding, the cost of food for a 3 day automobile trip* g*

Compute by adding the distance around the bases of a baseball diamond*

h*

Find the cost to your parents of sending you to school for one semester*



EVALUATION:

Tests that will help you to increase your

ability to add* 1*

TRUE - FALSE: true or false*

Place an X in the correct space for

5 a.

T F ( )( )

You should never search for easy com­ binations*

b.

( )( )

To increase accuracy, you should peck out each number with your pencil*

c.

( )( )

To increase accuracy, you should make all figures ithe same size•

d. ( )( )

It isn't necessary for you to mark down the amount of the ”carry over*1'

2*

COMPLETION:

Place the correct answer on the line

ahead of the item* _________a*

If you purchased a loaf of bread for 12 cents, a can of tomatoes for 17 cents and a dozen of eggs for 63 cents, how much would be the total cost? You are five feet tall, a friend is 6 feet tall, and another friend is 5j| feet tall, what is the height of all combined? On a trip you went 25 miles one day, 250 miles another day, and 9 miles the last day, how far did you travel altogether?

CHAPTER 2. SUBTRACTION HOW TO FIND THE DIFFERENCE BETWEEN NUMBERS A»

MOTIVATION:

Benefits that you will gain if you learn

how to subtract rapidly and correctly. 1*

ECONOMY OF MONEY:

You will be able to calculate

the amount of change you should receive from a~ transaction. 2#

MORE ENJOYMENT AT GAMES:

You will be able to fig­

ure the number of points your team is ahead or be­ hind in a sport. 5.

BETTER BUDGETING:

You will not spend more than

the amount alloted to certain items# B.

DIRECTION:

Some helpful suggestions for increasing

your speed and accuracy in subtraction. 1.

ARRANGEMENT:

How to align the numbers for better

efficiency. a.

Make all numbers the same size.

b.

Do not make the figures too small.

c.

Be sure that the right side of the numbers are even.

d*

Place the samller number beneath the larger numbers.

e.

Place a comma after every group of three fig­ ures, counting to the left. you to.align them properly.

This will help

INSTRUCTIONS:

How to subtract one number from

ano therCMhumber • a.

Begin at the right side by "taking away" the bottom number from the top number.

If one of

the bottom numbers is larger than the top num­ ber, you should proceed as follows: (1)

Borrow a number from the next top number on the left.

This increases the amount

of the number by ten. (2)

You are now able to subtract the two num­ bers •

(3)

The next bottom number can either be In­ creased by one or you can remember that you borrowed one from the top number.

(4)

If the next top number is also smaller, follow the same procedure.

b.

Continue subtracting until there are no digits left below.

c.

Do

not subtractby trying to find a number that

when added will equal the top number.

This

slows you down. d.

Do

not

peck outthe answer with your pencil.

If you know your combinations this will not be necessary.

3*

CHECKING- RESULTS: a*

How to keep from making mistakes*

Examine your computations to be sure there are no mistakes*

b • Make an estimate of the answer that you should g©t. c*

Add the result with the number that was sub­ tracted.

The answer should equal the top num­

ber* C*

PERFORMANCE:

Problems that will increase your ability

to subtract swiftly and accurately. 1.

DRILL EXERCISES:

Work the following subtraction

combinations*

2.

a*

9

g*

87 -69

b* h.

8 =1

c*

79 -69

LIFE PROBLEMS:

9 i.

d.

9 zl

846 - 98

k.

e. 698 -209

8

f. 1.

7

758 - 17

Solve the following practical sub­

traction problems: a*

Find the amount of change you should have left from a dollar bill after attending a Saturday movie that cost seventy-three cents*

»b*

Find the remainder of your one dollar weekly allowance, if you spent 10 cents for ice cream and 50 cents for a kite*

c.

Your mother sent you to the grocery store for a loaf of bread.

She gave you a quarter and

the bread cost twelve cents*

How much should

you return to her? d. In the last quarter of a football game your high school team had six points.

How many

points would your team have to get in order to win the game? e.

You had to be at work on your after-school job at 4s15.

How much time did you have to get to

work if school was out at 3:35? f# A bicycle cost #38*95* you have saved #29*86*

Out of your allowance How much more do you

need to save in order to purchase the bicycle? D.

EVALUATIONS

Tests that will help you to correct your

difficulties* 1.

TRUE - FALSE:

Place an X in the correct space for

true or false* a*

T F () ( )

If the top digit in a subtraction pro­ blem is smaller than the bottom num­ ber you should ^subtract the top num­ ber from the bottom number*

b*

() ( )

You can check your result by adding the answer with the number on top*

c.

() 0 )

It is necessary for you to borrow when the bottom number is the smaller*

e.

