Mathematics for Georgia schools, volume II [1962]

for Georgia Schools
VOLUME 2

2

3

4

5

DEPARTMENT OF EDUCATION

CLAUDE PURCELL, SUPERINTENDENT

ATLANTA,

GEORGIA

~

!1 eu, ])e.pt, 6f Ed/..'- tIC. J),'V,'SION 6.. rI"S'tr((.,t!"/'"

MATHEMATICS

FOR

GEORGIA SCHOOLS

VOLUME II

STATE DEPARTMENT OF EDUCATION DIVISION OF INSTRUCTION CURRICULUM DEVELOPMENT

Claude Purcell

State Superintendent of SChools

Atlanta, Georgia

1962

Copyright 1962
Georgia State Department of Education
Atlanta, Georgia

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 DISTRIaf SEVENTH CONGRESSIONAL DISTRICT EIGHTH CONGRESSIONAL DISTRICT NINTH CONGRESSIONAL DISTRICT TENTH CONGRESSIONAL DISTRICT

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

iii

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 mathemati cal knowledge. Whether it is John Glenn figuring a race through space or a young couple figuring their budget, they need mathematical accuracy and understanding.

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 forefront 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 indicales ways in which mathematics programs can be improved. Concrete materials, when used in the right way, aid in building understanding. Filmstrips, films, and educational television 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 ar: mged 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 me ant 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
vi

TABLE OF CONTENTS

SEVENTH GRADE

I. Introducting Sets .

.__________________________________________________________ 2

II. Numeration

.. __.

.

4

III. Operations with Whole Numbers

6

IV. Operations with Fractions

V. Relations

.

. ..__.

-------- 14 24

VI. Per Cent and its Application .

---.-- 26

VII. Measurement VIII. Geometry

..

.

.

---- 30 36

IX. Statistics and Graphs .__________________________________________________________________ 44

X. Problem SoIving

45

EIGHTH GRADE

I. Numeration

..__.

52

II. Operations with Non-negative Rational Numbers

56

m. Applications of mathematics in Business

Finance, and Daily Living

..

63

IV. Measurement

66

V. Geometry

-__________________________________

72

VI. Statistics and Graphs
vn. Introduction to Algebra

....__.. 81 .___________________________________________________________ 81

Seventh and Eighth Grade Vocabulary

88 ~__________________________

Seventh and Eighth Grade Materials

90

vii

....

ELEMENTARY ALGEBRA I. Sets and the Number Line II. Numerals and Variables III. Sentences and Properties of Operations IV. Mathematical Sentences and English Sentences V. The Real Numbers VI. Factors and Exponents
VII. Radicals and Roots VIII. Polynomial and Rational Expressions
IX. Solution Sets of Open Sentences X. Graphs of Open Sentences in Two Variables Xl. Systems of Equations and Inequalities XII. Quadratic Polynomials XIII. Functions XlV. Ratio, Proportion, and Variation
Vocabulary

96 97 99 100 101 ~______________________ 110 111 113 115 116 117 118 118 119 121

GENERAL MATHEMATICS I. Numeration II. Operations with Whole Numbers III. Operations with Fractions
IV. Relations V. Measurement VI. Graphs VII. Geometry VIII. Problem Solving
Vocabulary Books For Pupils References For Teachers Glossary

126 129 133 -'____________________________ 139

141

.

142

143

144 145

._____________________________________________ 146 147 149

II

SEVENTH GRADE

3

UNDERSTANDINGS TO BE DEVELOPED IN SEVENTH GRADE
Through a series of interesting, exciting experiences the seventh grade pupil is helped to discover ideas, facts, and principles for himself. Those pupils who need additional help are retaught and thus gain the security of understanding. Seventh grade pupils have opportunities to acquire an appreciation of the historical development of numbers and measures and relate these factors to the development of civilization. A review of the meaning of numbers in terms of place value, powers of ten, reading very large and very small numbers, and numeration with bases other than ten extends and reinforces the understandings of decimal notation. Both skills and concepts are essential. Skill must be based on understanding and not on rote memorization. More emphasis is placed upon pupil discovery and reasoning, reinforced by greater precision of expression. The meaning of operations is stressed, as well as the mastery of the fundamental operations with whole numbers and fractions at the adult level of skill and understanding. The seventh grade pupil sees mathematics in terms of structure and pattern. Generalizations are preceded by concrete illustrations. The language and basic ideas of sets are used advantageously. Extension of the understanding and use of equations as number sentences is provided, as well as increased use of mathematics symbols. Mastery of the ability to use ratios and per cents is important. Pupils at this level develop the ability to operate with and transform the several systems of measure, including the metric system and indirect measurement. They are familiar with the basic line, plane, and solid figures; they should become proficient in using the compass, the straight edge, the ruler, . and the protractor. The ability to interpret and construct simple graphs, understand the meaning of statistical averages, and recognize, understand, and use formulas is developed. The enrichment and extension of arithmetic vocabulary are important as is the ability to verify the accuracy of the solution of problems by tests of reasonableness and by arithmetical checks.
1

CONTENT
I. Introducing Sets
A. The idea of sets
B. The symbolism of sets:
C. Comparing sets 1. !-to-l Correspondence
2. Equivalent sets D. Relations between sets
1. Equal sets 2. Subsets

TEACHING SUGGESTIONS
Sets are introduced because of their usefulness in simplifying and clarifying many of the concepts of mathematics. It is not intended that sets is to be a separate topic but that the language and ideas of sets will be used through out future courses in mathematics.
Set is an undefined term but examples and synonyms should be given to strengthen the student's intuitive notion. A set is any collection. Sets of aishes, sets of books are examples of familiar sets. The items which belong to a set are called members or elements. For a set to be meaningful and use ful in mathematics, it must be well-defined. A set is welldefined if it is possible to tell whether an element belongs or does not belong to the set. Consider the following examples:
The set of natural numbers less than five
The set of big buildings
The first set is well-defrned; the second is not.
Capital letters are used to name sets. If the elements of a set are listed, they are enclosed in braces. A set may be described in words as in the preceding example, but the mathematical symbols are simpler to read and use less space.
Example:
The set of natural numbers less than five
= A {l, 2, 3, 4} = The set of natural numbers
B {l, 2, 3, ... }
The three dots are used to indicate that the elements continue in the same pattern.
Students should have practice in identifying well-defined sets, in changing word descriptions into mathematical symbols, and in describing in words, sets given in symbols.
In problems in which sets are used it is frequently necessary to compare two or more sets. Sets may be compared by matching the elements of one set with the elements of an-,ther.
Sets A and B are said to be in oneto-one correspondence if each element of
= = A can be matched with one element of B and each element of B can be matched
with one element of A. If A {a, b, c} and B {3, 4, 7}, then A and B can be placed in one-to-one correspondence.
abc
!!!
347
If two sets can be placed into one-to-one correspondence, they are said to be equivalent or to have the same number.
= = If two sets contain the same elements, they are said to be equal. If C = {2, 3, 7}
and D {7, 4/ 2 , 3}, then C D. Equal in this case is used to indicate that C and D are different names for the same set. Note that the order in which the elements occur is of no importance.
If every element of one set is an element of another, the first set is said to be a subset of the other. If the second set contains at lease one element
= = which is not in the first, then the first is a proper subset of the second and the
second is a superset of the first. If A {2, 3, 7} and B {3, 7, 2}, then by the
2

3. Intersecting sets 4. Disjoint sets E. Finite, infinite, and empty sets
F. Operations with sets G. Use of Venn diagrams

= definition of subset. A is a subset of Band B is a subset of A. (In symbols ACB,
which is read "A is contained in B" or "A is a subset of B.") If C {2, 3} and
= D {4, 3, 2, 7}, then C is a proper subset of D. (CcD)
If two sets have some members in common but each set contains at least one element which is not in the other, the sets are said to intersect and the set of elements which is common to both sets is called the intersection of the
= = two sets. If A {2,3} and B {3, 6}, then {3} is the intersection of A and B. = = Disjoint sets have no elements in common. If A {I, 2, 6} and B {3, 4, 7},
A and B are disjoint sets.
If the elements of a set can be counted with the counting coming to an end or if a set contains no elements, the set is said to be finite. All other sets are infinite.
= A {all natural numbers} = B {all whole numbers less than five}
C = {all integers less than seven and greater than ten}
Set A is an infinite set, and Band C are finite sets. C is an example of the set which has no elements. It is called the null set or the empty set, and the symbol used is or { }. The null set is considered to be a proper subset of every set.
We may perform operations on sets just as we do on numbers. Let A = {3, 5, 11, 6}, and B = {11, 6, 4}.
The union of the two sets (symbol U) is the set which consists of all the elements which are in A or B or in both. AUB = {3, 5, 11, 6, 4}.
The intersection of the two sets (symbol n) is the set which consists of all
the elements which are in A and also in B. AnB = {11, 6}.
Diagrams are helpful in simplifying facts about sets. These diagrams are usually called Venn diagrams although Euler and many other mathematicians have used diagrams to represent sets. Let us look at some examples:
1. A = {all isosceles triangles}
= B {all equilateral triangles}

B is a subset of A
2. C = {all rectangles}
= D {all rhombuses}
C and D are intersecting sets. Their intersection is the set of all squares.
3. E = {all even natural numbers}
= F {all natural numbers which are divisible by 2}
3

s
o E and F are equal sets
= 4. G {the even integers} = H {the odd integers}

G and H are disjoint sets.

I, II. Numeration

A. Early systems of numeration

Begin this unit with background material on the importance of mathematics in our lives. Lead into a discussion of how numbers probably began. For those students who do not have the proper background of information the following is suggested:
Primitive peoples kept records in various ways:
Knots tied in a rope
A pile of pebbles
Marks on a stick
Scratches on rocks or in the earth

1. One-to-one matching

These systems represented the first numerals or symbols for numbers. As man needed to know how many, he used the idea of one-to-one correspondence to develop symbols for groups of objects. The fingers were the foundation for the idea of the digit one, the concept of five, and the concept of ten.

The abacus, one of the oldest computing machines, has played an important

role in the history of the development of our number system. The beads

I

represent number symbols. An instrument of this kind helps the pupil understand the structure of the number system. Explain how the value of the

'I

bead depends upon its position. Point out that there are many different kinds

of abad, two major types being the Chinese and the modern. Make rod and

I

line abaci for illustrations.

I

Review the meaning of number and numeral.

I

2. Roman

Compare the symbols in our system of numeration with those of the Romans. Stress position of the symbols.

Write Roman numerals. Illustrate the difficulty of computation with the

I

Roman system of numerals.

3. Egyptian and Babylonian

Make charts, posters, board drawings, etc. showing the symbols for Egyptian

and Babylonian numbers. Some counting and writing of numbers may be done

I

with these symbols. Point out that the Babylonians used a base of 60.

I

B. Our system of numeration

Review the meaning of place value.

4

1. Understanding of the decimal system as powers of ten

Each digit in our number system has a place value based on 10 or a power of 10.

Place Name Units Tens Hundre<;ls Thousands Ten Thousands Hundred Thousands Millions

Place Value Power of 10

1

10

10

101

100

102

1,000

103

10,000

104

100,000

105

1,000,000

106

Notice that the zero power of ten fits into the pattern of powers and that 100 means 1.

Nine is the largest one-place number. Numbers larger than 9 are regrouped into ones, tens, hundreds, thousands, etc.

22 is 2 tens plus 2 ones
222 is 2 hundreds 2 tens
plus 2 ones

20 2 22
200 20 2 222

The value of any digit depends upon its position in the numeral.

Example:
+ + 444 is (4 X 100) (4 X 10) (4 X 1)

Zero has no value of its own but gives value to other numbers as: The numeral 5 means 5 ones. With a zero to the right it means 50 ones or 5 tens.

Illustrate with the odometer.

Explain exponent and base.
= 4'4.4.4'4 45

In this illustration the 4 is called the base, and the 5 is called the exponent or power.

Use practice exercises such as:

What is the value of 6 is each of the numerals:

106

61,223

601

906,000

4,262

621,123

What is the largest possible four-place number that can be written, using each of the digits 3, 6, 1, 4 once?

2. Reading and writing very large and very small numbers

Explain how the digits in numerals are grouped into sets of 3 called periods. Make charts, posters, etc., illustrating periods and position names through trillions.

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E

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456, 789, 123, 456 789

5

Make charts or posters to illustrate decimal place value to ten thousand and ten thousandths.

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C. Numeration in systems with bases other than ten.

Practice reading and writing these types of numerals from dictation. Place on the bulletin board clippings from newspapers and magazines in which large numbers and small numbers are used. Strive to develop the understanding that 6,573,289 means:
6,000,000 + 500,000 + 70,000 + 3,000 + 200 + 80 + 9
and is
6(1,000,000) + 5(100,000) + 7(10,000) + 3(1,000) + 2(100) + 8(10) + 9

However,

1,000,000 is 10 . 10 . 10 . 10 . 10 . 10 100,000 is 10 10 . 10 . 10 . 10 10,000 is 10 . 10 . 10 . 10 1,000 is 10 . 10 . 10 100 is 10 . 10 10 is 10 1

106 105 104 103 102 101
= 100

Thus 6,573,289 may be written:
6(10)6 + 5(10)5 + 7(10)4 + 3(10)3 + 2(10)2 + 8(10)1 + 9(10)0

Count collections of items and group them in sets of 10. Regroup to show systems with other bases such 5, 6, 7, etc. Explain the number of digits used and the symbols in each system. Count orally in the bases studied; fill in missing parts of a list of numberals; write numerals for numbers from 1 to 25. Numerals in other bases have the base indicated to the right and below the numeral as:
2 4 "
These indicators are called subscripts.
24" means (2 X five) + 4

Emphasize that the study of other number bases is undertaken to strengthen understanding of the base ten or decimal numeration system.

III. Operations With Whole Numbers

A. Counting numbers

Recall that the "counting numbers" are sometimes called "natural numbers," and that the counting numbers do not include zero. The counting numbers and zero are called "whole numbers."
Review the early ways of counting and matching such as a pebble for each sheep, a mark on the wall for each member of the tribe, etc.

6

B. The natural numbers have order

Point out present day methods of matching: a book for each student; a student for each desk; a pencil for each student. Develop the understanding of a one-toone correspondence between two sets.
Draw dots in a horizontal line and have students number them, writing below the points.
Example:

C. The commutative property for the addition of natural numbers

1234567
When there is no room to continue, point out that this naming of points could go on forever. This is a picture of natural numbers. They are lined up or ordered. There is a first number, but no last number.
Ask the students to arrange lists of numerals in their natural order.
Example:
111, 209, 322, 876, 701, 45, 563 or
IV, V, X, IX, VI, VII, VIII
Use the number line for developing concepts using the symbols <, >, #.
Examples:
5 < 10, 10 > 8, 4 # 14.
Illustrate "betweenness" of numbers: 5 is between 4 and 6; 3 and 4 are between 2 and 5.
Use various number combinations to show that the sum of two numbers is the same regardless .of the order in which the numbers are added.
"Add up" and "add down" columns of figures. This process is a check for addition examples.
= Lead to the generalization that for any whole numbers a and b, a + b b + a.
This is called the commutative property for the addition of natural numbers.
Point out that the commutative property of addition for natural numbers does not depend upon the kind of numeration system used.
Review numbers in numeration systems with bases other than 10. Make addition charts in other bases, and point out the commutative property of addition.
Example: Base 5
+I0 1 2 3 4
00 123 4 1 1 2 3 4 10 2 2 3 4 10 11 3 3 4 10 11 12 4 4 10 11 12 13

Add problems in base five such as:

+ 23tlve 2 fives 3 ones

14"ve

1 five + 4 ones
= = 3 fives + seven ones 4 fives + 2 ones 42"..

Subtraction is the inverse of addition. Is it commutative?
= Example: Does 6 - 4 4 - 6?

7

r

D. The commutative property for the multiplication of natural numbers

The commutative property holds for addition but not for its inverse, subtraction.
In the addition of two numbers, subtraction may be used as a check. In subtraction, addition is the check because these operations are opposites.

Give subtraction exercises with numerals in bases other than 10.

Example: 41 ft
-14ft
22 ft

Change 1 of the 4 fives to 5 ones. We then have 6 ones. Take 4 ones out leaving 2 ones. 1 five from 3 fives leaves
2 fives.

Arrange blocks, discs, stars, etc., in 7 rows with 9 in each row. Ask how many blocks are in the arrangement. Rearrange to 9 rows with 7 in each row. Point out that this is the same number as before but a different arrangement. The product is the same whether the first is multiplied by the second or the
= second by the first. Thus a b b. a becomes a true sentence whenever any
numerals for natural numbers are put in place of "a" and "b."

lntroduce the terms factor and product.
Factor X Factor = Product

Illustrate how the commutative property of multiplicati\ln may be used for checking:

5826 4723

4723 5826

Since the individual steps are different in each case, it is not likely that the same mistake would be made.
Make multiplication. charts with numerals in bases other than 10 and point out the commutative property of multiplication.

Example: Base 7

X01 000 101 202
=m 3 404 505 606

23456 00000 23456 4 6 11 13 15 6 12 15 21 24
I 11 15 22 26 33
13 21 26 34 42 15 24 33 42 51

E. The associative property for addition of natural numbers

What operation is the opposite of multiplication? Does the inverse of multiplication have the commutative property? Is it true that
= 8 -;- 4 4 -;- 8?
The operations of addition and multiplication are commutative, but the inverse operations are not.
With counters or on the flannel board or chalkboard, put 4 and 3 and 2: xxxx, xxx, xx
Push the first two groups together and count the total. Write:
(4 + 3) + 2 = 9
Rearrange and put the second two groups together. Write:
4 + (3 + 2) = 9

Explain that the parentheses are used for grouping numbers. Practice with other numbers.
8

F. The associative property for the multiplication of natural numbers
G. Natural numbets have the distributive property for multiplication over addition

= Point out that with the commutative property of addition the order in which
two numbers are added makes no difference in the sum, that is, a+ b b + a. The associative property of addition for natural numbers means that the way numbers are grouped to add does not change the sum, that is:
= (a + b) + c a + (b + c).
Does the associative property hold for the inverse of addition? Is this true?
=?
9 - (4 -- 3) (9 - 4) - 3
Put 5 X 4 X 2 on the chalkboard. Try various ways of finding the product, such as:
= = (5 X 4) X 2 20 X 2 40 = = 5 X (4 X 2) 5 X 8 40
Point out that the factors of a product may be grouped in any order without changing the product. This is called the associative property of multiplication for natural numbers.
= (a X b) X c a X (b X c)
Is this true?
=?
12 -;- (4 -;- 2) (12 -;- 4) -;- 2
The associative property holds for the operations of addition and multiplication, but does not hold for their inverses.
. . . . . With counters on the chalkboard, show 2 4 and 2 6: "
Put these together and show 2 rows of 10:

= Illustrate on the board as (2 4) + (2 6) 2. 10.
Repeat with other numbers. With objects layout 3 rows with 5 in each row:

Separate into:

= Il1ustrate on the board 3 5 (3 2) + (3 3)

Repeat using other numbers.

This idea links together the operations of multiplication and addition. This law 'is the distributive principle of multiplication over addition.
= a(b + c) (a b) + (a c) or
(b + c) a = (b a) + (c a) = + (a b) (a c)

Examples:
8. (4 + 9) = 8 4 + 8 9 = 104

32

30 + 2

X4

X4

128

120 + 8

= + (30 2) X 4 = + (30 X 4) (2 X 4)
= + 120 8 128

9

1

o...... Beans

Corn

1

+--5---+~

10

4

15



A garden plot 10 feet by 15 feet is divided into two plots as shown. The area of the whole plot is 10 X 15 or 150 square feet. The area of the bean plot is 5 X 10 or 50 square feet. The area of the corn plot is 10 X 10 or 100 square feet. The total area of the garden is the sum of the areas of the bean plot and the corn plot.
150 = 50 + 100 or = 10 X 15 (5 X 10) + (10 X 10)
Replacing 15 with 5 + 10 this follows:
10 X (5 + 10) = (5 X 10) + (10 X 10)

H. The set of natural numbers is closed under the operations of addition and multiplication

Put addition problems on the chalkboard. Point out that the addends are natural numbers and the sums are natural numbers.
7 + 8 = 15 = 35 + 49 84
232 + 629 = 861
When any two natural numbers are added, the sum is one and only one natural number. Thus the set of natural numbers is said to be closed or to have the property. ~ closure under addition.
Put multiplication problems on the chalkboard.
= 2 X 9 18
7x9=63 18 X 22 = 396
The above property is true for multiplication, since the product of two natural numbers is one and only one natural number.
Give exercises with subtraction problems, being sure that some answers are zero or less than zero.
10 10 9 13 15 17
Point out that it takes only one case to show that the system is not closed under subtraction.
What about division, the inverse of multiplication? When one natural number is divided by another natural number, the result is not always a natural number.
Examples:
= 6 -7- 2 3, a natural number, but = 7 -7- 2 3%. 3V2 is not a natural number.
Summarize with a discussion period, emphasizing that addition and multiplication over the natural numbers are basic operations having closure while their inverses do not.

10

I. The number one and the number zero. 1. One
2. Zero

Stress the importance of the number one. In the study of early systems of numeration it seems to be historically true that this number is the first one used by man and is the first natural number. The other natural numbers are formed by :repeated additions of the number one. Adding one to the natural number gives the successor to the number. Subtracting one gives the predecessor. The number one is the only natural number that does not have a predecessor.

Emphasize the fact that there is more than one way to represent the number one.

Examples:
+ 4 - 3, I, 1, one, 4 X 1/ 4 , 5/5' V - IV, 200/ 200, 1/ 2 1/ 2 , 100%
Add one to any even number and ask the pupils what kind of a number is obtained. Point out that such a procedure always produces an odd number. Do the same with an odd number to show that one added to an odd number gives an even number.

Ask the pupils to multiply any number by one and see what the result is. Point out that any natural number multiplied by one is that number, and thus one is called the identity element for multiplication or the multiplicative identity.
nl=ln=n

If we divide any number by one we will get the same number: 1/ 1 = 10, 5/ 1 = 5. Thus we can write n/ 1 = n. But one divided by any number does not give the number.

When the Roman and Egyptian systems of numeration were studied, it was found they had no symbol for zero. Man found that he needed this symbol. For example, when he needed to write an answer that he had calculated on the abacus and came to a wire that had no beads, he had no way of recording it. This problem was solved by filling the blank with the symbol 0 thus assuring the other numbers of their rightful place. Zero is a number that means not any. The invention of the zero simplified numeration.

Examine a ruler or yardstick. Usually the left end is not marked with a numeral. Since 0 is less than any natural number, this left end could be marked O. Zero is the number to the left of one and is the first whole number. Some of the meanings of zero are: a place holder, as in 304; progress of a game, as in a baseball score board; starting point on a scale, as on a thermometer; an exponent, as in 20.

The number zero is the number of elements in the empty set. If Susie has candy for each person in the group except for Mary, then Mary is aware of the "not any" aspect of the situation.

Ask the students to add 0 to any number and let them discover that they

get the same number Zero is called the identity element for addition.

Ask

the

students

to

expand

a

number

such

as

806

and

show

-
that

(8 100)

+

(0 10) + (6 1) = 800 + 0 + 6 = 806.

Discuss the idea that if any number is multiplied by zero, the product is zero. Point out that for a product to be zero, at least one factor must be zero.

for any n.

Division is the opposite of multiplication. 8/ 2 = 4, because 8 = 2 4. This is the way division problems are checked by multiplication. Thinking of / 4 in the same way which of the following is true?

The first cannot be true because 0 oF 4 4. The third cannot be true because 0 oF 4 1.

11

J. Factoring and primes 1. The meaning of prime and composite numbers
2. The meaning of factor

= = Since 0 4 0 the second is true. Thus 0/ n 0, where n in any non-zero
number.
= Try to find 4/ o' It would appear that 4 O . No such number can be
found, therefore 4/ 0 has no meaning. "n"/o is not the name of any number for any n.
Try to find the result of 0/ o' To do this it is necessary to find a number so
= that 0 o _~__. In the example 4/ 0, a number could not be found to fill
the blank; but here any number will fill the blank. This means that % has no definite value and is meaningless. % is not the name of any number. Emphasize that we are never to divide by zero.

Any natural number is either prime or composite except 1 itself. How does one tell whether any given natural number is prime or not? To factor a natural number is to find several natural numbers whose product is the given number. For example, a factorization of 12 is 3 4. Some others are 1 12, 2 1 3 2, and 2 . 6. Every natural number can be factored. Every number n has at least the factorization 1 n. The prime numbers are those
that cannot be factored without using 1 as a factor. For example, 2, 3, 5, 7,
11, 13, 17, 19, 23, 29. One is not considered prime because any number could
be factored into primes many ways if it were.

More than 2000 years ago Eratosthenes, a Greek mathematician, conceived a scheme for finding all the prime numbers less than a certain number. His device is called the Sieve of Eratosthenes.

It sifts out prime numbers. Use dittoed sheets with numbers 1 to 100 to illustrate Eratosthenes' scheme.

-l- 2

3 4- 5 -e- 7 -& -9- -l&

11 -l-a- 13 -l4- -l5- -l6- 17 -l& 19 -2&

~ ~ 23 -4!4- -25- ..a&- -2lr -28' 29 ~
31 -32" -aa- -34- -a5- -36- 37 -3a- -de- -4&

41 ...42- 43 .44- -45- -46- 47 48- -49- '-56"
-M- -62- 53 -M- -55'" --5fr" -M- -aa- 59 -66'

61 ~ -6S" -&i- --&.;- -e& 67 --M- -G9- $}-

71 2t2- 73 ..74- ..!f5- -!ftt --'Ff- ....!f8" 79 -00-

-at- -82' 83 -at- -85- -86- -8'r -aa- 89 -00-

.w- -9i- -92- -9a- -94" -95- -96- 97 -98-

-lOO-

Mark out 1; leave 2, but mark out every number that has 2 as a factor. Leave 3, but mark out every number that has 3 as a factor. The next number left unmarked is 5. Leave it, but mark out every number that has 5 as a factor. Proceed in this way and finally only the 25 prime numbers will be left, all composites and 1 having been sifted out. Notice that some numbers were marked out twice or three times. They were multiples of more than one prime.
The term factor has already been introduced. Instead of using the terms multiplicand and multiplier, each of them may be called factor.
= 3 X 6 = 18, 3 and 6 are factors of 18, also 18 = 3 X 3 X 2. 42 = 6 X 7,
also 42 2 X 3 X 7. The expression, "the factors" means" all the factors" of a number. The set of factors of 20 is 1, 2, 4, 5, 10, 20. The set of factors of 15 is 1, 3, 5, 15. The number 1 and the number itself are always factors of a number.

12

3. Factorization of whole numbers
4. Understanding of even and odd numbers

Consider the number 24. It can be written 4 X 6. Both 4 and 6 are composite numbers since they can be written as products of smaller counting numbers.
4 = 2 X 2; 6 = 2 X 3. Then 24 = 2 X 2 X 2 X 3.
Since 2 and 3 are prime numbers they cannot be expressed as products of smaller numbers. Thus 2 X 2 X 2 X 3 is a complete factorization of 24.
Every counting number greater than one can be factored into primes in only one way except for the order of the factors. This is called the unique factorization property of the counting numbers or the fundamental theorem of arithmetic.
When there is a set of numbers all of whose elements have factor 2, these numbers are called even numbers. Odd numbers do not have the factor 2.
Have the pupils work with numbers that emphasize the following:
A product is even if one of its factors is even.

5. Finding the greatest com mon factor
6. Finding the least common multiple

The sum of two even numbers is even.

The sum of an odd number and an even number is odd.

The sum of two odd numbers is even.

All whole numbers are multiples of 1. Thus 1 is a common factor of the members of any set of whole numbers.

However, in determining common factors one seeks only those other than one.

Consider the numbers 8 and 10. Both are even numbers, and both are divisible by 2 or are multiples of 2. Because 2 is a factor of 8 and also a factor of 10, 2 is a common factor of 8 and 10.

Consider 12 and 15. Is 2 a common factor? 15 is an odd number, so 2 is not a factor of 15. Thus it is impossible for 2 to be a common factor of 12 and 15.

However, 12 and 15 are multiples of 3; therefore 3 is a common factor of 12 and 15.

Do 18 and 30 have any common factors? The set of factors for 18 is {I, 2, 3, 6, 9, 18}. The set of factors for 30 is {I, 2, 3, 5, 6, 10, 15, 3D}. Notice there are several cominon factors: 2, 3, 6.

Do 8 and 21 have any common factors? The set of factors for 8 is {I, 2, 4, 8}.
The set of factors for 21 is {I, 3, 7, 21}.

8 and 21 have no common factors other than 1. Refer again to the common factors of the numbers 18 and 30, which are 1, 2, 3, 6. 6is the largest of the common factors. Such a factor is called the greatest common factor.

Explain that the least common multiple of several numbers is the smallest number which is a multiple of all of them. To find the least common multiple (L. C. M.) of a set of numbers, such as 9, 10, and 12, express each number using its prime factors.

9 =3 X 3
1O=2X5
12 = 2 X 2 X 3

two 3's one 2, one 5 two 2's, one 3

The L. C. M. must be a superset of the set of prime factors of each number.

Thus the factors of the L. C. M. are two 2's, one 5, and two 3's, or 2 X 2 X 5 X 3 X 3, or 180.

Find the L. C. M. of other sets of numbers.

13

7, Divisibility (base ten)

From the study of the Sieve of Eratosthenes we found that in the decimal system, a number is even if the last digit is one of the following numerals: 0, 2, 4, 6, 8. If its last digit is not one of these, it is odd.
It is easy to tell whether a number is divisible by 2, 3, 5, or certain other small numbers without having to carry out the division:
A number is divisible by 2 if it is even.
A number is divisible by 5 if its ones digit is 0 or 5.
A number is divisible by 10 if its units digit is O.
These rules were derived by considering the structure of a base ten numeral.
I m 1326 = 11 X 1000 + 3 X 100 + 2 X 10 +
Since 10 is divisible by 2, 5, and 10, so are 1000 and 100. All parts of 1326 are divisible by 2, 5, and 10 except the number of units, and this will be true with every number, not just the number 1326 used above. Thus if the units digit is divisible by 2, so is the number, and conversely. This is true in the case of 5 and 10.
A number is divisible by 4 if and only if its last two digits (tens, ones) form a number divisible by 4. The hundreds and thousands are always divisible by 4.
I I I 1326 = 11 X 1000 + 3 X 100 + 2 X 10 + 6

A number is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9. This rule depends upon the fact that every power of ten diminished by one gives a number containing all nines.

= 10 - 1 9;

100 - 1 = 99;

1000 - 1 = 999

Using the same numeral, 1326: 1326 = 1 X 1000 + 3 X 100 + 2 X 10 + 6 = 1 X (999 + 1) + 3 X (99 + 1) + 2 X (9 + 1) +6

=!IX999+3X99+2X91 +\1+3+2+6\

Since an "all nines" number is divisible by 3 and 9, the long boxed-in portion of the numeral above is divisible by 3 and 9, and this is true with every number. The remaining portion is the sum of the digits of the numeral.
The sum of the digits in the numeral 1326 is 1 + 3 + 2 + 6 = 12. Since 12 is divisible by 3 so is 1326; since 12 is not divisible by 9 neither is 1326.

IV. Operations With Fractions

A. History of fractions

Develop this section with the focus of attention upon the concept that fractions are names for numbers with properties which correspond to those of whole numbers.
Point out that man probably invented fractions when he began to measure as well as to count.

Encourage pupils to do research on the early usage of fractions.

14

B. Common fractions or the set of rational numbers 1. Meaning of fraction
2. Properties of rational num bers or fractions

Have pupils suggest measures which have come to us as a result of the use of fractions of Babylonians and Romans.
If a and b are numbers and b is not zero, then a/ b is called a fraction. If a and b are whole numbers with b not zero, the number represented by the fraction a/ b is called a rational number.
Place much emphasis upon the development of an understanding that a fraction is a name for a rational number in the same way a numeral is a name for a number and that there are many names for the same number.
Note: Whole numbers belong to the set of rational numbers in that a whole number can be expressed as the quotient of itself and one.
Example:
= 4 4/ l'
Allow pupils the opportunity of giving many examples of different names for the same number.
Examples: 4, IV, 4/ l' 8/ 2 , 12/ 3 , 2/ 5 ,
The names 4/ l' 8/ 2, 12/3, 2/ 5 are fractions which are names for the same number. These fractions are equivalent fractions.

a. Properties of whole num bers hold for rational numbers (1) Closure
(2) Commutative pro perty
(3) Associative property
(4) Distributive property
(5) Identity elements
b. Fundamental properties of fractions

Review properties of whole numbers and show by examples that closure, commutative, associative, and identity properties of addition and multiplication, and the distributive property of multiplication over addition hold for all rational numbers as well as for whole numbers.

Examples:

Closure with respect to addition and multiplication:

+ = + = 3/7

2/ 7

= = 1/2 1/2

5/ 7; 3/ 8

2/ 8

5/ 8

1/4 ; 5/ 1 4/ 1

2/ 1

Commutative property of addition and multiplication:
+ = + = 2/ 3 1/ 3 1/ 3 2/ 3 3/3 = = 2/ 5 3/ 5 3/ 5 2/ 5 6/ 25
+ + 1/4 (2/ 4 5/ 4) = + + (1/ 4 2/4) 5/ 4 = 8/ 4
= 5.(1/ 5 .1/ 5) = (5 1/ 5) 1/ 5 1/ 5

Distributive property of multiplication over addition: 43%=4(3+%)
= = =4-3+4-%
12 + 1 13

Identity for addition:
+ = + = 6/ 7 0 0 6/ 7 6/ 7

Identity for multiplication:

= = 1 7/ 8

7/ 8 - 1

7/ 8

Note: It is well to teach pupils to use symbols of multiplication and division which are generally used in science and mathematics. Thus 3 X 4 is written 3 4 or 3(4); a X b is written a b or ab; 7 -7- 3 is written 7/ 3,

Using the properties given above develop this property. It is not expected that all pupils will be able to reproduce the following development, but they should be able to see how it leads to the procedure they have used previously.

15

(1) H both terms of a fraction are muitiplied by the same counting number, the number represented is not changed

Example: Show that 3 3 X 3

= Let x 3/..

4 3X4

Then 4X = 3

= 4x and 3 are names for the same number.
:. 3. (~) 3 3

By the associative property
= (3 4) x 3 3

12x= 9
= x 9/ 12 = But x 3/,
And 9 _ 3X3

12-3X'

3

9 3X3

. '4= 12 or 3 X4

Note: "x" is used to replace such symbols as 0, ?, A, etc., in a number sentence.
= Example: 4. ? 16 4 0 = 16 or 4x= 16

The property of one may also be used effectively in developing this property of fractions.

Example:

33

3 3 3X3 9

"4=4 X 1 = 403=4X3=12

(2) H both terms of a fraction are divided by the same counting number, the number represented is not changed.

Use an example to show this property:
= Example Show 9/ 15 3/ 5 9 3.3 15 = 3":5 (Factoring 9 and 15)
By property developed above
3 3.3
5' -'"3:"5
9 3
15-5'

Note: The fraction 3/ 5 is said to be in simplest form because the numerator and the denominator have no common factor except one.

Give ample practice in changing fractions to equivalent fractions using these properties.

c. Using Ii number line to show properties of fractions.

Review the use of a number line in showing how whole numbers are related. Now expand its use to include fractions, showing that fractions are between whole numbers or are other names for whole numbers.

<o - - - - - - -1- - - - - - - -2- - - - - - - - -3 - - - - - - - -4 - - - -
, , 1/ 2/ 3/, 4./,

Use the number line to show relationships such a.s:
= = 8.1/4 == 1/, + 1/4 + 1/4 + 1/4 + 1/, + 1/4 + 1/, + 1/, = 11'/,.=2
15/, 12/, +3/4 3 + 3/4
+ Have a student note that 3 3/4 is usually written 33/,.
A number line may be used to show that both terms of a fraction may be multiplied or divided by a counting number without changing the number it represents.

16

3. Reciprocals
4. Multiplication of rational numbers a. Multiplying two rational numbers written as a whole number and a fraction

"1

2

o

""3

'3

1

<'

.>

o 1.2 3 4 5 6 7 8 9

999'999999

UiC the number line to show equivalent fractions such as:

= = = = = 1/3

3/9 ; 2/3

6/ 9;

3/3

1; 9/ 9

1; 9/ 9

3/3'

Use a number line to show products such as:
= = = 2 1/2 1; 3 1/3 1; 5 1/ 5 1, etc.

Thus

lh't-lh

0

1

2

3

r--A---..~

<

>

1/2 2/2 3/2 4/ 2 5/2 6/2 7/2

= = = + 1/2 1/2 2/ 2 1, or 2 1/ 2 1

Have students recall the definition of a rational number and show how
products such as 3 1/3 = 1 are related to the definition.

If a and b are whole numbers and b "1= 0, and x is a number which when multiplied by b equals a,

then and

bx x

==alab

1 1 and 3 are whole numbers and x is a number which when multiplied by 3 equals I,
= then 3x 1 = and x 1/ 3
1/3 is a rational number by which 3 may be multiplied to obtain 1.
Consider the sentence
= bx 1 (b is a counting number) Then x = '/. and b '/. = 1

'/. is called the reciprocal of band b is the reciprocal of'/ .

Definition: If the product of two numbers is one, the numbers are called the reciprocals of each other.

The understanding of this definition should be given much emphasis.

Have students give pairs of reciprocals such as: 1/4 , 4; 5, 1/5,

Show by using properties of rational numbers that the reciprocal of 4/ 5 is 5/ 4,

Follow this procedure:

= 4/ 5 5 = 4;
4/ 5 (5 1/4

4 1/4 = 1
(4/5 5) 1/4,

using the associative property.
= = (4/ 5 5) 1/4 4 1/4 1

This shows that (5. 1/4 ) multiplied by 4/ 5 is 1.

= But 5. 1/4 5/4

= ... 4/ 5 .5/ 4

1

Exercises of this kind will lead to the conclusion that reciprocal of the rational number a /. is the rational number / a when a "1= 0 and b "1= O.

Pupils at this level shoud extend their understanding of the processes in volving rational numbers or fractional numerals.
= = Review multiplication of these types 4 1/4 1 and 4 X 3/4 3.

17

A number line may be used to show these ideas.
= Show 4 X 3/4 3 in this manner:

1
o

2

3

%

1/4 , 2/ 4 3/ 4 4/ 4 5/ 4 6/ 4 7/ 4 8/ 4 9/ 4 10/4 11/4 12/4 13/4
40 3/4 = 3/4 + 3/4 + 3/4 + 3/4 = 12/4 = 3
That is, 40 3/ 4 = 3
The multiplications may be shown with diagrams also.