( ) ( ) You should arrange the numbers so that the smaller number is on the bottom.

MULTIPLE CHOICES

Place the correct number in the

parentheses. a.

( ) A baseball costs 64 cents.

If you had

$150 and purchased the ball, your change should bes (5) b.

( ) It

96 cents.

86 cents. (4)

(2)

68 cents.

82 cents.

costs 25 cents to swim in the local

pool. (1)

(!)

Out of 42 cents you should have: 17 cents.

cents.

(4)

(2)

71 cents.

23 cents.

(5) 13

11 CHAPTER 3. MULTIPLICATION HOW TO FIND THE PRODUCT OF TWO NUMBERS A»

MOTIVATION:

Rewards that will come If you can mul­

tiply correctly. 1.

ECONOMY OF TIME:

Multiplication is actually a short

cut method of adding like numbers• 2.

HAPPIER VACATIONS:

You will be able to calculate

the cost of all your travels. 3.

FUN AS A PILOT:

Aviators have to know how to mul­

tiply accurately and swiftly. 4.

HAPPIER SCHOOL LIFE:

Other students will admire

your ability to calculate the cost of club activ­ ities • B.

DIRECTIONS:

Suggestions to follow when learning how to

multiply. 1.

ARRANGEMENT:

How to line up the figures in order

to find the correct product. a.

Place the largest number on top (multiplicand.)

b.

Starting at the right hand side, place the mul­ tiplicand directly above the multiplier.

2.

PROCEDURES: a.

How to multiply two numbers correctly.

Beginning at the right hand side, multiply each of the digits in the top number by the extreme right dig^t in the bottom number.

b.

Place the right hand digit of the first answer

18 directly below the right band digit of the mul­ tiplier* c•

Garry the left band digit of the first answer over to the next answer*

d.

Repeat the process until all the top numbers have been multiplied*

e.

The same procdure should be followed for each of the digits in the multiplier.

Remember to

indent to the left one digit for each row of answers* f . After all the digits in the bottom number have been multiplied add the results together* 3.

CHECKING RESULTS:

How to prove your answer is cor­

rect. a* Before you begin to multiply, make an estimate of the correct answer. b.

Switch the bottom and top numbers around and multiply*

This should give you the same answea?

as before* C.

PERFORMANCE:

Problems that will increase your ability

to multiply* 1.

DRILL. EXERCISES:

Work the following multiplication

difficult combinations* a. 29 9

b.

87 9

c.

56 9

d.

34 9

e.

91 * f. 9

98 9

13 g.

87 8

h.

1. 659 7 2.

76 8

m.

LIFE PROBLEMS:

1.

423 89

65 8 n.

J.

4786 __ 54

48 8 o.

k.

897 67

5679 p. __ 52

8947 __66

Solve the following practical exer­

cises • a*

Your high school baseball team purchased 5 balls that cost $3.00 each.

How much was the total

price? b. Your boy scout troop can walk20 miles How farcan they travel c.

a day.

in a week?

A light plane can fly about 250 miles on one tank of gasoline.

How far can it go on 4 tanks

of fuel? d*

Your hot rod can go 89 miles an hour.

How far

can you travel in 12 hours? e.

You earn 95 cents an hour working after school# If you work 27 hours a week, how much do you earn in one week?

f•

There are 239 members in our school commercial club.

How much do they collect in dues each

month, if each member pays 75 cents? g.

Tires for your car cost $9.00 each.

How much

would you have to pay for 4 tires? h.

A jet propelled airplane can cruise at 475 miles

14 an hour.

How far can it travel in 5 days, if

it flies 6 hours each day? D.

EVALUATION:

Tests that will help you to increase your

multiplication ability. 1*

TRUE - FALSE;

Place an X in the correct space for

true or false* a*

T F ( )( )

You should place the larger number on top in a multiplication problem,

b*

( )( )

You should disregard zeros in a mul­ tiplication problem,

c.

( ) ( ) You should place each row of answers in a large problem^ directly below each other.

2*

MULTIPLE CHOICE;

Place the correct number in the

parentheses. a.

( ) A secretarial club of 28 members had a private roller skating party*

If each mem­

ber had to pay 25 cents the total cost to the club was;

'*.'12

(1)

6 dollars.

./

.lara.

(2)

7 dol-

.

(5)

7 dollars and 50 cents*

(4)

4 dollars and 50 cents* b*

if )

If you can run 25 yards in 3 seconds, the distance you could run in 12 seconds is: (1)

75 yards.

(2)

80 yards.

(5)

100

15 yards* c.

(4)

69 yards,

( ) Bread costs 12 cents a loaf.

If you had

to huy 24 loaves for a picnic, the cost would he: (5)

#1.88.

(1) (4)

#5.88. #2.88.

(2)

72 cents.

CHAPTER 4. DIVISION HOW TO FIND THE QUOTIENT OF TWO QUANTITIES A.

MOTIVATION: Benefits that will be yours If you can divide correctly* 1.

HIGHER INCOME:

Your ability to find the percent

of profit or loss will increase your business know­ ledge. 2»

RESPECT FROM FRIENDS:

People will notice your

ability to figure the amount each person should pay for a picnic or other activities. 3.

WISER INSTALLMENT BUYING: You will be able to cal­ culate the exact amount of interest on loans.

B.

DIRECTIONS:

Methods to follow in order to divide ef­

ficiently. 1.

ARRANGEMENT:

How to place the numbers to find cor­

rect quotient* a.

Make all digests the same size*

This will

help the alignment. b.

Examine your problem carefully to determine which number should be the divisor.

c.

Place the divisor to the left of the division sign, and the other inside the sign.

2.

PROCEDURES:

How to figure out the number of times

one number will go into another number. a.

Count tlie number of digits in the divisor.

b.

Count from the leftto the right, the same number of digits In the dividend

c.



Estimate the number of times thedivisor

will

go into the numbers counted* d.

If the numbers counted in the dividend are smaller, include one more number*

e.

Place the first answer directly above the last number counted to the right in the dividend.

f#

Multiply the divisor by the first answer.

g.

Place this answer directly below the numbers counted in the dividend.

h.

Subtract this answer directly below the number counted in the dividend.

i.

Bring down one more number from the dividend and place it on the right of the new dividend.

j• Repeat the same procedure until there are no more numbers left in the dividend to bring down or until the quotient digits are above all the remaining digits in the dividend. k.

Subtract the last numbers to find the remaind­ er.

CHECKING RESULTS:

How to prove your quotient is

correct. a.

Estimate the answer by glancing at the problem.

b.

Multiply the quotient times the divisor.

This

answer should equal the dividend* C*

PERFORMANCE;

Problems that will help you to divide

correctly* 1*

DRILL EXERCISES:

Work the following division

combinations• &•

408 divided by 8.

b.

410 divided by 8.

c.

682 divided by 22*

d.

6498 divided by 98*

e.

69768 divided by' 342*

f . 64296 divided by 3. g* 2.

54

h.

231

LIFE PROBLEMS:

i.

775

Solve the following practical div­

ision exercises* a*

You and a group of friends decided to “pitch in” on a picnic*

The group, consisting of 15

members had decided to collect a total of 45 dollars to finance the picnic.

How much does

each member have to contribute? b.

You averaged 52 miles an hour on a 440 mile automobile trip.

How many hours did it. take

to make the trip? c.

Your auto cost $1,000.00*

If you sold this

auto for $500.00,four years later, how much was the yearly depreciation?

19 c.

Your regularly salary on an after school job is #50*00 a month.

What would be the percent

of increase if you were given a raise of #6.00? e.

What rate of interest are you paying on a loan if you borrowed #20,00 and have to pay back #25.00?

D.

EVALUATION:

Tests that will increase your skill in

division. 1.

TRUE - FALSE:

Place an X in the correct space for

true or false. T F a. ( ) ( ) You should place the first digit in the answer directly above the last ,digit in the divisor that we will divide into. b.

(.) (

) You should multiply the divisor by each digit in the quotient.

This re­

sult should be added to the dividend. c.

(){

) You should never finish a division problem with a remainder.

d.

()(

) You should multiply the quotient by the divisor to prove the accuracy of your work.

2.

MULTIPLE CHOICE: parbh theses 8

Place tiie!correct number in the

20 a.

{ ) Your classroom has 448 seats*

If there

were 8 rows of seats each row would con­ tains

(1 ) 6 seats*

10 seats* b.

( )

(4)

(2 )

7 seats*

8 seats. (5)

56 seats.

Your front lawn is 120 feet wide.

The

• number of yards in the width is: yards. (4) c.

90 yards.

40 yards.

(5)

(1) 45

(3) 20 yards*

35 yards*

( ) Material for a dress costs #3.00 for 15 yards. (1)

The cost for one yard would be:

20 cents.

cents. a.

(2)

(3 )

(4)

(2)

60 cents.

40 cents.

(5) 25 cents.

( ) A board is 25 feet long.

You need 4 foot

braces for a club house. ly* you would get; 6 braces. (5)

(3)

9 braces*

(1)

(3) 15

If cut proper­ 5 braces.

21 braces.

(4)

(2)

7 braces.*

CHAPTER 5. FRACTIONS HOW TO ADD, SUBTRACT, MULTIPLY AND DIVIDE MIXED NUMBERS A.