Generalize: If a and b are whole numbers where b is not zero:
b 0 '/b = 1 and b 0a/b = a = Note: b 0 '/ b 1 shows the property of one pertaining to reciprocals.

Find the product of two numbers such as 3/5 X 4 in a manner like this:

Let x = 3/5 0 4

Since x and 3/504 are names for the same number,

= 5 0 x

5 (3/ 5 0 4)

Using the associative property of multiplication,
= 5x (50 3/ 5) 0 4 = = 5x 3 0 4 12 and x = 3.4 = 12
'-5- 5

= then 3

3.4

'5 04 Ii

Generalize: If a, b, and c are whole numbers and b ~ 0,

a coa C - b b =-~

a

a0 c

l)0 c ='-b-

(By commutative property)

b. Multiplying two rational numbers written as fractions

In a manner similar to that used above find the product of two numbers, such

= as 3/ 5 0 4/ 7 Let x

3/ 5 0 4/ 7

Since x and 3/5 0 4/ 7 are names for the same number
= 5x 5(3/5 04/7)
= 5x (50 3/5) 0 4/ 7 (by associative property)
= and 5x 3 0 4/7 = or 5x 3.4/ 7 Since 5x and 3.4/ 7 are names for the same number, = 7(5x) 7 (3.4/ 7) = and 7(5x) 3 0 4 = (7 0 5)x 3 0 4 (By associative property)

Then (7 0 5)x and 3 0 4 are names for the same number and 3,4 - 12
X=7:ij-35'
= x ~ (By commutative property) 5.7
... 3/50 4/7 =53:".47

This example shows that the product of two rational numbers written as fractions is a fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators.
= If the rational numbers are a / band e / d, then a / b 0 e / d ae/ bd.

18

c. Multiplying two rational numbers, one or both of which are written as "mixed" numerals
5. Division of rational numbers a. By the use of reciprocals
b. By the common denominator method

Consider examples which will give more meaning to the procedure of multiplication involving numerals commonly called mixed numbers.

Example:

5/s (2 1/2)

Before multiplying, change 2 1/ 2 to a single fraction.

= + Thus 2 1/ 2 2 1/2

+ = 4/2

1/2

= 5/2

= Then 5/s 5/2 25/ 16

Example:

4o3lf4

= + Since 31f4 3 1f4 = + 4 3lf4 4 3 4 1f4

Using the distributive property of multiplication over addition,
= = 4 (31f4) 4 3 + 4 If4 12 + 1 = 13
".431f4 = 13

In order that pupils may understand division of. rational numbers with the

use of reciprocals, review the properties of one and the effect of multiplica-

tion of a fraction by its reciprocal.

.

2
Consider 2/ 3 -;- 7/:;. Show that this may be written 3" .
-7-
"5

(Be sure the bar between the fractions is longer than the other two bars.)

Ask: By what number must you multiply 7/ 5 to get 1? Multiply both the numerator and denominator by that number.

2
3 Then
7
"5

2

5

3 X '1

7

5

5 X '1

2

5

3 X7

2

5

10

"3 X 7 - 21

Give sufficient practice in this method.

Point out that the "invert the divisor and multiply" rule is but 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 represent different numbers.

Consider the division of two numbers represented by like fractions, such as
5 3
8' -=- '8

Using the reciprocal method.

5

3

5

S

40

5

8 -:- s=-s-X 3="24= 3

Note that the quotient could be obtained by simply disregarding the denominators and dividing the first numerator by the second numerator.
In case of unlike fractions first change the fraction to fractions which have a common denominator and then divide the first numerator by the second.

19

6. Addition and subtraction of rational numbers a. Addition of two numbers whose fractions have the same denominator
b. Addition of two fractions having unlike denominators

Example:

== 51/ 3 ...;- 3/4

16/ 3 ...;- 3/ 4 64/ 12 ...;- 9/ 12

= 64/ 9 or 71/ 9

Consider an example such as 3/ 4 + 5/4

3
4=

3

0

41-=--1r+

1
4

-

+

1
4"

5

111111

4 - 5 0 4"=4+4+4+4+4"

3
:. 4 +

5
4" =

r
[4

Il
+-"4

l +

)
4J

r1
+ l4

1. 1

1

+ 4" + 4" +4

+

I)
4J

8
=4

+ = + Or 3/4

5/4

3 0 1/4

50 1/4

Using the distributive property of multiplication over addition,

= 3 0 1/4 + 50 1/4 (3 + 5)1/4

= = 8 01/ 4

8/4

+ = = and 3/4

5/4

8 0 1/4

8/4

Generalize in this way: If a, b, and c are whole numbers and b "1'= 0, "/. + C/ d

may be written a 0 1/. +c 0 1/.

Using the distributive property,
a 0 1/. + c 0 1/. = (a + c) 1/. or a + c.
-b-

This may be stated: The sum of two numbers whose fractions have the same denominator is the sum of the numerators divided by the common denominator.
+ Consider an example such as 3/Ii 2/3,
The least common multiple of the denominators is 15 and is called the lowest common denominator.

c. Addition of rational numbers which are "mixed" numbers '

= 3
Then"5

3X 3 3X5 -

9 15

= = 2 5 X 2 10
And3 5X3 15

3 2 9+10 19
Thens +3 = 15= 15

Give sufficient practice with examples of this type.

Generalize thus: If a, b, c, and d are whole numbers and band d are not zero,
=....,., + t. b
then "/. C/ d

Consider 34 as 3 + 4 and 2% as 2 + 0/8
+ Then find the sum of 31/ 4 23/ g
3 1/ 4 + 2 3/ g = (3 + 1/4) + (2 + 3/ g) = (3 + 2) + (1/4 + 3/g) = (3 + 2) + (2/ g + 3/ g)
= 5 + 5/g = 5 5/ g

Show that the sum could be found as follows:

== + 31/ 4 23/ g

13/4 + 19/9

26 26

/

g + 19/ + 19

9

g

45/g or 5 5/ g

20

I

1\

7. Subtraction of two fractions having the same denominator
8. Subtraction of two fractions having unlike denominators
C. Decimal fractions as. an extension of rational numbers 1. Decimal notation

= Since subtraction is the inverse of addition, show that if 3/4 + 5/4
then 8/4 - 5/4 3/4,

In symbols this may be expressed:

c/b_"/b=c-" when b =F 0 and c > a.

'-b-

-

Show that the fractions may be written as equivalent fractions with a <;om mon denominator and the same procedure as in 4 may be followed.

Thus

5/ 7 -

= 2/ 5

(5/ 7 .5/5) - (2/5 .7/7)

= 25/35 - 14/ 35

25 _ 14

=-3-5-

= 11/ 35

This may also be written in symbols in this way:

~ - 1> .= (a g) (t ~1

cb ad
bd - bd
cb - ad bd
if cb.2 ad and band d are not zero.

Give sufficient practice for pupils to be able to handle subtraction of various types of numbers involving fractions.

Show the following procedure in subtracting numbers of similar form:

Example: 7 1/5 - 3 3/ 4
= + = + + 7 1/ 5 7 1/ 5 (6 5/5) 1/5

= + + = + Using the associative property of addition 7 1/ 5 6 (5/ 5 1/ 5) 6 6/ 5

=r + r + Hence 7 1/ 5 - 3 3/ 4

(6

6/ 6

54)1-

(3

51 3/ 4)
3.

+ + = ,-6 5:4" J. L3 4:5.J

+ + (6 24/20) - (3 15/ 20) + = (6 - 3) (24/ 20 - 15/ 20)

== 3+ 9/ 20 3 9/ 20

Use of the horizontal form may be new to pupils. Encourage its use because

of the need for it in a later study of mathematics. However, some students may

understand more readily the algorism they have used in previous years. Thus

= = + 71/ 5

+ 6

6/ 5

6

24/ 20

= + = + -33/ 4 3 3/4 3 15/20

+ = 3

9/ 20

3 9/ 20

Review decimal notation of whole numbers and show the expanded form. Stress the meaning of place value.
Show that the expanded form in base ten shows that the value of each place immediately to the left of a given place is ten times the value of the given place and each place immediately to the right of a given place is one-tenth of the place value of the given place. Suggest that this pattern be continued in the notation of a number, such as 6732.58746.
= + + + + + + 6732.58746 6(103) 7(102) 3(101) 2(100) 5(1/10) 8 (1/102)
+ + 7(1/103) 4(1/10') 6(1/10')
Have pupils make a place value chart which gives the place name in words, in decimal notation, and as a power of ten.
Permit students to make up numbers and have some of these placed on the chalkboard for classmates to read. Those who make up the numbers should have them in written form.

21

2. Addition of numbers in decimal form
3. Subtraction of numbers in decimal form

To give more meaning to the decimal notation, illustrate changing rational numbers to other bases.
Examples: 1. Write 1/ 3 as a duodecimal. Since there are 12 twelfths in one unit, in 1/ 3 of a unit there are 12/3 or 4 twelfths, with none left over. Then 1/3 = 4/ 12 = 0.4twe,ve or 0:4 (In the duodecimal system the colon is used by some mathematicians instead of a decimal point.) 2. Write 5/ 9 as a duodecimal. There are 12 twelfths in one unit and 60 twelfths in 5 units. Then in 5/ 9 of a unit there are 60/9 twelfths, or 6 6/ 9 twelfths. The twelfths digit is 6 and there is 6/ 9 of a twelfth left over. There are 12 one hundredforty-fourths in one twelfth. Then in 6/ 9 of a twelfth there are 72/ 9 or 8 one hundred-forty-fourths, with none left over. . '. 5/ 9 = 0.68twelve
Note: Another method of changing is shorter but does not give as much emphasis to place value.
Example: 5/ 9 = 80/ 144 = 68/ 100 twelve
= 0.68twe,ve

Give students practice in changing expanded notations to decimal form and decimal forms to expanded notation.
Make students aware of the lise of the distributive property here.
Example: Add 0.46 + 0.245

0.46= 46 X 1/100 = 360 X 1/ 1000
0.245 = 245 X 1/1000
+ ... 046 0.245 = (460 X 1/1000) X (245 X 1/1000) + = (460 245) X 1/1000

= 705 X 1/ 1000 = .705
This method of addition is meaningful but not convenient. Therefore students should place one number below the other as they have been accustomed to doing.

Call attention to the' placement of the decimal points directly beneath each

other. digits

The in th

purpose e 1/ '00 p

of lac

this is e will

th be

at in

digits in the 1/ a column, etc.

'

0

place

will

be

in

a

column,

Example: Add 0.64 + 0.53

+ 0.64 = 6/ 10 4/ 100 and

+ 0.53 = 5/ 10 3/100

+ + 6+5 4+3
.'.0.64 0.53 = 1 0 "100

+ 11/ 10

7/ 100

+ + 10/ 10

1/10

7/ 100

= + + 1

1/ 10

7/ 100

= 1.17

Handle subtraction in the same way by placing numbers one beneath the other with decimal points one under the other. Explanations similar to that above may be used.

Consider a subtraction of this type: 1.4 - 0.53.

Note:
+ + 1.4 = 1 4/ 10 = 1 40/ 100 = 140

II'
22

4. Multiplication of numbers in decimal form .
5. Division of numbers in decimal form
6. Changing the form of a rational number. a. Writing common fractions in decimal form

Then
= 1.4 = 1.40
-0.53 0.53
--:87
Show this operation using the associative property.
Example:
= 0.4 X 0.45 (4 X 1/10) X 45 X 1/100 = (4 X 45) X 1/10 X 1/100 = 180 X 1/1000 = .180
Lead pupils to discover or recall that the number of decimal places in the product is the sum of the decimal places in the two numerals.

Consider an example such as 0.128 -:- 0.4

This may be expressed as 0.128
0.4

Find another fraction whose denominator is a whole number and which is equivalent to the first fraction.

0.128 0.128 X 10 1.28
OA =0:4X1O=~
= Then 1.28/ 4 1/ 4 X 1.28 = 1/4 X (1/ 100 X 128) = 1/ 100 X (1/4 X 128) = = 1/100 X 32 0.32
Then show that the usual form for division will give the same result and is much shorter.

Note:

(divisor) X (quotient) = (dividend)

4x

.32

1.28

This generalization may be made: Whenever the divisor is a whole number, the dividend and the quotient have the same number of decimal places.

Encourage students to estimate the answer. This will often prevent an error in placing the decimal point.

Have students review the meaning of fraction.
Then ask: if % means 1 -:- 4, what is the result if the division is performed? Then stress that .25 is the decimal equivalent of %.
Use examples which show that all such divisions do not come out exactly.
= Show that 3/ 7 0.428571 ...
Have the students discover what happens if the division is continued, thus developing the idea of a decimal numeral that never ends. Note that three dots are used to indicate this.
= Show that divisions such as 1 -:- 4 can be expressed in the same way.
Thus % 0.2500 -...
Use enough examples to enable students to conclude that any rational number can be expressed in decimal form which either repeats as a single digit or a block of digits over and over.
If the fraction has a denominator which is a factor of ten, or powers of ten, the property of one furnishes an easy way to write decimal equivalents.

23

b. Writing fractions in com mon fraction form
7. Rounding decimals
V. Relations
A. Equivalence 1. Equivalent fractions
2. Rational numbers in decimal form

Example: 3 25 X 3
--4-= 25 X 4

75 100

=

75

Review the meaning of place value in a number in decimal form. Then show
that a decimal may be written as a common fraction with a power of ten as a denominator. -

Consider a number in decimal form, such as 0.036, and ask how it could be expressed to the nearest hundredth.

Use a number line to show the principles involved.

I . I I

.03

.035

.04

.045

I
.05

Locate 0.036 on the number line. Ask: Is it nearer 0.03 or 0.04? Then 0.036 to the nearest hundredth would be written how?

Use other examples, including the rounding of .045 to the nearest hundredth. Since 0.045 is halfway between 0.04 and 0.05 a decision needs to be made as to whether 0.04 shall be accepted or 0.05. A recommended procedure is to choose the decimal whose last digit is even. If only one number is involved, it makes little difference which is chosen. However, if several numbers are involved, the suggested procedure would be preferable.

Other number lines may be needed to show rounding to other decimal places.

Review the idea that there are many names for the same rational number. Example: 2 f 1 0 , 3/ 15, 6/ 30
Review the basic properties of fractions; that both terms of a fraction may be multiplied or divided by the same counting number without changing the number represented. (Property of one)
These fractions are members of a set containing an infinite number of elements (equal fractions) and this set is called an equivalence class.
Recall that any rational number may be expressed as a common fraction or a decimal. The decimal and the common fraction which represent the same number are equivalent.
Have pupils make a chart which will include equivalents for .which they will have frequent use. A column for per cents may be included to be filled in after a study of per cents.
These headings may be helpful:

Fraction Simplest Form
1 2 1 3
etc.

Hundred as Denominator
50 100 33 1/3 100

Decimal 0.50
0.333 ...

Per Cent 50 %
331/,.%

B. Inequalities

Use a number line to show inequalities

< .....

>

01234 567

Recall that numbers on this scale are arranged in order of increasing value from left to right.

24

C. Ordering 1. Whole numbers
2. Ordering' fractions
D. Ratio 1. Meaning

Review symbols for inequality. Give examples and have pupils make others.
For example: 8>6 is read "8 is greater than 6." 7<12 is read "7 is less than 12." 5#4 is read "5 is not equal to 4."
Now, using the number line have pupils make statements about the numbers on the line. Have pupils make mathematical sentences and have others tell if they are true or false.
Examples: 5+4>7 20 <j: 5
6 + 4 # 12 - 2
Use the number line to show:
The set of whole numbers is ordered.
There is a smallest whole number (zero), but there is no largest whole number.
Every member of the set has an immediate successor.
Thre is not always a whole number between two whole numbers.
A number is less than a second number if the first is to the left of the second.
The number line may be extended in both directions, but the numbers to the left of zero are not whole numbers. (Note: pupils can be shown that there .are numbers to the left and these are the negative numbers.)
Review the two methods of determining whether two fractions represent the same number.
Using the property of one, find two fractions with equal denominators which represent the given numbers. lf the numerators are the same the given fractions are names for the same number. lf the numerators are not the same, the fraction with the larger numerator represents the larger number.
Convert both fractions into decimal form and compare results.
A number line may be used to illustrate this relation unless the number of intervals required is too great to show on the line.
lf the two fractions do not represent the same number, it is important to know which is greater. On the number line the fraction representing the larger number lies to the right of the other.
Ask pupils how they can compare two numbers? Point out that the comparison by division has a special name, ratio.
Definition: The ratio of a number c to a number d, d # 0, is c/d. Ratios may also be written as c:d, and (c,d).
Ask: When is cld a rational number?
In set language the definition becomes: a ratio is any element of a set of equivalent ordered pairs of the form alb where a and b are whole numbers and b#O.
Pupils will need an explanation of the meaning of ordered pair. 51 6 is an ordered pair of natural numbers and could be written (5,6). The ordered pair (6,5) is different from (5,6).
Most textbooks limit the meaning of ratio to the comparison by division of two numbers expressed in the same unit of measure.
Explain that ratio may express a comparison by division of two numbers which are measures of different kinds of quantities. In such cases "per" indicates division.
Examples: miles per hour, cents per dozen, dollars per 100
25

2. Usage E. Proportion

Help pupils to see that the second quantity represents the standard of comparison.

Lead pupils into discussion of experiences and problems using ratios.

Examples:

Let some students measure their heights and then measure the lengths of their shadows. Have each measurement made several times for accuracy. Construct a table showing results. Form ratios of shadow to height for each student.

Other students who earn money at home by doing chores may be paid at a certain rate per hour. Suggest that several prepare tables showing their earnings of the week. Show the ratios of amounts earned to number of hours worked.

Class discussion will show the equality of ratios in each table.

Refer to the ratio table showing the measure of heights and shadows.

Help the class to understand that in a case like this the physical quantities measured by the numbers are proportional to one another. Then form the definition of a proportion: A proportion is a statement of equality of two ratios.

Use many examples which show equality of two ratios.

<X>nsider:
= a. 4/ 5
c. 5/ s

= b. 2/ 7 = d. 31 6

12/ 42 ; 21 / 42

Note:

= = In (a) 4 X 15 69 and 5 X 12 60 = Therefore, 4 X 15 5 X 12

= = In (b) 2 X 42 84 and 7 X 12 84 = Therefore 2 X 42 7 X 12

= = In (c) 5 X 24 120 and 8 X 15 120 = Therefore 5 X 24 8 X 15

Have class find if the same relation holds in (d).

Have them try examples of other pairs of rational numbers which name the same number and se of the relation holds.

Consider this generalization:
= If alb cld and b =1= 0 and c =1= 0,
then, a d _ c b (Using property of one)
bOd-dOh

and ad cb bd - db

ad bc bd -b([

Therefore ad = bc

Note: If two fractions with a common denominator are names for the same number, their numerators are equal.

Call attention to the property developed above and to its usefulness in solving problems dealing with proportions.

VI. Per Cent And Its Application

A. Understanding the meaning of Ask pupils if they have received "grades" of 80, 90, etc., on test paper. If so,

per cent

what did the grade mean?

Explain that if the paper had 100 answers and 90 of them were correct, the ratio 90/100 or 90% could be used to express the rating of the paper.