MOTIVATION:

Benefits that will be yours if you learn

to work with fractions* 1.

HAPPIER HOME:

You will be able to measure your

floor spaces for rugs and linoleums* 2*

SAFER DRIVING:

Your automobile gasoline gauges

are divided Into fractions* 3 . BETTER INVESTMENTS:

Reading stock and bond quot­

ations accurately will earn you money* 4.

SATISFIED FRIENDS:

Your guests will not feel cheat­

ed if you can divide cakes and pies proportionately* B . DIRECTIONS:

Rules for you to follow when you are work­

ing with fractions* 1.

ADDING FRACTIONS:

How to find the sum of two or

more mixed numbers. a*

Find the lowest common denominator for all the numbers*

b.

Change each denominator of the mixed number in­ to this lowest common denominator*

c*

Add the numerators of the fractions.

This to­

tal represents the new numerator* d.

Add the denominators of the fractions.

This

total represents the new denominator* e*

If there are whole numbers present, add these

together.

The total of the whole numbers plus

the new denominator and numerator is the result • SUBTRACTING FRACTIONS:

How to find the difference

between mixed numbers. a.

Convert the denominators into one common de­ nominator.

b.

Subtract the numerator of the bottom fraction from the numerator of the upper number.

c.

If the upper numerator is smaller, borrow one from the whole number.

This one should be

changed to a whole fraction and added to the upper fraction. d.

The new numerator should be placed above the common denominator.

e.

Subtract the whole numbers if necessary.

MULTIPLYING FRACTIONS:

How to compute the product

of two mixed numbers. a.

Multiply the denominator and the whole fraction of each number.

b.

Add the numerator to each of these products*

c.

Cancel opposite numbers if possible.

d.

Place the product of the numerators over the product of the denominators.

DIVIDING FRACTIONS: two mixed numbers*

How to find the quotient of

23 a*

Change the mixed numbers to improper fractions as in multiplication*

b.

Invert the divisor and treat as a multiplica­ tion problem.

C*

PERFORMANCE:

Problems that will assist you in becom­

ing proficient in dealing with fractions* 1* DRILL EXERCISES;

Work the following problems

that involve mixed numbers* a*

1 8

4j

b* q1 I



d* 4

3 8

e* 4-1 4

h. | X6|

1. 7l x |

k. 42 2 8 5*9

1 * 7I • 33

m.



* 8

5 w 2 8 6

4i

g.

5

f*

7-|

5 * 4I 6 * 3

n*

8 x 6 7 £

2. LIFE PROBLEMS; Solve the followingpractical

ex­

ercises. a*

Two and one-half candy bars were purchased by you and two of your friends*

What portion

would each receive if the cahdy was divided equally? b.

On a camping trip you hiked 1 mile the first day, 1 3/4 miles the second day, and 2 l/8 miles the third day, how many miles did you hike altogether?

c*

Your sister ate 1 1/8 pieces of pie.

There

24 were six pieces altogether.

How many pieces

of pie were left? d.

Your bicycle cost #11.00.

A friend*s bike cost

3 l/4 times as much as yours.

How much did he

pay for his bicycle? e.

A recipe for candy called for 1 1 / 4 teaspoons full of cocoa. as much.

You decided to make 2 1/2 times

How much cocoa would you need in the

recipe? D.

EVALUATION:

Tests that will improve your ability to

work with fractions. 1.

TRUE-FALSEs

Place an X in the correct space for

true or false. T F a. ( ) ( ) You may add the same number to both the numerator and the denominator with­ out changing the value of the fraction. b.

( ) ( ) You can multiply the numerator and th© denominator by the same number without changing the value of the fraction.

c.

( )( )

You must find the lowest common de­ nominator when multiplying fractions.

d.

( )( )

You must find the lowest common de­ nominator when dividing fractions.

e.

( )£ )

You should.invert the divisor when multiplying fractions.

MULTIPLE CHOICES

Place the correct number in the

parentheses* a.

( ) You can pass a football 1/3 times as far as a friend.

If your friend can throw the

ball 18 yards, you can throw the ball:

b.

(1)

8 yards.

(4)

30 yards•

(2)

36 yards.

(3)

6 yards.

( ) The blueprint of a model airplane you were making was scaled down to 1/4 it's actual size.

If the propeller was 3/4 inches long

on the drawing, the actual length would be:

c.

(1)

5 inches long.

(3)

1 inch long.

(2) (4)

3 inches long.

2 Inches long.

( } You purchased 3/4 of a tank of gasoline for your car.

If your tank held 15 gallons of

gasoline, you purchased: (2)

12 gallons5.

11 1/4 gallons.

(3)

(1 )

11 gallons*

2,0 gallons.

(4)

CHAPTER 6 . DECIMALS HOW TO MAKE FRACTIONS EASIER A.

MOTIVATIONS

Rewards that will be yours if you can-

work with decimals* 1.

WISER BUYING; decimal.

Prices are usually written as a

You will be more efficient in figuring

costs* 2*

EFFICIENT SHOP WORKS

Turning wood or metal on a

lathe requires you to be accurate to the smallest degree* 5.

EASIER COOKINGS

You will be able to visualize

fractional parts of recipes* B.

DIRECTIONS:

Rules to follow when you are solving frac­

tional problems* 1*

ADDING AND SUBTRACTING DECIMALS;

How to find the

sum and difference of problems that require the use of a decimal point. a*

Always align the numbers so that the decimal points will be In a vertical line.

b.

Add the column of figures exactly as you would In any addition situation*

c.

Place the decimal point directly below the other decimals in the answer.

d*

Align whole numbers as If the decimal is on the extreme right.

e.

Follow the same procedure when subtracting decimal problems.



MULTIPLICATION OF DECIMALS:

How to place the de­

cimal point when finding the product of fractional numbers• a.

Add the number of digits to the right of the decimal point in both the multiplicand and the multiplier#

b.

Count to the left in the product, an equal number of digits.

c.

Place the decimal point to the left of the last digit counted in the product#

DIVISION OF DECIMALS: How to place the decimal point when finding the quotient of fractional num­ bers# a.

If only the dividend contains the decimal, place the decimal directly above in the quot­ ient •

b.

When only the divisor contains the decimal, use the

following rules:

(1)

Imagine that there is a decimal oh the right of the dividend.

(2)

Move the decimal, in the dividend, the same number of places that are to the right of the decimal in the divisor.

28 (3)

Add as many zeros to the dividend as nec­ essary.

(4)

c.

Place the decimal point in the quotient *i directly above the moved decimal point*

When "both the divisor and dividend contain a decimal point, the following rules are applic­ able: (1)

Move the decimal to the right in the div­ idend the same number of digits that are to the right decimal in the divisor*

(2 ) Place the decimal in the quotient direct­ ly above the new decimal point. 0»

PERFORMANCE:

Problems that will increase your ability

to work with decimals* 1.

DRILL EXERCISES: blems.

Work the following decimal pro­

Be sure to place the decimal point in the

right location* a.

2.45 4 .298

b.

8976 4 .2 *

c.

855 - '.92

d.

3.48 - .39

e.

598.6 -■ 67

f.

9432 i 4.2

§*

6.75 i 5.

h*

4.365 4 2.5

29 2*

LIFE PROBLEMS: Solve tlie following practical ex­ ercises involving decimals * a*

How much, farther do you have to ride your bi­ cycle for the speedometer to register 10 miles, if it now registers 3.8 miles?

b.

If you live .6 of a mile from school.

How,far

»

do you walk to and from school each day? c.

You are turning out a screw driver handle on a lathe.

If the handle Is supposed to be 3/4

of an inch in diameter, plus or minus .003, what is the largest diameter expressed as a decimal that the handle can be turned? d.

If your car averages 15.5 miles per gallon of gasoline, how many miles should you be able to go with 3*5 gallons?

D.

EVALUATION:

Tests that will improve your ability to

work with decimals. 1.

TRUE - FALSE:

Place an X in the proper spaces for

true or false. T F a. ( ) ( ) A case of 24 bottles of Coca Cola.at #1 .95 a case is a better buy than 3 bottles for 25 cents. h.

( ) ( )

Writing numbers as a decimal is just another way for you to write fractions*

30 c.

( ) ( ) You should count all the digits to the right of the decimal in subtraction problems to find the correct location of the decimal point.

2.

MULTIPLE CHOICE:

Place the correct number In the

parentheses• a.

( ) Your classroom wall is 9.5 feet by 30.3 feet.

The number of square feet of paper

needed to decorate the wall for a parentteachers party is: 2878.5. b.

( )

(3)

287.8.

(1 ) (4)

28.785.

(2 )

287.

If you have a piece of ribbon 6 yards long, the number of pieces you can cut 1.5 yards long is: (3)

40.. (4)

.04.

(1).

.4.

(2)

4.0*

31

PART II. PRACTICAL MATHEMATICS HOW TO SOLVE EVERY-DAY ARITHMETIC PROBLEMS An attempt Las been made In the following chapters to show you how to use mathematics in your daily life.

The

chapters have been separated into home, business and voca­ tional problems.

If you wish to have the ability to apply

the previous six chapters, study the remaining chapters care­ fully.

CHAPTER 7. PERSONAL MATHEMATICS HOW TO SOLVE COMMON HOME PROBLEMS A.