26

B. Problems of per cent 1. Per cent as a ratio

Explain that per cent, or percent, comes from the Latin phrase per centum which means "by the hundred."
The symbol % is used for per cent and is a short way of saying 1/100' The
=per cent form of a fraction is readily interpreted as a ratio. For example, 90% 90 per 100; 90 compared to 100.
Ask the class to express the ratio of the number of games won to the number of games lost in a certain sport: 6 out of 10. Discuss other ways this may be written: 6/ 10, .6.
Ask how many games they would have won if they had played and won twice as many; five times as many. Write the above facts in four ways:
60 out of 100
6/ 100, as a common fraction .60, as a decimal fraction 60%, as a per cent.
Use cross-ruled paper and represent given per cents.
Develop the understanding of per cents of more than 100.
Show pupils the meaning of per cents less than 1%. Lead them to see % % is not the same as % but is % of 1%.
Lead pupils to discover that since fractions represent a comparison of two numbers, % means one out of every four, or 25 out of every 100, or 25%.
4/4 means one whole, or 100 out of 100, or 100%.
Do similar exercises with other fractions.
Use the number line to help students discover s is % of V4 or % of 25% or 12% %.
Develop the per cent equivalents:
1/4 to 4/4; l/ S to 8/ s; 1/ 3 to 3/ 3 ; 1/ 6 to 6/ 6 , From experiences of students provide problem situations which use per cents.
These may include: Per cent of pupils in the seventh grade who are boys; who are girls; who are present; who are absent. Per cent of lunch money collected. Per cent of library books read; not read.
Ask pupils to translate problems into ratios, thus showing the relationships in a rate or comparison situation.
Ask the pupils to translate sentences, such as "How many books out of each 100 books in the library are checked out today?" into sentences containing the language of per cent, such as "What per cent of the books in the library are checked out today?"; and "What per cent of your allowance do you save?" into "How many cents do you save out or" each 100 cents of your allowance?"
Wh~n pupils understand how to express per cents as ratios, they are ready to solve problems of per cent. The method is no different from the method used to solve ratio equations. The solution depends on finding the missing term in an equation of equal ratios, one of which has 100 as its second term.
Problem:
500 pupils are enrolled in the Elm Street School. 350 of the pupils usually walk to school. The number of pupils who usually walk to school is what per cent of all the pupils?
Number of pupils who walk is 350; number of pupils enrolled is 500. 350 (number of pupils who walk) 500 (number of pupils who enrolled)
27

To find the per cent of pupils who walk to school you need another ratio. This ratio must express a comparison for 100 pupils that is the same as the com parison of 350 pupils with 500 pupils. You do not know how many pupils walk for each 100 enrolled, but you can use a letter in your ratio to hold a place for the numeral that tells how many: n/ 100' n is the number of pupils who walk, and 100 is the number of pupils enrolled.

The two ratios are equal because they make the same comparison. That is,

350 n . 500 - 100

Divide 500 by 5 to get 100 and 350 by 5 to find the numeral to replace n. 350
divided by 5 is 70. 70 replaces n. The comparison of 350 pupils to 500 pupils
is the same as 70 pupils to 100 pupils. 70/ 100 can be expressed as 70%. The 350 pupils who walk to school are 70% of the 500 pupils enrolled in the school.

Problem:

One day 12% of the 25 in a class of seventh graders enrolled in Elm Street School were absent. How many seventh graders were absent that day?

12% means 12/ 100
12 (number of seventh graders absent) 100 (number of seventh graders enrolled)
The other ratio compares the number of seventh graders really absent with the number really enrolled.

n (number of seventh graders absent) . 25 (number of seventh graders enrolled)
12 n 100-"25

Divide 100 by 4 to get 25, and divide 12 by 4 to get the numeral to replace n. 3 replaces n. 3 pupils were absent.
I

Problem:

I One day 95% of the pupils in the fifth, sixth, and seventh grades went to

a program in the auditorium. 190 pupils went to the program. How many

pupils are enrolled in these grades?

II

95% means 95/ 100

95 (number who went to the program)

100 (number of pupils enrolled)

It is clear that 190 pupils went to the program. So the other ratio will compare these 190 with the number really enrolled.

190 (number who went to the program) n (number of pupils enrolled)

The two ratios are equal because they make the same comparison. Therefore they form the proportion:

95

190

100

n

Multiply 95 by 2 to get 190, and multiply 100 by 2 to get the numeral to replace n. 200 replaces n. 200 pupils are enrolled in the three grades.
= = Review the principle of proportion that if a : b c: d, then ad be. (De-
veloped in V, E, of this guide) Then use this principle in finding the missing term of a ratio in problems involving per cents.

The illustrations which follow may be used to help pupils understand the use of proportion in solving percentage problems.

28

2. Base, rate, percentage
C. Applications 1. Commission
2. Interest 3. Discount 4. Increase and decrease

Problem:

Finding a per cent of a number:

75% of 48 is what number?

75 n
100-48

100 n 3600 n = 3600 100 n = 36
Problem:

Finding what per cent one number is of another: 36 is what per cent of 48?

36 n
48 100

48 n n n

3600 3600 -7- 48 75

75

75</0

100

Problem:

Finding a number when a per cent of it is known: 3 is 20% of what number?

20 100

=n3

20n 300

n

300 --;- 20

n 15

Develop an understanding of the three basic numbers of the per cent relationship with their names: percentage, rate, and base.
= = = Examine and discuss the formulas p br; r p/b; b p/r. Show that r is
the ratio of a number to 100; p and b are the terms of an equivalent ratio.

Discuss the commissions made on selling greeting cards, candy, magazines, and other items, as club projects. Relate commissions to occupations in which people work on a commission basis and those on a salary plus commission basis. Solve problems in computing commissions and net proceeds.
= Develop the formula i p X r X t. Use the formula to find the amount of
interest paid on various loans.

Have pup~ls make bulletin board displays of newspaper clippings illustrating discounts. Give practice in finding discounts and selling price.

Per cent is used to indicate an increase or a decrease in some quantity. Develop the idea that the per cent of increase (or decrease) is found by comparing the actual increase (or decrease) with the -original quantity. Use equivalent ratios fo determine the per cent.

Example:

The population of a certain town was 24,000 in 1950. If the population increased to 32,000 in 1960, what was the per cent of increase in popula-
tion?

32,000 -24,000
8,000

actual increase

29

----------~ -

If n% represents the per cent of increase, then
8,000 _ n 24,000 - 100
24,ooOn = 100 X 8,000
n 100 X 8,000 24,000
= n 331/ 3
There was an increase of 331/ 3 %.

VII. Measurement
A. Counting and measuring

Ask questions in such a manner that pupils see that while answers to questions such as "How far?" or "How many?" involve numbers, some such answers are found by counting and others are found by measuring.
How many pupils? (counting)
How long is the stick? (measuring
Do reasearch on the origin of some ancient units of measure.
Emphasize that all measures are comparisons and all measures are approximate and involve some error.
How tall is the tree? (comparison)
Give practice in estimating lengths and distances. Help pupils develop a mental picture of things of various sizes which may be used in making comparisons.
Show that sizes of line segments and of simple closed curves can be compared without actually measuring them.
Example: Compare WX with YZ

W

x
Y

z

Compare the size of closed region X with closed region Y by cutting out a copy of closed region X and placing it over closed region Y.

X

Y

I

Study the ruler. Notice the lines that mark the different lengths. Which are longest? Which are shortest? What lengths are measured between these two? Classify as to exact or approximate such amounts as the following:

8 inches

9 oranges

2 dozen

3 ounces

5 miles

6 pounds

(8 inches is approximate; 9 oranges is exact; etc.)

30

To add strength to the pupils' ability to measure with a ruler, give problems in addition, subtraction, and division and require that the computations be done using the ruler.

Example: Draw a line segment. Mark off 5/ 16" three times. Measure the total length marked off.

(1)

(2)

(3)

B. Standardization of units of measure
C. The English American system

Check the work without the use of the ruler:

= (3 X 5/ 16

15/ 16)

Ask .pupils to look up the story of Archimedes, the Greek mathematician, and of King Kiero of Syracuse and to report on the new method of measuring which was discovered then

Emphasize criteria for a standard unit of measure as depending on two things:

Everybody knows what it is. Everybody agrees to use it in making certain kinds of measurements.

Give demonstrations of linear measurements (Call attention to the fact that all measurement of line segments is linear measurement.)

Prepare with pupils a list of of units of measures used by ancient man. It would be interesting to show the lengths indicated by these measures.

Unit
Cubit Span Fathom Barleycorn

Length
Length of a man's forearm Half of a cubit Distance across a man's outstretched arms Three in a row and round and dry equals an inch

Prepare a list similar to the one described in the previous suggestion. Use units of measure common today but not standard. Such a list might include items such as:

a handful a mite a pinch a city block

a country mile a sackful quick as a flash a little bit

Teach appreciation for the standard units of measure now in common usage. Ask pupils to use length of forefinger to measure lengths of various objects such as tablets, pencils or crayons. Compare answers and note the many differences. Vary the exercise using the foot or the arm as units of measure.

Add interest to the study of non-standard units of measure by using pupil-made charts and posters. An interesting one might be of the Ancients "Rule of Thumb" showing the following:

Picture of

Representing Measurement of

Thumb Foot Three feet Both arms outstretched

1 inch 1 foot 3 ft. or 1 yd.
2 yds. or 1 fathom

Make a list of some standard units used for measuring distance, time, and weight.

Example:

Distance Time Weight

mile, yard second, hour ton, ounce

31

There are many names for the same length. Example:
= = 3 feet 36 inches 1 yard
Help pupils to become proficient in the changing of a name of a measurement from one form to another. Using a ruler marked in 16th of an inch measure the closed curves and add all of measures together:
b
LJ
Measure lengths of each closed curve below. Can this be done with a ruler?
b
a

D. Precision and error in measurement

Lead pupils to see that they will use number systems with many different

bases in regrouping smaller units of measure in changing larger units. Show

examples, such as:

6

17

Ti ft. '/J in.

(using base of 12)

3 ft. 9 in.
.3ft:- ~

2 pt. = 1 qt. (base of 2)
= 3 ft. 1 yd. (base of 3) = 7 da. 1 wk. (base of 7)
30 da. = 1 mo. (base of 30)

Help children to understand that "precision" and "accuracy," when referring to measurement, do not mea~ the same thing.

Prepare exercises similar to the one below.

Example: Give the precision of each measurement:

12 miles (precise to the nearest mile) 12.0 miles (precise to the nearest tenth of a mile)

Bring out the fact that no matter how precise a measurement is made, there is always a difference between the true measur of th object and our measure. In general, the greatest possible error is one-half of the smallest marked interval on the scale being used.

32

E. The metric system.

Example:

Measurement made to nearest
1/ 2 inch 1/4 inch

Greatest Possible Error
1/4 inch 1/8 inch

Have pupils measure lengths and widths of various figures to (1) nearest 1/ 2";
(2) the nearest 1/ 8 "; (3) the nearest 1/ 16," Give greatest possible error. Chart findings.

Ask pupils to bring to class measuring devices used in business or professions.

Display pupil-made charts which all pupils might use in answering questions concerning tolerance. Point out that an allowance for error in measurement is called tolerance, and that states have set tolerances for weights and liquid measure in order that customers might be protected.

Pupils should know that the notation of a measurement must be written so that anyone looking at it can tell just how precise the measurement is. If a measurement is made with a ruler graduated in 16th of an inch, the answer must be expressed in 16ths. The fraction would not be changed to lowest terms.

For example, if a length is found to be 3 1/ 2 inches, it would be written 38/ 16 inches.

Since metric units are used to constantly in seventh grade science, pupils need an understanding of and skill i the use of this scale.

Help pupils to become familiar with the various uits of metric linear measurement. Display a table which shows the plan of linear metric units.

Metric Measures Length and Distance

= 1 kilometer 1000 meters = 1 hectometer 100 meters = 1 decameter 10 meters

1 meter

1 decimeter

.1 meter

1 centimeter

.01 meter

1 millimeter

.001 meter

Ask pupils to examine the chart closely. Do they understand that metric meas-
use is convenient to use because it is based on 10 just as our number system is based on 1O?

Have reports on the origin of the metric system and its world-wide acceptance.
Except for the English-speaking countries, the metric system is the official system of measurement.

Lead pupils to explore the conveniences of the metric system but postpone attempts at changing from metric to English measures or from changing from English to metric measure.

Have pupils make a 10 centimeter ruler, using cardboard and a straightedge. Use this ruler to measure line segments to the nearest centimeter.

Find the meaning of the prefix "milli." When pupils understand it to mean one-thousandth, ask them to compare the millimeter with a meter.

Example: A length 1000 times the size of the millimeter is the length of the meter, for 1 mm equals 1/10 of a centimeter and 1 cm equals 1/ 1000 of a meter.
1/10 X 1/ 100 of a meter is 1/ 1000 of a meter.
Study the divisions on a metric ruler. Using this ruler answer questions such as the following:

33

F. Measuring angles.

What is the length of the larger divisions? (em) What is the length of the smaller units? (mm) How many millimeters are there in 1 em.? (10)
in 1 decimeter? (10) How many decimeters in 1 meter? (10)

Using a metric ruler, draw a line segment 4.5 em. in length. Divide the segment into 3 smaller sections each to be 1.5 em. Give length of each segment in millimeters (15mm.)

Notice that with the exception of "meter," all names of the metric units use the word "meter" with a prefix.

Chart and display meanings of these prefixes since they are to be used to name other units of measure in the metric system.

Prefix

Meaning

KiloHectoDekaDeciCentiMilli-

1000 or 103 100 or 102 10 or 101
1/10 or 1/10' 1/100 or 1/ 10, 1/ 1000 or 1/10"

Help pupils to see the relationship of various measures of capacity and measures of weight to our number system. To aid them in this, the teacher should show that the gram and the kilogram are used for measuring weight.

1 gram equals .035 ounces 1 kilogram equals about 2.2 lbs. Gram is used for very light objects (letters, small medicinal pills) Kilogram is used for measuring heavier objects (meat and butter) 1000 grams (g.) equals 1 kilogram (kg.)

Ask children to check by using an example that to change smaller units to larger units you must divide, but to change larger units to smaller units you must multiply.
= Example: How many ounces are there in 2 pounds? (16 oz. in 1 lb. 2 X 16
32 oz.)
= = How do you change 40 inches to feet: (divide by 12) (12" 1 ft.; 40 ...;- 12
3 1/6 ft.)
Emphasize that only units of the same size may be added together.
= Example: 5 ft. 8 inches changed to feet would be 5 8/ 12 feet 5 2/ 3 ft.
Develop conversion tables with pupils and have them use these tables in converting metric measures to English measures and English measures to metric measures.

Define an angle as the union of two rays with a common endpoint.

A

~

~

Use a drawing of an angle to show that rays AB and AC are the sides of an

angle.

Use a protractor, show pupils that the standard unit for measuring angles is one hundred eighty-one rays all drawn from the same point. These rays determine 180 congruent angles, and they form a scale numbered from 0 to 180.

34

G. Measuring circles.

Practice measurement of many angles drawn in various sizes and in awkward position so that pupils may become proficient in estimating sizes and in use of protractor with its two scales.
Separate various angles into sets according to their measure.

right angle (measurement is

acute angle (measurement less
than 90)

obtuse angle (measurement more than 90 less than 180)

Define a circle as a set of points, each point of which is equidistant from a fixed central point.
Exercises:
Draw a circle and label its parts.
Draw a circle with 0 as a center. How many degrees are there in all angles about O? (me..lsure with protractors.)

H. Scale drawing.

Using a protractor, find the number of degrees between various directions. A NORTH

.J

D WEST

C EAST

B SOUTH
Through class discussion lead pupils to understand that a scale drawing has the same shape and proportions as the object represented. Scale drawing may reduce or enlarge the actual size of the subject.
Examples: 4:3 means enlargements. 4 refers to the scale drawing and 3 refers to the subject reproduced. 2.3 means reduction. 2 refers to the scale drawing and 3 refers to the subject reproduced.
Perhaps some pupils have made scale drawings or models of cars, boats, etc. which they will bring for display. Show other examples of scale drawings: maps, pictures, etc. Projects: Take road maps and estimate distances between cities; measure the classroom and make a scale drawing of it.
35

VIII. Geometry
A. Non-metric geometry 1. Points, lines and space

Do not define point, line, space; merely study their properties and get a general intuitive idea through the use of many examples.

An example of a point is the tip of a pin or a dot on a paper. We think of the geometric point as being so small it has no size.

Space is a set of points. These points have many properties.

A line is a set of points and means a straight line. A line will be draws like

this <

>.

Examples of lines: edge of table and the intersection of the ceiling and one wall of the room.

A geometric line may be extended in either direction without limit.

Have students draw figures to let them discover for themselves the following ideas about points:

There is an unlimited number of points in a line

.

An unlimited number of lines may be drawn through a point.

Two lines may intersect in one and only one point.

Only one line may be drawn through two points

_ _ _ _ _ _0 __

There are infinitely many points on this line.

Two lines which meet to form 90 angles are perpendicular lines.

A line segment has definite length.

A line may be extended or limited.

Representations of a geometric line are edge of teacher's desk, a chalk mark on the board, intersection of two walls, a crease when a piece of paper is folded, latitude and longitude.

A line segment is determined by two points on the line.

A

c

B

Point C separates the line segment AB into two segments. Point C divides the line into three sets: the points to the right of C, the points to the left of C, and C itself. When C is joined to either half line, a ray is formed.
Exercises: On a line label two points M and N. Name four half lines that have been formed. Into how many sets of points is the line divided? Name a rayon the figure.
Give an example of a set of lines on a point.

36

2. Planes
3. Rays 4. Angles 5. Polygons 6. Names and symbols for
points, lines, rays, planes and angles

Draw a line segment. Label the end points of the segment M and N. Label A, a point of the set MN. Label B, a point of the set AN. Is B a point of the set MN? Is B a point of the set MA? Label C, a point of the set MA. Name the segments in the picture.

A plane is a particular set of points in space.

Examples of planes: top of desk, floor, walls, ceiling.

The floor represents a plane that is part of a larger plane which has no boundaries. A plane is flat and extends indefinitely.

A line lies in a plane if two points on the line are in. the plane. Many planes may. contain the same two points. An example of this is a revolving door. When two walls come together the line formed may be in either plane.

Identify planes in the room such as tops of desks, wall ,windows, door, sheets of paper, posters, and chalkboard.

A lir.e lies in a particular plane if it contains two different points in the plane. Consider the sheets in a tablet as planes and where they are bound together as a line. The ends of this line are two points. These two points are in many planes.
Ask the question, "Why will a three-legged table stand solidly against the floor while a four legged table will not always do so?"

Three points are in one and only one plane if they are not in the same straight line (collinear).

If three points in space are on the same straight line they will be in many planes. However, if the three points are not in the same line, they will be found in one plane only.
Exercises:
One point may be contained by how many different lines? A pair of points may be contained by how many different lines? One point may be contained by how many different planes? A pair of points may be contained by how many different planes? A set of three points may be contained by how many different planes? Why do photographers usually use the tripod to hold the camera?

A ray is a set of points with an endpoint.- - - - - - - - >

Make student aware that a ray extends in one direction only. Distinguish between ray and line.

An angle is a set of points on two rays from a common point. These rays are called the sides of the angle.

Identify these angles:
Straight Right Acute Obtuse Complementary

A

B is called a segment. If AB A are points not all on one line

the set of points of the segments AB, BC xA is called a polygon.