MOTIVATION:

Benefits that will come to you If you can

solve personal difficulties. 1.

HAPPIER HOME LIFEi

You will be a help to your

parents In their budgeting. 2*

HIGHER ALLOWANCE:

Your parents might increase

your Income if you keep records for them* 3.

MORE SELF ASSURANCE:

It will be unnecessary for

you to ask your friends to solve your problems. 4.

HIGHER SAVINGS:

You will be able to calculate

the cost of home repairs. B.

DIRECTIONS:

Suggestions to follow in solving various

home problems. 1.

FINANCING: a*

How to budget your money.

When selecting and sorting items in a budget, consider the following suggestions. (1)

List under food all groceries, and in­ clude meals bought away from home.

(2)

Include under rent or home upkeep; in­ terest, taxes, insurance, repairs and upkeep.

(3) Under clothing include the cost of clean­ ing and repairing as well as new clothes# (4)

List under operating expenses the cost of

fuel, telephone, care Tare, and other similar expenses* (5)

List under entertainment such items as hooks, travel expenses, vacations, club dues, and picture shows*

(6) Such items as dividends on stocks, in­ terest on savings and salary should be listed under income* b*

Decide the amounts to be alloted to each ex­ pense by the following method. (1 ) Estimate your income. (2)

Consider how you wish to live and how much you want to. save.

(3 )

Construct several types of budgets until you have one to your suiting*

BARGAINING; a*

How to shop for best food prices*

Select items that compare in price with your food budget.

b.

Select foods high in vitamins but lower in cost if the budget must be watched*

c.

Find the price per pound when deciding be­ tween buying 3 pounds of beans for 18 cents or 4 pounds for a quarter.

d*

Select canning foods by weights and prices when the same goods are sold in different size cans.

e.

Buy in quantity when there is a saving*

f*

Choose unpackaged, goods for lower prices*

ERRORS: How to prevent mistakes in change making, and sales charges* a*

Prove the correctness of sales charges hy the following ruless (1)

See that the amount listed on an article and the amount on the sales slip corres­ pond.

(2)

Check all addition.

(3)

Compare the number of items on the sales slip with the number of items you pur­ chased.

(4)

Multiply all items that have been extend­ ed*

b.

Give change to customers or friends in the following manners (1)

Begin counting with the amount of the other person's bill*

(2)

Add pennies -until it reaches the nearest multiple of five.

(3)

Use the next largest coin possible to reach a dollar*

(4)

Hand the dollar bills out until five dol­ lars has been reached, then give a five

dollar bill. (5) e.

Count all change aloud,

Receive change from people in the following manner: (1 )

Repeat to yourself each amount the clerk gives you.

(2 )

Remember to begin counting at the cost of your purchase.

RECIPES:

How to decrease and increase recipes for

cooking. a.

Determine the amount of increase or1decrease in cooking by observing the following rules: (1)

Look up the number of servings the print­ ed recipe will make*

(2 ) Determine how many servings you will need. (3)

Decide how much the recipe be increased.

will have to

(For example, a recipe

serves 8 people and you want to serve 12. Since 12 is 1^ times as great as 8, you will have to multiply each of the ingred­ ients by 1§.) (4)

Decide how much the recipe is to be de­ creased*

(For example^, if a recipe will

make 8 hot cakes and you want only 6 , you will have to use just 3/4 of each ingred-

lent•) PAYINGS

How to reduce the co&h of making a drive­

way. a.

Determine the width and length of the drive­ way by measurement,

to.. Prepare the forms for the cement* c.

Use right angles to get parallel lines and square corners.

d.

Determine what thickness of cement you should have for the expected weight.

e.

To find the amount of cement needed you shoulds (1)

Multiply the width times the length times the depth to get the volume of cement ne­ eded.

(2 )

Change the amount of cement to cubic yards, which is the way it is sold.

f.

Call the ready mixed company on the telephone and they will deliver to you the proper amount of cement.

PLOORINGs a.

How to measure a room for a rug.

Determine how the linoleum is sold. (1)

By the square foot.

(2) By the running foot, six feet wide. to.

Find out the number of square feet in the roam if it is sold by square feet.

37 c.

Find out how many strips there are six feet wide in the room if it is sold by the running foot*



PERFORMANCE:

Activities that will increase your abil­

ity to 3olve home problems*

1*

Make a budget of your expenses for one month*

2*

Bring to class several sales slips and check the totals for errors•

3*

Help out at the candy booth during the lunch hour*

4.

Bring to class your favorite recipe.

Write it out

in proportions large enough to serve the whole class* 5.

Determine the amount of paint needed to paint the inside of the classroom*

D.

EVALUATION:

Tests of your ability to solve common home

problems » 1*

TRUE - FALSE:

Place an X in the proper spaces for

true or false* T F a. ( } ( ) You should always pu^bfeise a r u g b y the cubic foot* b.

(

) { ) If a recipe serves 8 people and you wish to serve 12, you will have to divide to find the amount of each ingredient•

c*

(

) ( ) In changing a recipe from ,8. servings

to 6 servings you should use 3/4 of the amount of the ingredients given* MULTIPLE CHOICES

Place the correct number in the

parentheses* a*

( )

The number of cubic yards of cement ne­ eded to make a driveway 9 feet by 20 feet by 6 inches is: (3)

b*

4.

(4)

(1)

40*

(2)

3.33.

10.

( ) If one pint of paint covers an area of 20 square feet to cover an area of 90 square feet you should purchase: (2)

4 pints.

(3)

(1)

5 pints.

4 gallons* (4)

2 quarts.

CHAPTER 8 . BUSINESS HOW TO MANAGE YOUR MONEY A.

MOTIVATION:

Benefits that you will receive If you are

thrifty. 1*

LESS WORRY:

A “nest egg" in the bank will make

you feel more secure. 2m

RECORD OP CASH PAYMENTS:

Your check stubs are an

excellent form of receipt. 3.

PROVISION FOR EMERGENCIES:

Many insurance policies

allow you to take out loans on them. 4.

BETTER CREDIT RATING:

A savings account with a

bank will increase your reliableness. B.

DIRECTIONS:

Suggestions to follow to increase your

efficiency in money matters. 1.

SAVINGS ACCOUNT:

How to use a bank for saving

money. a.

Select a bank that is conveniently located.

b.

Choose a bank that belongs to the Federal Reserve System.

c.

Ask your friends about a bank before making a choice.

d.

All banks have the same rate of interest.

Ask

them what the rate happens to be. e.

They will record your depositand give you- a deposit book.

Keep this book In a safe place*

f*

Always add the amounts to check for errors*

CHECKING ACCOUNT:

How to place money in a bank for

current use* a*

Decide how much money you need for every-day expenses*

Remember that money in a checking

account does not draw interest* b*

Take the amount to the bank selected.

c.

The bank will open an account for you and give you a check book*

d.

The following are some of thd reasons why you should open a checking account* (1)

Provides a safe place for money.

(2)

Makes payments to others easy and inex­ pensive.

(5 )

Affords a record of these payments.

PREPARING DEPOSITS:

How to take large amounts to

a bank* a.

Bills should all face the same way with the highest denomination on top*

b*

Large amounts of coins should be wrapped.

The

bank furnishes coin wrappers* c*

Count all money before going to the bank and again as the cashier counts it*

BANK FORMS: a.

How to fill out checks and bank drafts,

Fill in the stubs first*

b.

Subtract the amount of the check from the pre­ vious balance*

c.

Begin words or numbers close to the left mar­ gin.

d*

Don*t place a future date on checks nor a Sun­ day date*

INSURANCE: a.

How to provide for your future.

Determine from your budget the amount you can afford for insurance.

b.

Determine what type of insurance policy you desire.

c.

Consider the following points. (1)

Does the policy have a cash surrender value ?

(2)

Would I be better off to invest the money rather than buy insurance?

(3)

What is the amount of the premium?

INVESTMENTS: a.

How to earn more for your money.

Determine if you are financially able to spec­ ulate on the market.

b.

Determine the type of securities you wish to huy.

c.

Compare the rates of interest on various stocks and bonds with the rate of savings account9 and United States Savings Bonds.

C.

PERFORMANCE: Problems that will help you to keep track of your money* 1.

Bring some deposit slips and checks to school to be filled out in class.

2*

Check the accuracy of an old bank statement keeping withdrawals and deposits separate.

3.

Determine your monthly income from three jobs, a paper route which brings .50 cents a night, baby sitting three nights from 9 to 12 P. M. at .25 cents an hour, and mowing two lawns at .75 cents an hour one of which took 1^ hours and the other 2 hours. If you have a job of your own use those figures.

4.

Compute the amount in dollars of several bonds drawing various interest rates.

5.

Compute the difference in the amount earned on *

$1 ,000.00 invested in bonds, stocks, insurance, and savings account. D.

EVALUATIONi

Samples of check-ups on your ability to

take care of your money. 1.

TRUE - FAISEi

Place an X in the proper spaces for

true and false.

T

a.

F

( ) ( )

It is correct to list all checks to­ gether on deposit slip.

b.

( ) ( ) You should keep small amounts of money in,a checking account.

43 c.

( ) ( )

Savings accounts earn more interest than bonds.

2.

MULTIPLE CHOICE:

Place the correct number in the

parantheses. a.