A point is represented by a dot. The dot is called or named by a capital letter; thus' A.
A line is designated in three ways:

A.-----B called AB sometimes written with arrows as

<

>

AB

37

called "b" or - - - - - . l called "f." b Designate a plane by 3 letters as
Called plane ABC

Exercise:

Show how to draw and letter a point, line, line segment, ray, plane.

= A a (point)

< - 0 - _ - 0 - -

A

B

AB

> (line)

Ao

oB AB (line segment)

~

> AB (ray)

A

B

.A .C

plane ABC (plane)

Teach pupils to read angles in the following ways:

B<:

Read: "LABC" (angle ABC) or "LB" (angle B)

< b Read: "Lb" (angle b)

<2 Read: "L2"

(angle 2)

B. Working with points, lines, planes, and space
1. Angles formed by two lines in a plane passing through a common point a. Vertical angles b. Adjacent angles c. Supplementary angles

Draw two intersecting lines which will illustrate the angles to be identified. Have pupils discover the relationships of vertical angles and supplementary angles by measuring the pairs of angles.

2. Three lines in a plane a. Concurrent lines
b. Angles formed by a transversal cutting two lines

Show by illustration that concurrent lines are two or more lines which pass through a common point. Draw two lines cut by a transversal. Show non-parallel lines and parallel lines cut by a transversal. Identify the various kinds of angles formed.
38

(1) Corresponding angles
(2) Alternate interior angles
(3) Alternate exterior angles
3. Construction
a. A perpendicular bisector of a line segment
b. A bisector of an angle c. An angle equal to a
given angle.
d. Inscribed regular polysides gons of 3, 4, 6, 8 and 12 (1) Regular hexagon
(2) Equilateral triangle
(3) Square. (4) Regular octagon
(5) Regular dodoecagon
4. Plane geometric figures and their measures
a. Polygons of 3, 4, 5, 6, 7, 8, 10, 12 sides
b. Perimeter of polygons and circumference of a circle

Prepare sheets for distribution to pupils. Show on the sheets several illustrations of two lines cut by a transversal. Include one pair of parallel lines cut by a transversal. Have pupils name all types of angles and measure them. Lead pupils to discover relationships of pairs of angles formed by the transversal cutting the parallel lines.
Use details given in the textbook for constructing perpendicular bisectors of line segments, the bisector of an angle, and an angle equal to a given angle.
Encourage students to use sharp pointed hard lead pencils. Demonstrate how to use the ruler and how to hold the pencil against the ruler.
Invite the drafting teacher to visit the class and give some suggestions about construction.
Point out that the bisector of a line is also perpendicular to the line but that a perpendicular to a line is not necessarily a bisector. The concept that the converse of a true statement is not necessarily true should begin with this simple idea.
Bisect angles drawn in all directions, both acute and obtuse.
Constructing an angle equal to a given angle is a prerequisite to constructing parallel lines in the eighth grade. Review measuring angles in all positions (acute, obtuse and right). The given angles should be of all types.
Define regular polygon. When a polygon is inscribed in a circle all the ver tices are points on the circumference, and all its sides are chords of the circle.
Draw a circle using any radius.
Using the radius, divide the circumference into six equal arcs. Letter the ends of the arcs A, B, C, D, E, F. Draw chords AB, BC, CD, DE, EF, FA.
Use the same directions as above except draw chords AE, EC, CA.
Connect the ends of two perpendicular diameters.
Divide the circle into 4 equal arcs and bisect each arc. Connect the ends of the arcs to form a regular octagon. (To bisect an arc bisect the chord of the arc.)
Draw a circle. Divide the circle into 6 equal arcs Bisect each arc. Connect the ends of the arcs to form a regular dodecagon.
All of these regular polygons may be inscribed by the use of central angles.
Ask students to develop designs of their own with the ruler and compass.
Ask pupils to identify geometric forms to be found in the classroom.
Discuss geometric forms which are found in highway signs.
Have pupils bring to class illustrations of the various forms found in pictures, in books, magazines, etc.
Call attention to the distinction between geometric concepts and the models used to represent them.
Practice drawing and constructing these polygons. Have pupils make and complete charts like that in figure 1. Discuss relationships.
Discuss the Quadrilateral Family (Fig 2).
Textbooks contain adequate material in developing an understanding of perimeter and circumference. Have pupils develop formulas which are to be used in solving problems involving perimeters of the various polygons and the circumference of a circle.

39

c. Areas of polygons and circles

Allow pupils to derive formulas through experimentation and observation.

Name of Polygon Sides

Triangle

3

Quadrilateral

4

Pentagon

Hexagon

Octagon

Decagon

Dodecagon

Angles
3 4

I Number of Diagonals Total No.

Vertices from One

of

Vertex . Diagonals
I

3

0

0

4

1

2

Figure 1 The Quadrilateral Family

QUARDRILATERAL

TRAPEZOID

RECTANGLE

ISOSCELES TRAPEZOID

'-- r----.I SQU~ I

Rectangle

10" 3 II

= The rectangle is divided into I" squares. Count the squares (30), A 30 sq. in. = Therefore A 1w.
40

Square

4"
D r - -. . . .- r - -. . . .- - - , C
s t---t--t--+--t 41/

A '--........--JsL-.........~ B

Count the squares
= = = A 16 sq in. Therefore A lw or A s2

Triangle

Divide the square into two triangles.

D

C

""""

"""""

A

"""" B

Count the squares in each triangle.
The number of squares in ABD = 4Vz
= Therefore if A of square ab then A of triangle

Vzab.

D

C

7

B

Draw parallelogram ABCD. Draw h perpendicular to AB. Cut out the parallelogram. Cut off triangle AED, placing AD so that it coincides with BC. A rec-
= tangle is formed. The students are familiar with finding the area of a rectangle.
In this case the width or height is the same as the altitude. Therefore A bh.

Trapezoid

Draw two trapezoids the same size and shape.

b

b

A E '-----'L-------~B

'----------~F

b'

~

41

Put them together like this:

R b

P bl

b'

b

L

M

No

\

From this point on let the students discover the area of a trapezoid.
= = The base of the new figure which is a rectangle = b + b'; h is the height.
Therefore A of LOPR h(b' + b). Therefore A of one trapezoid lh h (b' + b).

Circle
Draw a circle of any radius. Divide the circle into sectors. Cut out sectors and place them to form a parallelogram as shown in the figure below.

5. Solid geometric figures a. Introduction (1) Recognizing geometric forms in familiar objects
(2) Extension from two dimensions

The base of the parallelogram is % C of the circle. The altitude of the parallelogram is the radius of the circle.
= = = Area of parallelogram % Cr but C 217T. Therefore Area of circle = % 2 7T r r. That is, A of circle 7Tr2.
Topics for report: "Moebius Strip" "Primitive Pythagorean Triples"
Give examples and present models of familiar objects which are geometric solids - balls, boxes, blocks, paper cups, cans, hat boxes, etc. Ask pupils to name others. It might be well to separate these into groups: prisms, cylinders, cones, and spheres.
Name the several classifications, showing examples, but do not give formal definitions. Help pupils to recognize the differences in the classes of solids. Provide practice in classifying by having pupils indicate to which particular group objects belong.
Explain, through the use of models, the meaning of vertex, edge, and face.
The extension from two dimensions to three can be made without difficulty. Lines were studied first, then plane figures which were made by lines. Now solid figures are composed of plane figures. Pupils should have models before them in order that they may discover this fact for themselves. Just as a plane figure was defined as being the lines and not what was inside, so the solid
42

(3) Recognition of solids b. Surface area

figure consists of the faces, not the inside. In plane figures the measurements considered were perimeter and area; in the solid figure they are surface area (the number of square units of material required in constructing a model) and volume (the number of cubic units which can be placed inside a model).
No formal definition of solids should be given. However, pupils should under stand intuitively the difference between plane and solid figures.
In a plane figure all points of the figure lie in one plane.
In a solid figure all points of the figure are not in the same plane.
Since pupils have already worked with areas of plane figures it might be well to develop first the surface area. It is suggested that the rectangular solid and cube be studied at this level, leaving other solids for study in the eighth grade. Superior seventh grade students might study all the solid figures as a special project.
A model made of cardboard, but not fastened together, is very useful in developing this concept. Use letters 1, w, and h for the number of units in the length, width, and height, respectively, and mark all edges with the appropriate letter.

c. Volume

1

,,,--------------

,/

h

/

/

/

Fold the solid and hold it is position, or fasten it with tape, so pupils can see that is a rectangular solid. Then flatten it so that they can see all six faces. Allow them to develop the area formula for themselves. They will probably obtain something like this:
S 2 lw + 2 Ih + 2 wh
S 2(lw + Ih + wh)
If the formula they find is longer, ask if it can be written in shorter form. Stress again the fact that square units of measure are used.
= = Develop the formula for the cube as a special case where 1 w h and use
the letter "e" to represent edge.
S = 2(e e + e. e + e e) = 2(e2 + e2 + e2)
= 2(3e2)
= 6e2
In developing a concept of volume use a box which is one unit in height and have pupils fill the box with cubic units. Ask them to compare the number of cubic units in the box with the area of the base of the box.

Number of cubic units in box one unit high equals the number of square units in the area of the base. Now suppose the box were three units high. It would
43

hold how many times as many cubic units as the first box? Pupils will discover that the area of the base equals the number of cubic units in one layer and that the height tells how many layers can be placed inside. Then they should
= be able to derive the formula, V I w h for the volume of a rectangular
solid and its special form
= V e3 for the volume of a cube

IX. Statistics and Graphs

A. Collecting data

Graphs should be treated as an important part of statistics.
Have students collect some data for the class to study, such as: ages, heights and weights of pupils in seventh grade.

Ask students to bring to class data that has already been collected, such as: passenger car sales for ceatain years, census for several years, U.S. Petroleum production for certain years, and weights of Explorer Satellites.

B. Types of graphs

Discuss with students how to organize and interpret this material and how to present it to the public.
Give an over-all picture of different ways of presenting data. (Let students name all the types of graphs they have seen in books, magazines and newspapers.)

Discuss points to remember in making graphs.

These include: Details to be shown. Space on which the graph is to be made. Position of title. Labels for easy interpretation. Scale to be used. Graph to be lettered or numbered from left to right.

1. Broken line

Use this type graph to compare rates of increase and to emphasize changes that occur in some item. Warn students not to try to interpret the graph between points.

2. Bar graphs

Remember the following when making a broken-line graph: Draw lines perpendicular to each other as you would draw the x and y axis. Choose a scale. Look at space and numbers you are to represent. It is not necessary to use the same scale both horizontally and vertically. Show on the graph the unit used. Choose and print the title in a prominent place.
Use this type of graph to show comparison between similar items.

Point out that the number scale must begin at zero.

a. Horizontal bar graphs b. Vertical bar graphs c. Divided bar graphs

Remember the following when making these three graphs: Show on the graph the chosen scale and start numbers from left to right, and from bottom to top. Decide which would be better, horizontal or vertical graph. Draw the bars and the spaces between the bars uniform. Label the bars. Choose a title and print it in a prominent place.

3. Pictographs

Ask students to gring pictographs to class.

4.Histogram

Call attention to the fact that histograms are used to show how frequentlY each score occurs.

5. Circle graphs

Remember the following When making a histogram: Arrange items in order. Make a frequency distribution. Choose a proper scale. Show on the horizontal axis all the items in order. Draw the vertical bars next to each other. Let the midpoint of each vertical bar correspond to the item shown on the hori zontal axis. The height of each bar gives the frequency corresponding to the item shown on the vertical axis.
Use this graph to show a comparison between parts of the whole and the whole.

44

c. Summarizing data 1. Kinds of averages a. Mean b. Median c. Mode 2. Application
X. Problem Solving
A. Readiness
B. Reading

Review the construction of central angles and applications of per cent.
Remember the following when constructing circle graphs: Find the sum of the quantities to be used in the graph. Find fractional parts of the whole. Find a central angle to match these fractional parts. Draw a circle and mark off central angles. Label and print title. Change the fractional parts to per cents on the graph.

Write in formula form for students the meaning of mean:

= sum of items

mean

number of items

Define median for students. It represents that number which has half of the items above it ahd half of the items below it. It is the middle one of a set of numbers.

Define mode as the item which occurs more than any other one.

Solve problems which involve the finding of mean, median, and mode.

Problem solving is not a separate unit but is taught throughout the course. It is placed here for the convenience of the teacher.
Provide activities, situations, materials, and instruction that will motivate and assist the student in developing experiences needed in solving problems.
Through testin, observation, and evaluation determine the student's ability to carry out the fundamental operations with numbers. A student who is skillful in arithmetic and understands the number system and its relationships will be more skillful in solving problems than one who lacks this knowledge.
Help the student to develop skill in a simple logical method of procedure. Direct questions will help in leading the student to make discoveries for himself.
Develop understanding of words used in the problem. Help the students develop comprehension skills, critical reading skills, and locational skills such as those used in reading graphs and tables. Basic concepts should be presented before the student is asked to read from a book.
Use many types of reading situations and materials to give the student a feeling of success in reading problems.
Give exercises in vocabulary study such as finding meanings; matching words with objects; grouping words; dramatizing concepts, such as buying and selling; classifying and identifying words and concepts; locating and discussing unfamiliar words and expressions.
Begin with easy problems and have the pupils state what the numbers stand for.
Have the pupils discuss a problem telling everything the problem contains.
Aid the pupils in visualizing the problem situation by dramatization, changing large numbers to smaller numbers, or by a diagram of the problem.
Read many types of problems: one-step problems, two-step problems, problems with irrelevant facts, problems with missing facts, problems with hidden questions, problems without numbers.
Solve problems orally: John has $1.60. How many 4 cent stamps can he buy?
Sam is 16 years old. His father is 3 times as old as Sam. How old is Sam's father?
45

C. Selecting the right process D. Estimating answers

Ask the children to make and read problems related to their own activities. Give some problems easy enough for all to achieve some degree of success and others which are difficult enough to challenge the more able students. Use the chalkboard to guide discussions and bring out meanings. Stress the development of concentration as the student reads a problem.

Help the pupil to read the problem and understand it as a whole, rather than a series of parts. Guide in the understanding of the various relationships. The essence of problem solving is recognizing how what is wanted is dependent upon or related to something else.
Aid the pupil in selecting the right process. Encourage the use of a diagram or setting up of a mathematical sentence to discover the situation described in the problem.
Example of diagraming:
If it takes 3f4 yard of cloth to make one dish towel, how many dish towels can be made from 9 yards of material?

,--'---v---A---v--" -,--'---v---A---v--".--i,---"----,~~.

oI .

.I
1

. I ... I .

2

3

I.
4

I.
5

I.
6

I.
7

I. I
89

= 12 towels

Guide the pupil to see that a mathematical sentence is simply a way of translating the problem into arithmetic language or symbols Written like any sentence from left to right, it shows the true relations between the groups and the actions involved.

Example:
Joe has some minnows. A good friend gave him 18 more. How many minnows did Joe have at first if he now has 29 minnows?
N + 18 = 29 or D + 18 = 29 or
29 = N + 18

Be sure the sentence reflects the action in the problem; for instance, if the
= student expressed the above situation as 18 + N 29 it would be incorrectly
expressed since 18 is added to the unknown quantity, not the unknown quan-
tity added to 18.

Point out that the sentence for a problem can be an inequality or an equality.
Example:
Joe had some minnows. A friend gave him 18 more. Now Joe has fewer than 29 minnows. How many minnows could Joe have had at first?
x + 18 is less than 29 or 29 is greater than x + 18

Teach estimation by doing simple mental arithmetic.
Make comparisons of known amounts to aid in developing skill in the ability to estimate reasonable answers. For example, in multiplying 19 X 5, 20 is compared with 19. It is one more than 19. The student can easily compute
= = = 20 X 5 100. Since 19 is one less than 20, 1 X 5 5; thus 100 - 5 95.
$5.98 - 2.19 is about $3.00, $4.00, or $5.00 72 - 37 is about 50, 40, or 30 6 X 38 cents is about $2.00, $1.80, or $2.40
Aid the student in seeing how rounding off numbers can aid in estimating !'easanable answers.

46

E. Understanding the sequence. of steps in problem solving

Stress the fact that no definite set of steps should be followed for all problems. The following procedure, however, may be helpful in many problem situations:
Read the problem silently. Discuss the problem orally Sketch, dramatize, or illustrate the problem. Discuss what is wanted, what is given, or what is needed, as you select the
process. Estimate the answer. Discuss reasons for the estimate. Solve and check the problem.

47

-- ----

--'-'~------:'I

EIGHTH GRADE

UNDERSTANDINGS TO BE DEVELOPED IN EIGHTH GRADE
Eighth grade mathematic~ has been designed to enable the pupil to attain an understanding of and skill in computation with the four fundamental operations on rational numbers; to gain a more complete knowledge of ratio and proportion, of per cents, of graphs, of the metric system, and of problem solving; and to sample topics in algebra and geometry which he will study more thoroughly later. Methods of presenting these topics are of great importance. A vocabulary of words and symbols used in mathematics is essential to success in the course. This should be a time of discovery for the pupil. Intuition and experiment should aid in discovery. The pupil should have an opportunity to work with mathematical games and puzzles and to engage in meaningful individual research projects. An adequate library of mathematics books and periodicals should be available for his use. Understanding of the basic laws which govern the number system and their application to computation will strengthen the pupil's ability to use numbers and will make the transition from arithmetic to algebra one of logical continuity.
51

CONTENT
I. Numeration
A. Early historical development
1. Caveman's numeration, oneto-one correspondence.
2. Ancient systems of numeration. a. Egyptian

TEACHING SUGGESTIONS
Review ancient symbolism and systems of numeration with bases other than 10. The most important reasons for these are to contrast them with the decimal system and to give a better understanding of the role of symbols and place value in numeration. Refer to seventh grade guide (1. Numeration) for a review of the historical developments. This additional information will add interest. It may be obtained through a library assignment.

The Egytian hieroglyphics, or picture numerals, have been traced back as far as 3300 B. C. The Egyptian system combined two principles addition and repetition. The symbols used are as follows:

Our Numeral 1

,Egyptian

Object Represented stroke

10

II

heel bone

100 1,000

?

coiled rope

,.t:

lotus flower

10,000

(

bent reed

100,000 1,000,000

%

burbot fish astonished man

Example:

( ( 61""n III = 20,265 r """ II

b. Babylonian
c. Roman d. Other systems B. Decimal notation 1. History of Hindu-Arabic numerals.
2. Basic characteristics.

The Babylonian system of numeration has been tra~ed back to about 2000 B. C. The writing was done on clay tablets with a stylus made of a wedge-shaped
< piece of wood. Their symbol for one was 'and for ten was The Babylonians
used sixty as their base. The second place had a value of sixty, third place, sixty X sixty, etc.

Examples:

:' (l)

~

(2) <''""

25
+ (16 X 60) (2 X 1)
= 960 + 2 = 962

Review and extend knowledge of the Roman system.

Pupils may enjoy a research assignment of other systems, such as Mayan, Chinese, or Greek.

Review contributions of Hindus and Arabs in developing symbols we use.
Show film on history of decimal notation.
Stress that the decimal system is so named because it uses groups of ten. The word decimal is derived from the Latin word decem which means "ten." Since we group by tens, we say ten is the base of the decimal system of numeration. It is most important that pupils know these characteristics of decimal notation:
Each symbol is the name of a number.
The position of a symbol tells the size of a group.