( ) If you have #239.77 in the bank and write 3 checks, one for #9.17, one for #53*25, and one for #23.50, your check stub bal­

h.

ance will be:

(1)

#154.85.

(3)

(4)

#154.75.

#164.85.

(2)

( } Tour wage is #1.10an hour for 40

#155.85.

hours

a week and time and a half for overtime. If you work 44 hours one week, your Social Security deduction will be: (2)

#1.52.

(3)

#1.45.

(4)

(1 ) #1.51. #2.18.

CHAPTER 9. SHOP HOW TO SOLVE PROBLEMS IN YOUR HAND WORK A«

MOTIVATION:

Rewards that will be yours if you can

calculate vocational problems. 1,

HIGHER INCOME:

The more diversified problems you

can work the better chance for promotion. 2*

HIGHER SCHOOL MARKS:

Your shop courses require

you to figure costs and measurements. 3.

ECONOMY OF MONEY:

If you can measure accurately*

home repairs will be cheaper. B.

DIRECTIONS:

Procedures to follow to increase your

ability in mechanical mathematics. 1.

BOARD FEET: a.

How to find out the cost of lumber*

Consider the

following when measuringthe size

of board you desire. (1)

In measuring the width of lumber a frac­ tion of an inch equal to or greater than is counted as a whole inch.

(2)

A fraction less than ^ is neglected.

(3)

Count a thickness less than 1 inch as an inch.

b.

Multiply the

length times

c.

Divide by 12if the width is in inches.

d.: Multiply by 5£he thickness

the width.

in inches.The re­

sult gives the number of board feet.

e.

Find out the cost per hoard foot and multiply to find the total coat*

e

ALLOWANCE FOR WAST:

A

How to estimate enough wood

to offset loss in jointing. a*

For flooring and ceiling, from about 2

inches

up to 5^- inches in width, the amount allowed is one fourth. b.

For ”Adams” flooring, usually about 1^ inches wide, the amount added is one third.

c.

In most localities it is customary to give the lumber dealer the actual size of the surface to be covered, and let him add the proper am­ ount.

HEIGHTS OF OBJECTS:

How to find out how tall a

tree is without climbing it. a.

Measure the shadow of the tree.

b.

Measure a shadow of a stakeor otherobject near the tree.

c.

Measure the height of the stake.

d.

Set up the proportion; the height of the stake to it's shadow equals the height of the tree to‘it's shadow.

PULLEYS AND BELTS:

How to find

thenumber of re­

volutions a machine is turning. a.

The following rules apply in finding the speed.

46 (1)

Measure the diameter of the living pulley*

(2 )

Determine the revolutions per minute of the driving pulley.

This is usually

known. (3)

Measure the diameter of the pulley being driven*

(4)

Set up the proportion; the diameter of one pulley times it’s speed is equal to the diameter of the other pulley times1 it’s speed*

G.

PERFORMANCE:

Problems that will improve your ability

to do shop problems* 1.

Figure the cost of building wooden objects around your home.

2*

Determine the height of the school flag pole.

-

3*

Compute the revolutions per minute that the school band saw is turning.

4*

Measure the diameter of circular objects in the classroom*

5.

Go to the local lumber dealer and find out the current cost of white pine lumber.

6*

Figure the amount you should allowfor wastage

in

building a desk. D.

EVALUATION:

Tests to determine your ability to work

shop problems*

47 1.

TRUE - FALSE:

Place an X in the proper spaces for

true and false* T F a* ( ) ( ) If a board is less than ^ inch thick, you disregard the thickness in figur­ ing board feet. b.

( ) ( )

You divide the width in inches, by 12 in order to change it to feet*

c.

( ) ( )

If you set up the proportion; the height of a tree over the shadow of a stake times the height of the stake over the trees shadow, the answer will be correct.

2*

MULTIPLE CHOICE:

Place the correct number in the

parentheses* a*

( ) Ina piece of lumber 1 inch thick, 12 in­ ches wide, and

5 feet long, the number of

board feet is:

(1 )

5 board feet.

(3)

60 board feet* (2 ) 12 board feet.

(4)

6 board feet* b.

( ) A tree that has a shadow of 12 feet is be­ side a 4 foot high stake with a shadow of 1 foot,, therefore, the height of the tree is: feet.

(1) (4)

48 feet. 40 feet.

(2)

36 feet.

(3) 12

48 c.

( )

If the diameter of a driving pulley is 12 inches and it*a speed is 400 revolu­ tions per minute, the speed of the driven pulley whose diameter is 5 inches would b© s

(1 )

volutions.

4800 revolutions• (3)

(2)

1600»revolutions*

36 revolutions*

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130 re­ (4)