52
~I

3. Place value 4. Exponents
C. Numeration with. bases other than ten

The symbol for zero is used to fill places which would otherwise be empty.
Provide opportunities for pupils to re-discover generalizations concerning place value of numerals, such as:
Each period contains these places.
A thousand is a thousand ones.
In moving from right to left each new place has ten times the value of the preceding place.
In mov;ng fro:n left to right each new place has one tenth the value of the preceding place.
Zeros are necessary as place-holders.
Extend pupils' understanding of place value in large numbers with charts showing periods through trillions.
To show place value in the number 8,432,563,654 analyze it in this way:
8,000,000,000 400,000,000 30,000,000 2,000,000 500,000 60,000 3,000 600 50 4
Now show the place value of each digit.
Provide opportunities for pupils to find and read large numbers from books, magazines and newspapers related to science, government, current events, etc.
The study of exponents here serves to emphasize the role of "base" and "position" in numeration.
Show:
= 10 X 10 100 = 10 X 10 X 10 1000 = 10 X 10 X 10 X 10 10,000
Ask how many times ten is used in multiplication in each expression. There is
= = a need of a shorter way of expressing 10 X 10 X 10 X 10 10,000. It is 104
10,000.
Then:
10 X 10 X 10 = 103
= 10 X 10 102
4, 3, and 2 as used above are called exponents. 10 is called the base. 104 is called the power.
In using powers encourage pupils to discover the relationship between the exponent and the number of zeros in the expansion.
Have students select a number and show the place value of its digits in this manner:
= 4328 4(103) + 3(102) + 2(101) + 8(1)
Point out that the superiority of the decimal system over ancient systems is in the use of place value and the zero symbol and not that the base is ten.
Review numeration in bases given in the seventh grade guide.
53

1. Base eight a. Place value
b. Counting in base eight
c. Converting from base eight to base ten

In the base eight system the symbols 0, 1, 2, 3, 4, 5, 6, and 7 are needed.
The values of the places are: one, eight, eight2, eight3, etc. That is, the place values are one and the powers of eight.
Make a place value chart showing powers of eight:

Eight4

Eight3

EighP

Eight!

One

8X8X8X8 8X8X8

8X8

8

1

4096

512

64

8

1

Counting in the base eight system starts as follows:

Base Ten 1 2 3 4 5 6 7 8 9
10

Base Eight 1 2 3 4 5 6 7
10 11 12

Note:

10"gh' is read "one zero" and not "ten"; 12"gh' is read "onetwo," not "twelve."

Take a number such as 324".", and show expanded notation. Thus:
= 324"gh, (3 X eight2) + (2 X eight) + (4 X 1) = 192 + 16 + 4 = 212'.n

c. Converting base ten to base eight

Consider a number such as 81 ton. The largest power of eight contained in

8L'n is eight squared.

1

The division shows that there is one eight2 group in 81 and

64 i81

17 ones left over.

64

17

2
8 i17
16
1

The division shows that there are two eight groups in 17 and one ones group left over.

Therefore 8L'n = (1 X eight2) + (2 X eight!) +1 X one = 121 01g",

A shorter way of changing a numeral in base ten to base eight is to make

continued division. 8L.n may be converted to base eight in this manner:

8 / 81 8 / 10 group of eight

Remainders
> lone left over

1

group of eight2 - - - - >

l' 2 eights left over

---'1

~1'

Write the remainders in the reverse order, obtaining:

2. Duodecimal numerals

In this system grouping is by twelves. When writing numerals in this base, you must make use of two new symbols. Usually "T" is written for ten and "E" is written for eleven.
54

a. Counting in base twelve b. Converting base twelve ,to base ten
3. Binary numerals a. Counting with binary numerals b. Converting base two to base ten

Make a place value chart and counting chart similar to those used in teaching base eight numeration.
Convert numerals from base twelve to base ten and from base ten to base twelve.

Examples:

Convert 2T4,welvo to base ten
2T4,we,ve = (2 X twelve2) + (10 X twelve) + (4 X one)
= 288 + 120 + 4 = 412'eo

Convert 568'eo to base twelve.

3
144 / 568 432
U6

This division shows that there are 3 twelve2 groups in 568 and 136 ones left over.

11
'12 /136
132 '-4-

This division shows that there are eleven twelve groups with 4 ones left over.

= Therefore 568'eo 3E4,welve

The continued division method may be used as follows:

12 / 568

12 / 47 groups of twelve

>4 ones left over

--3 groups of twelve2 ---> E ones left over

I

t

= Therefore 568,,0 3E4twelvo.

-/

v

The symbols used in the binary (pronounced bi na ri) systems are 1 and O.

These numerals are based on groups of two.

Make a place value chart and a counting chart similar to those used for base eight.

Convert numerals from base two to base ten and from base ten to base two.

Examples:

Change 1101O,wo to base ten
+ + + 11010 (l X tw04) (1 X tw03) + (0 X tw02) (1 X two) (0 X one)
= 16 + 8 + 0 + 2 + 0 = 26'eo

Change 46ten to base two.

1
32 /46
32
14

This division shows that tw05 is the largest power of two contained in 46 and 14 ones are left over.

1
8 T14
8
6
1
416
4
2

This division shows that the largest power of two in the remainder is tw03 and 6 ones are left over.
This division shows the largest power of two in the remainder is tw02 and 2 ones left over.

55

4. Operations with numbers written in bases other than ten.

1
2/2
2
o

This division shows that two! is contained in the remainder with 0 ones left over.

= Therefore 46 101110two

The continuous division method may be used. Thus:

2 I 46 2 / 23 groups of two 2TTI groups of two2 2 / 5 groups of two3 2 / 2 groups of two4
--1 group of two5
= Therefore 46'00 101110two

0 ones left over 1 two! left over 1 two2 left over 1 two3 left over
o two4 left over

Help pupils construct addition and multiplication tables for number bases studied at this level. Make up simple exercises involving the basic operations in these bases. Use the exercises for the purpose of helping pupils to understand better the structure of a place value numeration system.

A number of books are available which give detailed explanations of operations in other number bases.

II. Operations With Non-Negative Rational Numbers

It is important that a mastery of operations with whole numbers and fractions be attained through an understanding of the principles governing the operations.

Using cumulative records, including individual performance records of the previous grade, and prognostic tests determine how much review is needed and refer to suggestions presented in the guides for grades six and seven. These references should be used extensively if it is found that students have not developed an understanding of the processes involved.

A. Meaning of a rational number

Review definition of rational numbers: If a and b are whole numbers with b not zero, the number represented by the fraction alb is called a rational number.

Give examples, such as 3/ 1 , 9/ 3 , 12/ 4 , which will show that whole numbers and fractions may be classified as subsets of the set of rational numbers. Be careful
not to give the impression that these are the only subsets of rational numbers.

B. Using the number line 1. One-to-one correspondence
2. Density of fractions 3. Ordering C. Operations with whole numbers

The number line gives one of the most successful ways of picturing numbers.
Show with the number line that we can associate each rational number with a point of the line giving a one-to-one correspondence with some points on the line. Be careful to bring out in discussion that there are many unlabeled points on the line.-
If pupils are inquisitive concerning points to the left of 0, negative rationals may be introduced (See VII, A, Introduction to Algebra).
Construct a line to show halves, thirds, fourths, etc., leading pupils to see that another fraction can be placed between any two.
Show that the number line gives location of rational numbers in order of increasing size. The larger of two numbers lies to the right of the other. The smaller of two numbers lies to the left.
For detailed suggestions concerning the teaching of operations with whole numbers refer to the guide for seventh grade. The suggestions made here are to re-emphasize the importance of the properties of whole numbers under the basic operations, and to give a greater depth of understanding of the processes with which students have worked for several years.

56

1. Addition a. Commutative property
b. Associative property 2. Multiplication of whole
numbers a. Commutative property

Let pupils try an experiment which leads them to discover this important principle in the following way.
On two strips of paper make number scales, A and B, letting each interval or line be one-fourth inch.
Place the scales as shown in this diagram:

. I? ~ ~ ~ : ~ ~ : 8 9 10 11 12

A

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

B

Ask how many intervals there are between 0 and 4 on Scale B. How many are there between 0 and 7 on Scale A? Then ask if the positions of the scales show
4 + 7 - 11. (7 on Scale A is directly over 11 on Scale B.)

Now, place the scales so that 0 on Scale A is directly over 7 on Scale B. Ask what number on Scale B is directly under the 4 on Scale A. Ask if this shows
that 7 + 4 = 11.

Pupils may enjoy making similar scales for other bases of numeration.

Practice with base ten scales and/ or others should lead to a recognition of this property of addition for whole numbers. The sum of two numbers is the same regardless of the order in which they are added. This property is called the commutative property of addition of whole numbers.
= It may be stated: If a and b are whole numbers then a + b b + a.

The number line may also be used to show this property. Its use requires less time, but perhaps it is not as interesting as the use of the sliding scales.

For example:

3

4

)

I

)

--)

0 1 2 3 4 5 6 7 8 9 10

4+3=7 or

4 3

0 123 4 Consider this example:

5678 3+4=7

--)
9 10

26 + 15 + 25 = 66 (26 + 15) + 25) = 41 + 25 = 66 or 26 + (15 + 25) = 26 + 40 = 66
Other examples in base ten and examples in other bases may be used to show the associative property of addition for whole numbers. It is stated: The sum of 3 or more numbers is not affected by the order in which they are grouped.
That is, if a, band c represent any whole numbers,

(a + b) + c = a + (b + c)

Ask the pupils to give examples to show whether or not the product of two numbers is the same regardless of their order.
Example: 12 X 9 = 108 9 X 12 = 108
57

b. Associative property c. Distributive property of
multiplication over addition
3. Using the properties
4. Inverse operations a. Subtraction as the inverse of addition

This property is the commutative property of multiplication and may be stated as:
If a and b represent whole numbers, then a b = b a
Ask what is meant by 54 6? Does it mean (5 4) 6 or 5 (4. 6)? Since both ways of grouping give the same answer, the conclusion is that either meaning is correct.
This shows the associative property of multiplication and may be stated as:
= If a, b, and c represent any whole numbers (a b) c a (b. c)
Ask pupils if this is a true sentence:
15 X 16 = (15 X 10) + (15 X 6)
Have pupils perform the computation to show whether or not the statement is true. Then show that the statement could be written as:
= 15 X 16 15 (10 + 6)
= + (15 X 10) (15 X 6)
Use other factors of 16 and have the pupils determine whether or not the statements made are true.
Show that the multiplication algorism makes use of the property illustrated above.
Example:
45 X8
4{)=8X 5 320 = 8 X 4{) 360
Thus 8 X 45 = (8 X 40) + (8 X 5)
Remind students that the property used here is the distributive property of
= multiplication over addition. It is stated: If a, b, azidc represent any whole n1J~
bers, then a (b + c) (a b) + (a c)
Lead students to become aware that the commutative, associative and distributive properties are the bases for deriving most of the important properties of systems of numbers.
Give problems which illustrate their use.
Examples:
Find the cost of 144 pencils at 5 each.
Cost = 144 X $.05
144 is the multiplier, but it awkward to use this number as such. The commutative property allows the use of .05 as the multiplier.
645 X 120 X 5 = ?
Using the associative property, 645 X (120 X 5) = 645 X 600
USing the commutative property,
= = 645 X 600 600 X 645 387,000

Use the number line.

.3

4.

o 123 456 789
Ask: If one adds 3 and 4 by counting on the number line, where dues he stop?

58

b. Division as the inverse of multiplication
5. Closure
6. Classes of whole numbers according to factor ability a. The number one b. Zero c. Prime numbers d. Composite numbers
D. Operations with fractions 1. Meaning
2. Common fractions a. Expressing a fraction in simplest form
b. Reciprocal
c. Properties of rational numbers expressed as fractions

Ask: To find the difference between 7 and 4 could one begin with 7 and count four intervals to the left? Where does he stop?

Have students rewrite as true sentences such statements as:

= 7 X 19 ?

= ?-;.- 6

18

= ? ..;- 24 576

These should show that division "undoes" what multiplication has done.

Ask: If one knows two factors of a product can he tell what the quotient divided by one of the factors is?

To redevelop the concept of closure give an illustration, such as: Pretend all the whole numbers are in a box. Ask: If any two numbers are removed, will their sum be a number found in the box?

Then state that the set of whole numbers is said to be closed under addition.

With further discussion and examples help pupils to know that the set of whole numbers is closed under multiplication also but not under division and subtraction.

Pupils should develop an understanding of the concept that a set is closed under an operation if the combination of any two elements of the set gives an element of the set.

Detailed suggestions are given in the seventh grade guide. Refer to them in reteaching these ideas.

For remedial work with fractions the sixth and seventh grade guides should be used as needed.
Re-emphasize these concepts:
A fraction is a symbol, / b, where a and b are numbers, with b~O.
A fraction is a name for a rational number. There are different names for the same number.
Two fractions which represent the same number are equivalent fractions.
Review the meaning of a fraction in simplest form, which most pupils know as a fraction in lowest terms.
Review the development of the principles that if both terms of a fraction are multiplied or divided by the same numbers, the number represented is not changed.
Review definition and proof of the definition as found in the guide for seventh grade.
Lead students to give examples which will show that, if the product of two numbers is one, neither number is zero.
Review properties as developed in the seventh grade guide.
Pupils at this level of learning should be able to state the properties and illustrate their use.

59

d. Addition of fractions
e. Subtraction of fractions f. Multiplication of frac-
tions

Consider an example such as:

+ + + + 1/ 5 1/ 5 1/ 5 1/ 5

1/5 = 1

Have students show that this is true by using the distributive property.

= + + + + 1 1/ 5 1 1/ 5 1 1/ 5 1 1/ 5 1 1/ 5
(1 + 1 + 1 + 1 + 1) .1/ 5 1

Review finding the least common multiple which is the lowest common denominator.

Guide pupils in the method of selecting the L. C. D. Consider:

5/8 +7/ 12
Begin with the larger denominator which is 12. Multiply it by 2. Since 2 X 12
= = 24 and 3 X 8 24, 24 is a multiple of 8 and 12 and is therefore the L. C. D.
Guide pupils to understand that, in adding three or more unlike and unrelated factors, the same method is used in finding. the L. C. D.
Review and extend material covered in previous years. Recall that subtraction is the inverse of addition.

Show that the set of non-negative rational numbers is not closed under subtraction.
Example: 3/ 6 - 5/6 does not give a positive fraction. Review and extend the ideas developed in sixth and seventh grade. Use diagrams if necessary.
Examples:

6X 1/ 2 =6/ 2 =.3

Show:

=== ++ + + ++ ++ + 6 X 1/2

1/ 2

1/2 X 1/ 2

(1 1 1 1

6/2 or 3

1/ 2

1/2

1/2

1 1).1/2

= and 1/2 X 6 3, since multiplication is commutative.

= 1/ 2 X 1/ 2

1/4

Show by diagram:

2 /3 . 3 /I)

_
-

~3 _5

3 -

' a :2 5

_ -

2/

5

(Usm. g

commutative

property

and

the

property of one)

= + 3 X 3 1/ S 3(3 1/~) (Using distributive property) = + 33 3 l/S = 9 +1
=10

= + ++ l/S X 3 1/ S

1/ 3(3

l/ S)

1/3 3

1/3 1/3

1

1/ 9

11/9

60

g. Division of fractions
E. Operations with fractions m
dedmal form 1. Rational numbers in deci-
mal form 2. Reading and writing deci-
mals
3. Addition and subtraction of numbers expressed as decimals

or

+ 1/:> X 3 1/:> = 1/:> (9/ 3 1/3) = X = = 1/:> 10/:> 1/ 9

11/9

7/1~ X 4/, = 1/3

By inspection determine that 4 is a factor of both numerator and denominator and that 7 is a factor of both. This becomes:

70 4
703~

11

~X-.!.= /io<J

1

12

7

3 0 ;I 0 7

3"

Caution: Do not call this "cancellation." It is division. Encourage mental computation.

If necessary, review and extend ideas developed in grades six and seven.

Stress the use of the reciprocal and not the' artificial method of inverting the divisor.

Example: 3/ 8 -7- 3/ 12

----.....3,./:8=:-
3/ 12

3/ 8 X 12/:> 3/ 12 X 12/ 3

= 3/2 or 1 1/ 2

3/ 8 X 12/3

Review and extend ideas developed in the sixth and seventh grade.

Teach for a better understanding of rational numbers equivalent to repeating decimals. Help pupils to understand that every rational number may be changed to a decimal which terminates (repeats zero), or which repeats.
Give examples which show that the decimal will repeat a pattern of numbers or repeat zeros.
Recall decimal fractions as a natural extension of whole numbers in relation to place value of digits.
Determine pupils' understanding of tenths, hundredths, etc. as decimal fractions. Checking their ability to:
Locate decimal fractions on a number line.
Explain decimal and common fraction equivalents.
Arrange a series of decimal fractions in descending or ascending order.
Read and write decimal fractions and mixed numbers. Include decimal place value through millionths.
Compare values of decimal fractions.
Have pupils try to change a repeating decimal, such as 0.666 .... to the rational number which it represents. Superior students should be able to understand the procedure which follows.
Let n = 0.666 ... then 10n 6.666 .
= Subtract 1n 0.666 .
9n = 6.000 .
= n 6/ 9 or 2/ 3
.'.0.666 = 2/ 3 Review, if necessary, suggestions given in sixth and seventh grade guides.
Extend understanding of addition and subtraction of decimals using the number line. Also illustrate in the following way:

61

J>

4. Multiplication and division of decimal fractions
5. Rounding decimals F. Scientific notation
1. Definition
2. Procedure

8.43 means 8 + .4 + .03 5.69 means 5 + .6 + .09
= Then 8.43 + 5.69 8 + .4 + .03 + 5 + .6 + .09

Regroup, using associative property. Then
= (8 + 5) + (.4 + .6) + (.03 + .09) 13 + 1. + .12 = (13 + 1) + .1 + .02 = 14.12

Then
8 + .4 + .03 5 + .6 + .09 13 + 1.0 + .12 = 14.12

Subtract:

8.43

5.69

8 + .4 + .03 = 7 + 1.4 + .03 = 7 + 1.3 + .13
Then 8.43 7 + 1.4 + .13 5.69 5 + .6 + .09
2 + = .8 + .04 2.84

Give practice with this type work showing that this type of regrouping is done mentally when the usual algorisms are used.

Recall and show that subtraction is the inverse of addition.

Review and extend understanding using sixth and seventh grade guides as needed.

Review the method used in seventh grade guide, (Section IV). Use the number line to show meaning.

Call attention to the scientist's use of very large and very small numbers. Have students find examples in textbooks and reference materials.

Any number expressed as the product of two factors, the first being between 1 and 10 and the other being a power of 10 is said to be expressed in scientific notation.
= = = = Review meaning of powers of 10: 10 1; 101 10; 102 100; 103 1000, etc.

Have students discover the relation between the exponent and the number of zeros.

Have students practice with whole numDers ana then with decimals, finding the required factors for scientific notation. Lead them to discover relation between the placement in the power of ten.

Example:

5638 may be expressed: 5638 units or 5638 X 10 1 563.8 tens or 563.8 X 10 56.38 hundreds or 56.38 X 102 5.638 thousands or 5.638 X 103
= Call attention to the fact that 5638 5.638 X 103. The latter number is said
to be in scientific notation.

0.786 may be expressed:

786

thousandths

or

786

X

1

/

3 10

78.6 hundredths or 78.6 X 1/ 102

7.86 tenths or 7.86 X 1/ 10

Call attention to the last expression 7.86 X 1/10' as the scientific notation of 0.786.

Give sufficient practice so that students can handle this notation accuratelY and are able to interpret numbers which they may need to read.

62

III.' Applications Of Mathematics In Business, Finance, And Daily Living

A. Percentage L Understanding per cent

Reteach, if necessary, principles of percentage as developed in the seventh grade guide.
Ask the students to devise a chart indicating per cent, decimal, and common fraction equivalents. This may be used for reference.

2. Multiplication principles and percentage

Have students reproduce a given per cent on cross-ruled paper. Express the amounts in various ways to show the ratio between the quantities as: 1/ 4 , 25 out of 100, 25/100' .25.
Students should recognize that per cent refers to a comparison whose second component is 100. 881/ 100, 88 per cent, and 88%.
Use the experiences of the students to make problem situations involving per cents:

Per cent of examples computed correctly Per cent of boys or girls in a class; per cent of class present or absent Per cent of rainy days or sunny days within a certain period of time Per cent of money collected Per cent of games won or lost Per cent of library books in use; per cent not in use Reading and interpreting social studies data

Have the students translate problems into ratios showing relationships in a rate or comparison situation.

Compare one group with another using ratios or rate.

3. Percentage problems and proportion

Example: The eighth grade softball team won 7 out of 10 games played. Expressed as ratio this is 7/ 10, What per cent of games did they win?
Have the class make a chart on the chalkboard showing the number of games won for each multiple of 10 if the team had played 100 games. Lead them
= to discover 7/ 10 n/ 100. Since the ratio n per 100 expresses the same ratio
as 7 per 10 they will see it takes 10 groups of 10 games to make 100 games. Since 10 is multiplied by 10 to get 100, so the numeral that replaces n can be
= = = found by multiplying 7 by 10. n 70. Thus 7/ 10 70/100 .70 or 70%.
Make a collection of the uses of per cents found in newspapers and magazines. These may be classified under the three uses of per cent:

Finding the per cent of a number.

Examples:

A dress is advertised for $14.95 plus 3% sales tax. A watch is advertised for $59.50 plus 10% federal tax plus 3% sales tax.

Finding what per cent one number is of another.

Examples:

Batting averages, pitching records, and team rankings from the sports section.
Advertisements of sales, giving the former price and the sale price, as a dress formerly priced at $19.95 now selling for $10.95.

Finding a number when a per cent of the number is given.

Example:

An advertisement for a ring for $44.00 includes the 10% federal tax. The buyer would like to know the selling price before the tax was added.

Develop the understanding that the method of solving these problems is no different from the method used to solve proportions. The solution depends on finding the missing term in an equation of equal ratios, one of which has 100 as its second term. (Refer to VI. B of the seventh grade guide.)

63

4. Per cents less than 1% and greater than 100%
5. Using the percentage formula

Show the students the meaning of a fractional part of 1%. Use a visual aid if necessary. The average student should be able to find 1 per cent of a number by dividing the number by 100. He should know that this gives the same result as moving or placing the decimal point two places to the left in the number. Then he can find the fractional part of the 1%.

If pupils lack an understanding of per cents greater than 100%, form two stacks of squares or discs and find the ratio of the two quantities in both orders. Express each ratio as a per cent.

Example:

000
a

0000
b

The ratio of the number of discs in (a) to the number of discs in (b) is 3/ 4 or 75%. The ratio of (b) to (a) is 4/ 3 or 133 1/ 3 %. Lead to the following generalizations:

When a smaller number is compared with a larger number, the smaller number is less than 100% of the larger number.
When a larger number is compared to a smaller number, the larger number is greater than 100% of the smaller number.
When two equal numbers are compared, one number is 100% of the other number.

Review the meaning of the three basic numbers of the per cent relationship: percentage, rate, and base.

The percentage is the product of the rate and the base. In any percentage problem, if two of the three numbers are known, the third one may be found. If the percentage is unknown then the process of multiplication is used to solve the problem. p = br.

If the percentage is one of the two known numbers, then the process of division is used. The percentage is divided by the other known number to find the unknown number.

B. Banking 1. Interest on loans

= Show that in the formula r P/ h, r is the ratio of a number to 100, and p and b
are the terms of an equivalent ratio.
The base is always the number which is the basis or standard of comparison, and the rate is always labeled as a per cent.
= When the student understands how to use the percentage formula, p rb,
he should be given problems to solve. He should identify the values of two of the letters in the formula and substitute these values in the formula. Then he should solve the equation for the missing value.
Take the class to visit a bank.
Teach the meaning of checking account, savings account, signature card, deposit slip, check and check stub. Help students to learn how to make out a signature card, a deposit slip, a check and check stub, and how to endorse a check.
See that students know the services rendered by banks and that they understand how banks make their money.
Review the meaning of interest, principal, and annual interest rate. point out that a bank pays interest on money deposited in savings accounts and charges interest on money loaned. Show that finding interest on a loan is the same as finding a per cent of a number. The principal is the base, the rate is the per cent, and the amount of interest is the percentage.
= Use the interest formula, i prt, with understanding. Point out that; in 'find-
ing simple interest, the year is considered to have 360 days. If the time is le~ than one year the amount of time is expressed as a fraction of a year an
64

2. Interest on savings 3. Charges for special services C. Investment l.Real estate 2. Stocks and bonds D. Loans E. Taxes
r. Installment buying

the interest for one year is multiplied by the fraction. I
Explain the meaning of promissory note, personal loan, and bank discount.
Have the students find the difference between simple interest and compound interest, compounded annually for a period of years on a given principal. The difference between simple interest and compound interest can be shown very effectively by comparing the amounts from two equal principals placed at the same rate for a long period of time, if one principal is placed at simple interest and the other principal at compound interest. Point out that because of the time consumed in the computation of compound interest, interest tables have been devised. Teach students how to use these tables.
Assign a committee from the class to find what charge is made at local banks for a checking account. Explain service charge. Solve problems using the information gained from local banks as to minimum monthly balances, services charges, etc.
Point out that real estate is a form of investment chosen by many people. Explain mortgage, property tax, down payment, and assessed value. Compare the costs of renting and buying a house. Solve problems which include these topics.
Explain that a person who buys a share of stock becomes a part-owner of the company that issues the stock, and he receives dividends from the profits that are earned during the year.
A person who buys a bond is lending money to the company or branch of government that issues it and receives interest on the face value of the bond. Solve problems dealing with buying and selling stocks and bonds.
Ask the students to find an advertisement of a local loan company. Point out that a loan company makes a business of lending money to persons who may have difficulty borrowing from a bank. Compare interest charged on bank loans with interest charged on money borrowed from loan companies.
Lead students to understand the relationship of government services to the cost of these services. Point out that governments as well as individuals and families prepare budgets and that the money required to meet government budgets comes from taxes. Discuss sales tax, income tax, property tax, excise tax, custom duties or tariff, gasoline tax, automobile license fees, tolls, and parking meter fees.
Have pupils solve problems which involve:
Sales tax on purchases Using a federal income tax table Finding the amount of property tax when the tax rate and estimated value are given Writing a property tax rate in several different ways Finding the amount of excise tax on a purchase Finding the tax rate when given the total assessed value of property and the total cost of government Finding the total gasoline tax on a given amount of gasoline
Internal Revenue Service furnishes units on the teaching of Federal income tax. Contact nearest district office for free material for teachers and for each pupil.
Discuss what is meant by installment buying, down payment, installments, and carrying charge.
Develop problems from sources such as a mail order catalog to show the application of the above terms. Point out the advantage of the largest possible down payment. Figure the rate of interest after the difference between the cash price and the installment price is computed.
Note: For suggestions concerning problem solving, refer to X, Problem Solving, in the guide for seventh grade.

65

IV. Measurement
A. Counting and measuring

Review as needed material in seventh grade section.
Emphasize to pupils that some answers are found by counting and others are found by measuring.
Example: How long is the hall? (measuring) How many made charts? (counting)
Help pupils see that all measures are comparisons and all are approximate and involve some error.
For xample: 3.5 is the approximation to the nearest tenth for all numbers between 3.45 and 3.54. One method for abtaining this approximation is to drop the digits after a certain significant place. When the first digit dropped is less than 5, the preceding digit is not changed; when the first digit dropped is greater than 5 or is 5 with succeeding digits not all zero, the preceding digit is increased by 1; when the first digit dropped is 5, and all succeeding digits are zero, the commonly accepted rule (Computer's rule) is to make the preceding digit even, (add one to it if it is odd, and leave it as it is if it is already even.)
Pupils will readily see that when one measures a line segment, he compares the length of the segment with the number line marked on the ruler.
Have pupils prepare rulers (if these are not available) that are graduated into 1/ 12", 1/4", 1/8", and 1/6" segments.

1

2

3

4

B. English and metric measurements

Help them to become proficient in the use of these rulers by answering questions concerning them.
Example: Draw a segment 3 inches long and divide it into sections 3/8 inches long.
or Draw a line segment the full width of your paper. Mark these segments on your line segment being sure each new one begins where previously drawn one left off.
2 1/ /'; 3/ 8"; 5/ 16"; 1 1//' What is total length of these segments? Read your answer on your ruler.
Point out to pupils the relationship of various measures of capacity and measures of weight to our numbering system.
Compare the metric system with our English-American system.
Note that multiplying metric units is often easier than multiplying fractions.
Have pupils experiment with this by working sample problems.
Note that to change units in the metric system it is only necessary to move the decimal point. Compare this with our decimal system.
The United States, Great Britain, and a few other nations are the only countries of the civilized world that do not use the metric system as the officiaL system of measure. In the United States, however, the metric system is used in some areas.
Ask pupils to name some areas in which the metric system is used to measure in our country.
(Science, armed forces, medicine, sports, photography and others.)
Ask pupils to cite examples of measurement used in these areas, for example, in photography, the 16mm. film.

II
66

r

1. History of the metric system
2. Metric measures of length

Have a report on the development of the metric system in France about 200 years ago. After pupils understand that the French defined the meter as a distance equal to 1 ten-millionth of the distance from the North Pole to the equator, ask them to express this in different ways.
= Example: As a fraction: 1 mm 1/10,000,000 distance from North Pole
to equator.
= As a decimal: 1m. .000 000 1 distance from North Pole to equator
Using exponents: 1m = 1/10" distance from North Pole to equator.
Help pupils to become familiar with the various units of metric linear measurement. Display a table which shows the plan of linear metric units.

Metric Measures Lengths and Distances
= 1 kilometer 1000 meters = 1 hectometer 100 meters = 1 dekameter 10 meters

1 meter

= 1 millimeter .001 meter

= 1 centimeter = 1 decimeter

.01 meter .1 meter

3. Metric measures of weight 4. Metric measures of capa-
city

Lead pupils to explore the conveniences of the metric system. They will note after careful study of the above chart that the metric measure is convenient to use because it is based on 10 just as our number system is based on 10.

Where needed review problems in the seventh grade section of the guide. Some of these include measuring with metric rulers and understandings of the prefixes helpful in comparing other metric measures with the meter.

Work problems which show a comparison between our system and the metric system.

Draw a line segment 1 decimeter long. Measure this with a foot ruler. How long is it to the nearest inch? (about 40 inches-more exactly, 39.37 inches).

Have pupils participate in the working of problems in which our official system of measure is changed to the metric system.

Example: In order for the team to layout a 60 meter track, how many feet of ground would they need?

Reinforce student understanding of the pattern in metric units of measure. Have pupils participate in setting up a table of metric weights.

Unit Milligram (mg) Centigram (cg) Decigram (dg) Gram (g) Kilogram (kg)

Important Metric Units of Weight
Decimal Relation .001 gram .01 gram .1 gram 1. gram
1000. grams

The milligram and centigram are most often used for weights less than 1 gram. Use chart to answer such questions as the following: One milligram is what part of a gram? (.001 g.)

Establish the concept that the weight of an object depends on its distance from the center of the earth. A person's weight is less in space than it would be on earth.

Check to be sure pupils know the meaning of the prefixes "milli_" and "kilo-" by asking them to express various quantities as decimal parts of a liter.

67

C. Direct measurement 1. Measuring time

Example: 3 centiliters (.03 liter) 5 deciliters (.5 liter) the sum of 6 milliliters and 5 deciliters? (.506 liter)
Familiarize pupils with the basic metric measure of capacity as being the liter. Point out that for measure less than 1 liter the milliliter is often used. For very large measures of capacity, the use of the kiloliter is customary. Students need to know also that unlike our system, the metric system uses the same measures of capacity for both liquid and dry measures.
Use the above information to answer questions such as:
To the nearest tenth, 1 liquid pint is what part of a liter? (.4732 L.)
A time cone made by pupils can be placed on a globe as an aid to developing an understanding of time zones.
The pattern consists of a sector of a circle whose radius is equal to the radius of the globe. Make the central angle 240 with an added 10 for overlapping. Mark off the circumference of each sector every 50, making the 10 marks longer and adding the hours of the day as indicated in the sketch.

Use a world map to show the 24 hour time zones. Each zone of 15 longitude would be slightly more than 1,000 miles.
Use the map to show how standard time zones agree wIth meridians and that two places 15 apart have a time difference of 1 hour.
Use a globe to make it clear to pupils that the earth turns 360 on its axis in 24 hours, then 1/ 24 of 360 or 15 is 1 hour. 15 of longitude equals 1 hour of time.
Use the globe to show pupils that places located to their east receive an earlier sunrise and th~t when people travel east or west they change time accord :ngly.
Use a pupil-made 24 hour clock to express time in many ways. It is easier for travelers to avoid confusion between A. M. and P. M. t,ime.
0000 /zoo Z ... OO

0600 1800

0300 is 3 A. M. and 1500 is 3 P. M.

Between noon and midnight figure time by adding 12 to the hour, multiply by 100, and add minutes.
68

2. Measuring temperature
3. Measuring electricity D. Indirect measurement
1. Ratio and proportion

Indicate important role of the clock in navigation.
Make a chart showing comparison of a Fahrenheit and a Centigrade thermometer. Be sure pupils know that the Fahrenheit thermometer is the one used in home, school, and outdoors. Centigrade scale, also called the celsius scale, is used extensively in work in science. Ask pupils to reason their way to necessary answers for converting from Centigrade to Fahrenheit and vice versa.
Using the chart to show a comparison of Fahrenheit and Centigrade guide pupils into the discovery of the formulas:
= + C = 5/ 9 (F - 32)
F 9/ 5 C 32
There are three basic units employed in the measurement of electricity: the volt, the ampere, and the watt.
Students may read various wattage requirements for home appliances and calculate cost of operating these for various periods of time at rates which electric company bulletins will have listed.
Encourage alert students to further their study of measurement by doing research on atomic weights, the speed of light, and astronomical distances.

Review the concept that a mathematical sentence that says two ratios are
= = equivalent is called a proportion. It may be written as: 4/ 9 12/ 27 or 4:9
12:27.

Explain that this sentence may be read as "4 is to 9 as 12 is to 27."

Show on a diagram the special names for the terms of a proportion:

4:9

12 : 27

--.r L =---.J i means
~ extremes

i

Redevelop by use of informal questions the distinction between a ratio and a fraction.
Pupils need to know that in a true proportion, the product of the extremes is equal to the product of the means. To give strength to this understanding, use the associative and commutative principles of multiplication to conclude that these products are equal. (Use guide for seventh grade for review, if necessary)
i.e.
= 3:4 15:20 = 15 5 X 3 = 20 5 X 4 = 3:4 (5 X 3) : (5 X 4)
Product of extremes is 3 X (5 X 4). Product of means is 4 X (5 X 3).
Substitute letters for numbers in order to provide many different illustrations of the principle.
= a:b c:d
If c:d is equivalent to a:b, c is product of a and some multiplier, while d is product of b and the same multiplier. Call this multiplier x.
c = x-a d = x-b
Substitute and get
= a:b (x-a): (x-b)

69

2. Square root 3. Scale drawing

= Product of extremes a(xb) = Product of means b (xa)
Products are equal.
v Finding the square of a number and finding the square root of a number
are inverse operations. When the symbol is used, it is defined as meaning the positive square root.
Example:
To find the square root of a given number find the number which when multiplied by itself will produce the given number.
Draw a square. Mark it off into 1" blocks. Help children to see that finding the square root of a number is the same as finding the length of the side of the square.

5"

5"

Strengthen the understanding that square root is a special kind of division in which the divisor and the quotient are the same number.

Example: 7
v'49

Try simple problems of estimating square root.

Example:
v'50 = 7.1 approximately
= 71 X 71 5041

Add interest to square root by using problems such as:

How far will you travel:
(8 - 2)2 + 2 v'4
= = + + (8 - 2)2 2 v4 62 (2 X 2) 40

Newton's method, the process of dividing and averaging the divisor and quotient, is suggested as a method of finding square roots.

Sometimes the scale of a drawing is given as a ratio. This means 1 unit of lerigth in this drawing represents a number of the same units in the actual

object.

i. e.

Actual Length 75 mi

Scale
1" = 25 mi

Scale Length 3"

Have pupils find actual lengths of various drawings such as:
c

D E
70

4. Pythagorean property

c

Use scale such as 2:5

A simple way to explain to pupils the meaning of scale is to say "2 inches in the drawing represents 5 inches of the real object."

Develop an appreciation for the use of scale drawings. Ask pupils to list various situations in which scale drawings are helpful.

The list might include such things as blueprints needed in various types of construction, maps of various kinds, and the like.

Select appropriate scale for your drawing. Then to find the length needed solve as follows:
= Scale: 11 16" 1 yd.; length to be drawn to scale is 50 yds.

- w 1/16

X

1

= 1x 1/ 16 X 50

X = 3 1j8

Therefore, the 50 yds. may be represented by a drawing 3 1/ 8" in length.

Review meaning of hypotenuse. Construct a right triangle ABC. On each side of the triangle construct squares. Do this on graph so that the squares may be counted.

BC AC

= =

3 4

AB = 5

G

F H

I

D

E

a. How many squares in CIHB?
b. How many squares in DECA? c. How many squares in ABFG?
Is this sentence true?
(Answer to a) + (answer to b) == (answer to c)

71

What is the area of CrnB? What is the area of DECA? What is the area of ABGF? Complete the sentence:
+ = Area of CIRB ~~~~~~~~~~~~ ~~~_~~~~~

EXERCISE:

B

a

A
Would these sets of numbers make a right triangle? 28", 21", 35"? 6", 8", 10"? 7", 9", 11"
= = Find C if a 6/1 and b 9/1 = = Find b if c 15/1 and a 1" = = Find a if c 35" and b 10 2/ 3"
Topics for reports: 1. The Life of Pythagoras, 2. Other ways of proving the Pythagorean property.
Another way to prove the Pythagorean property:

v. Geometry
A. Non~metric geometry
1. Identification a. Points, lines, planes

A

B

ABCD is a square. ABCD is divided into 25 congruent squares. Each square is divided into two congruent triangles. The triangles are right triangles. Look at the triangles MNO. Compare the number of triangles in the square on the hypotenuse with the sum of triangles in the squares on the two sides.
Rave pupils solve problems in which this property is used.

Review and reteach, if necessary, the material given in VII, A, in the guide for seventh grade.

Review concepts of points, lines and planes developed in the seventh grade. Stress the difference in picturing a line and a line segment.

Example:

~----------'>

A

B

The arrows show that the picture represents a line. AB, written AB, represents a line segment.

72

b. Angles c. Polygons 2. Intersections a, Intersection of sets
b. Intersecti(}nS of lines and planes Two lines

Review the properties of line and ray. Be sure pupils are aware that a ray extends in one direction only. Distinguish between a line and a ray.
Have pupils identify other sets of points which are called lines such as curved, broken, horizontal, vertical, oblique, parallel, perpendicular, intersecting. Call attention to the fact that in geometry "line" means a straight line.
Review definition. Have pupils identify various types of angles (straight, right, acute, obtuse, complementary, supplementary, adjacent, central, vertical).
Review ways of reading angles. Point out that the length of the sides of an angle does not determine the size.
Discuss formation of polygons and kinds of polygons. Save further development for a study of geometric forms.

() means intersection. Sometimes referred to as cap.

Example of intersection of sets:
= Set K {I, 2, 3, 4, 5, 6} read the set K consisting of the numbers 1, 2, 3, = 4, 5, 6. These numbers are called elements of the set. Set L {2, 4, 6, 8}
= The intersection of set K and set L is the set M {2, 4, 6}, written as
KnL is {2, 4, 6},

Make up other examples.

Example of an empty set.
= Set K {I, 2, 3, 4, 5, 6} = Set L {7, 8, 9}

= The intersection of sets K and L is a set with no elements, called the
empty set, cp. If set M is the intersection of K and L, M cp and sets K and L are disjoint sets.

Give the following problems:

= = If A {John, Tom, Gilbert, James} and B {Tom, James, Ray, Waldo}
= Find the set of boys who belong to both A and B. An B {Tom, James}

= = If A {2, 3, 4} and B {4, 5, 6}

Find A()B.

AnB = {4}

= If A {7, 8, 9} = B {2, 3, 4}
Find AnB

Make up other examples.

Additional material may be found in a number of books.

~

~

AB and CD are two lines. There are three possible intersections of these lines.

~ ~= AB n CD one point
The lines intersect. They are in the same plane (coplanar).

AC:<B

~

~

ABnCD

plane.

D

cp A<----> B They do not intersect. They are in the same

C<

>D

= ~

~

AB n CD cp They do not intersect. They are not in the same plane. They are

called skew lines.

73

A line and a plane

The intersection of a line and a plane may contain no points, one point, or many points.
Examples: Let the cord represent a line and a sheet of paper represent a plane.
.elL
li-A---.B -.
= L n ABC a line composed of many points
c.
B.

= L n ABC one point
AB is a line in plane KLM (top of box) Plane xyz is the bottom of the box
= AB n plane xyz <p. They have no points in common; therefore their inter-
section is the empty set. (See Fig. II)
The intersection of two different planes is a straight line if their intersection does not form an empty set. I
A
........- - - o t - -...... B

Pland P2 are planes
= P l nP2 AB
c

")-

B -10_ .....

1 I

I

I

I

- ------"I ,,,

f!g. u.

Ml represents the top of the box and is a plane parallel to M2 which repre-

sents the bottom of the box.

= M1 ('\ M2

<p

Exercises:

Give three ways in which two lines may intersect. Give three ways in which a line and a plane may intersect. Give two ways in which two different planes may intersect.

74

3. Separations
4. Unions a. Union of sets

There are three planes in the figure ABC, DEF and Gill. XY represents the fold of the paper. Only the fold touches the table.
Answer the following questions:
How many planes in the figure? Do the planes intersect? How? Is XY in all three planes?
A plane separates space into two half spaces.
The set of points on the right side of plane P belongs to one half space and the set of points on the left side belongs to the other half space. The plane does not belong to either half space.
Plane P separates space into half spaces. P is the boundary of each half space.
If two points X and Yare on the same side of plane P the intersection of XY
= and plane P is empty. XY n P
The plane P may be divided into two half planes by a line.

ex

z.

.y

= If X and Yare on one side of L then XynL <p. If X and Z are on oppisite
sides of the line then xzn L is not .

The union of two sets contains all the elements which belong to at least one of the two sets. The symbol U means union. It is sometimes called cup.

Example: X

II
YZ w

XY U YZ = all points of XY together with all the points on YZ which is the
segment XZ.

If A = {2, 3, 4, 5, 6} and B = {2, 4, 6, 8} AU B = {2, 3, 4, 5, 6, 8} If A = {Amy, Venia, Bess, Ruth}

75

b. Angles

= and B = {Flo, Myrtice, Sue, Ruth}
then A U B {Amy, Venia, Bess, Ruth, Flo, Myrtice, Sue}

Exercises:

= = C {a, e, i, 0, u} D {b, d, f, h}

CUD =

.

= = S {Mary, Jane, Jo} T {Nancy, Gracie}

SU T =

.

Draw two lines whose intersection is a segment. Draw two lines whose intersection is empty. Draw two lines whose intersection is not a segment. Recall that a ray is a set of points with an end point. The union of two rays forms an angle.

~

~

BC and BA are rays intersecting at B

to form L ABC. The middle letter names

the angle. Point B is called the vertex.

C BA and BC are called sides of the angle.

A

c. Triangles
B. Working with points, lines, angles, circles, planes, and space. 1. Constructions
a. Perpendicular bisector of a line segment.
b. Bisector of an angle c. Perpendicular to a line
at a point not on the line. d. Perpendicular to a line at a point on the line.

The angle separates the plane into two parts as in the picture. X, Y and Z are three points not in a straight line. A triangle is formed when there is a union of 'XY, YZ and ZX. (Review meaning of union.) X, Y and Z are called vertices. XY, YZ, and' ZX are called sides.
Distinguish between the terms "construction" and "drawing." The former should be used only when a straight edge and compass are used. The latter term is used when measuring instruments such as a protractor and a ruler are used.
Review constructions studied in the seventh grade. The constructions suggested are those given in most eighth grade textbooks. Refer to the book for details of procedure. Review meaning of bisect. Point out that the bisector of a line may also be the perpendicular to the line. Bisect angles drawn in all directions, both acute and obtuse. Angles should be drawn in many positions in order to avoid associating the perpendicular with vertical lines only.
A perpendicular to a line is not necessarily the bisector of the line.
76

e. Inscribed polygons
Regular hexagon
Equilateral triangle Square Dodecagon
f. An angle equal to a given angle
g. Parallel lines

Define regular polygons. When a polygon is inscribed in a circle all the vertices are points on the circumference and all its sides are chords of the circle
Draw a circle of any radius. Using the radius divide the circumference into six equal arcs. Letter the ends of the arcs A, B, C, D, E, F. Draw chords AB, Be, CD, DE, EF, FA.
Use the same directions as in (1) except draw chords AB, BC, CA.
Connect the ends of two perpendicular diameters.
Draw circle. Divide circle into 6 equal arcs. Bisect each arc. To bisect an arc bisect the chord of the arc. Connect the ends of the arcs.
Ask students to develop designs of their own with ruler and compass.
Constructing an angle equal to a given angle is a prerequisite to constructing parallel lines.
A transversal is a straight line intersecting a set of lines. In the figure T is the transversal.

A----~~~-

B

C _-----h.:------D

2. Measuring angles 3. Concepts of symmetry

Angles 1 and 2 are corresponding angles Angles 2 and 3 are alternate interior angles Angles 4 and 5 are alternate exterior angles
= If AB n CD then AB is parallel to CD.
Construct several sets of parallel lines cut by a transversal and measure the angles.
Let students discover which angles are equal or supplementary if lines are parallel. Point out the vertical angles, alternate interior angles, alternate exterior angles, supplementary angles, corresponding angles.
Review the use of the protractor. Have pupils measure and construct angles of all sizes.
Have pupils measure the three angles of a triangle to discover the sum of the angles of a triangle.
Fold figures to show there are two identical halves. Example: Draw on board to show axes of symmetry.

/ /
Draw the axes of symmetry of the following polygons: equilateral triangle, isosceles trapezoid, square, rectangle, rhombus, regular pentagon.
77

4. Concepts of congruency
5. Similar triangles
a. Use of similar triangles in indirect measurement
C. Understanding geometric forms and their measures 1. Polygons of 3, 4, 5, 6, 7, 8, 9, 10, and 12 sides a. Relationships in polygons

Have students copy some pictures from board or book that have symmetry. Some figures will have more than one axis of symmetry. If there is a point common to all axes this point is called the center symmetry.
Look for symmetry in nature, architecture, materials for clothes and draperies.

~

I

Before congruent (pronunced kong groo ent) triangles are defined show

students two objects which are alike, such as two basketballs, two number

two cans of beans, two textbooks of a certain kind.

Ask the question "Is there a time in the business world when items of the

same size and shalJe are necessary?" Example: parts for a television washing

machine.

'

Construct two triangles with base 1" and two base angles = 55 and 45. Cut out one, place it over the other to show they fit exactly in every way.

Construct two triangles which are 2" on a side. Cut out one triangle and place it on the other triangle to show that they fit exactly in every way.

Ask the questions, "Is it necessary to know three sides and three angles in order to construct a triangle?" "What determines a triangle?"

Discuss marking equal parts in congruent triangles.

Triangles are similar if their corresponding sides are proportional and their corresponding angles are equal.

Point out differences between congruent figures and similar figures by asking the following questions: How does a boat compare to a scale drawing of the same boat? Are all squares similar? How do the backs of all eighth grade arithmetic textboods compare? Are all rectangles similar? Are all isosceles triangles similar?

Discuss with pupils the need for finding distances or lengths which cannot be found by direct measurement. The class may think of examples of this need.

Example:

A tree casts a shadow 50 feet long when a 17 foot pole casts a shadow 25 feet long. By using similar triangles, find the height of the tree.

Have on hand pictures and models of geometric figures. Ask pupils to identify them.

Ask pupils to identify geometric forms which they see in everyday life.

Pupils have become familiar with some of these polygons in making constructions and will remember others which they studied in the seventh grade. Help them make charts similar to the one on polygons in VI, B in the guide for seventh grade.

Lead pupils to look for relationships in polygons, such as:

b. Triangles c. Quadrilaterals

The number of diagonals drawn from one vertex of any polygon is three less than the number of sides of the polygon.
The number of triangles formed in any polygon by drawing diagonals from one vertex is two less than the number of sides of the polygon.
The number of degrees in the sum of the angles of any polygon is ISO" multiplied by a number which is two less than the number of sides of tbe
a=: polygon.
Discuss the different types of triangles: isosceles, equilateral, scalene, obtuse, right and equiangular. Pupils should be able to identify or draw type.
Discuss the different types of quadrilaterals: rectangle, square, paralle.1ogtll!lt rhombus, trapezoid, isosceles trapezoid. Pupils should be able to IdeJI and draw each type.

78

If

d. Parallelograms
2. Perimeter of polygons and circumference of a circle
3. Areas of plane figures
Rectangle Square Triangle Parallelograms. Trapezoid Circle Regular hexagon

Have pupils construct parallelograms and measure the sides and the angles in order to discover properties of parallelograms.
Reteach the meaning of perimeter. Use material given in the textbook to review the formulas for finding perimeters of various plane figures and solve problems which involve the use of these formulas.
Review the vocabulary needed for working with circles and reteach the formulas for finding the circumference of a circle.
Refer to VI, B In the guide for seventh grade. Reteach this material concerDlng area, if necessary.
Extend pupils' understanding of the use of area formulas. Give sufficient experience in solving problems which require the use of these formulas.

Pupils will need to develop the formula for finding the area of a regular hexagon, since this has not been introduced previously.

E

D

F~----~f------~a

4. Solid geometric figures a. Introduction
b. Regular Polyhedra

A

B

= A 6. lh sa
Superior students should be able to generalize concerning the area of any regular polygon.
= Let n number of sides = p p perimeter = a length of altitude = s length of one side
= Then A n.% sa = But n.a p
."" A = % ps

Review material presented in seventh grade guide. Be sure that students can recognize all the familiar solids. Explain, using models, the difference between right and oblique prisms. Review vocabulary: vertex, edge, face, base, surface area, volume, etc.
Whenever it is possible to do so, allow students to experiment and discover for themselves formulas which they will use to solve problems. Give enough problems to be sure that students can use the formulas with ease and understanding.
The faces of a regular polyhedron are congruent regular poylgons. These faces may be equilateral triangles, squares, or regular pentagons.
79

_

. ._ _-

.a.......- - - - - - _
There are only five regular polyhedra (tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron). Projects: (1) Make models of regular polyhedra of straws and cord. (2) Make models of cardboard.
Example: Regular tetrahedron
o

c. Surface areas Cylinder

M

N Fold on the dotted lines and fit thl' vertices M, N, and 0 together with tape.

Prisms: In addition to the rectangular solid and cube which were developed in seventh grade, treat prisms that have other bases. Stress the fact that surface area of a solid is the sum of the areas of the various plane figures which make up the solid. Instead of developing formulas here it might be well to help students to discover what dimensions they would need to know in order to find surface area.

In the case of the cylinder, a model which can be separated into three parts will be helpful.

Sphere
d. Volumes Prism Cylinder Pyramid Cone

If students can be helped to discover that the base of the rectangle equals the circumference of the base of the cylinder, they should have no trouble
= arriving at the formula: S 21lTh + 2".r2 = By applying the distributive law, can also be written S 2".r(h + r).
Development of the formula for the surface area of a sphere is too difficult for this age student, but an intuitive idea of its correctness can be gained from the following experiment:
Take a solid ball and cut 1t into two hemispheres. Wind heavy cord around the curved surface of one hemisphere and the base of the other. Compare the lengths of the two pieces of cord. Since the piece wound around the curved surface is twice the length of the other, that area must be twice the area of the circular base. Therefore the area of sphere must be
S = 4".r2
In developing the formulas for volumes of prism and cylinder stress the idea that the area of the base gives volume for "one layer" and the height gives the number of layers. So the formulas are:
Prism: V = Bh (where B represents area of base)
= Cylinder: V ".r2h ("'r2 represents area of base)
If students have models so that volumes of pyramid and prism and of cone and cylinder can ve compared by filling pyramid and cone and pouring into prisms and cylinder, they can easily derive formulas for pyramid and. cone:
Pyramid: V = 1/3 Bh
= Cone: V 1/ 3 ".r2h
80

Sphere

An intuitive idea of the correctness of the formula for the volume of the sphere can be gained from the following experiment:

The materials needed are a hollow sphere and cylinder with the cylinder

sphere with fine sand or some similar substance. Pour the sand from the sphere

to the cylinder.

Note that the cylinder is 2/ ~ full.

Therefore the volume of the sphere is 2/3 the volume of the cylinder or

= so V 4/ ~1rrs.

= = V 2/s1rr2h, but h 21'

VI. Statistics And Graphs

A. Collecting data

Refer to IX in 7th grade material.

B. Types of graphs

Refer to IX B in 7th grade material. Discuss and construct these graphs in a little more sophisticated manner.

Take up the frequency distribution along with the histogram.

1. Broken line 2. Bar graphs
a. Horizontal bar graph b. Vertical bar 3. Pictographs 4. Histograms 5. Circle graphs

C. Summarizing data

Refer to IX C in 7th grade material.

1. Kinds of averages

Call attention to the phrases upper quartile, lower quartile.

a. Mean b. Median c. Mode

VII. Introduction To Algebra

A. The number line

Review the positive integers on the number line Section lIB, 1, 2

1. Properties of the number line

Show that the rational numbers (non-negative) on the number line are located in order of increasing size.

Examples: 6 > 4 .'.6 is to the right of 4
2 < 5 .'.2 is to the left of 5

2. Graph of a set of numbers

Explain that 0 is the point of reference called the origin. It divides the line into two half-lines, the half line to the right is the positive one. All numbers >0 are on this half-line and are called positive numbers.
Emphasize that for every number there is a point on the number line. Now bring out the fact that for every set of numbers there is a set of points on the number line.
Examples: A {2, 4, 6, 8}
0 123 45 67 8 90 all numbers> 2
0 123 45 67 8 90 all numbers between 2 and 5
0 123 45678 9 0 3 < all numbers <8

81

3. Addition on the number line
4. Applications of the number line
B. Negative rationals

~

~

0 1 2 3 4 5 6 7 8 9 10

Refer to C1 of this section.
= Examples: 2 + 4 6

2

4

~-..A---,

0 1 2 3 4 5 6 7 8 9 10
List the uses of the number line (ruler, thermometer, profits, losses, altitudes, longitudes, time before and after a certain event).

Discuss profit and loss to lead students to think about negative rationals which follow in B.

Lead the students to see that there is a need for negative numbers. School Mathematics Study Group (S.M.S.G.) recommends that a raised hyphen be used to denote a negative number, as 1, 2, 3. Read C2) as "negative 2."

Point out that the negative numbers are as real and as useful as the positive numbers. Positive numbers may be written without sign or with plus signs preceding them.

Graph a set of numbers on the number line.
2 < (all numbers) < 6

C. Operations with rational numbers
1. Addition of rational numbers

210 123 4 5 6 7 8 9 -2 ::;:, (all numbers) ::;:, 6
-2 -1 o 1 2 3 4 5 6 7 8 9
Refresh the memory of students by adding positive numbers by use of the number line.
= = Recall that a + 0 0 + a a for any rational number (called additive
identity), that 2 + (-2) = 0 (called additive inverse), and that a set of integers
includes the negative number, 0 and the positive integers (counting numbers).
Show by examples how to add a positive and negative number.

Example:
2 + (-7) = 2 + [-2 + (-5)J = + + [2 (-2)J (-5) == + 0 (-5)
-5

+ -2 (-5) is another way of
writing -7 Associative property additive inverse additive identity

Generalize: x + (-y)
and y>x.

-(y - x) if x and yare positive rational numbers

= + Example: 7 (-2)

(5 + 2) + (-2) 5 + 2 is another way of writing 7

= + + 5 (2 (-2) associative property

= 5 + 0 additive inverse

= 5 additive identity

= Generalize: x + (-y) x - y if x and y are positive rational numbers and

x> y.

Example:
= (-5) + (-2) -7. Show this on the number line.

'-
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
= Generalize: (-x) + (-y) -(x + y) if x and yare positive rational numbers-

82

1

2. Multiplication of rational bers a. Multiplication of a negative number by a posi tive number
b. Multiplication of a negative number by a negative number
3. Division of rational num bers
4. Subtraction of rational num bers
D. Equations and inequalities
1. introduction

Review the fact that any number multiplied by 1 is the number itself if the number is positive.

Show that this is also true of negative numbers. Therefore, 1 is called the identity for multiplication.

= Show that -7. (+ 1) -7 by
1 7 = 7 1 = 7 ... 1 (-7) = (-7) 1 -7

= Show that -7 (-1) 7 by
7 + (-1) 7 which can be written as
+ = + 7 (-1) 7 1 7 (-1) 7 = + 7 (1 (-1)
= 70

= .".

7

+

(-1)

7

=0 0,

then,

(-1)



7

is

the

additive

inverse

of

7.

The

additive

= inverse of 7 is -7 .". it is true that -7 (-1) 7

= Generalize: -x 1 -x
.". -x Y = -xy

= + Show that -1 (-1) + = + -1 [(-1)

1 (-1)]

(-1). (-1)

(-1).1

= -1 (-1 + 1)

= ~ 1 (0)

.". (-1) (-1) is the additive inverse of -1

= + .". (-1) (-1) must

1

= Generalize: (-x) (-y) xy

Teach division of rational numbers from the standpoint that division is the inverse of multiplication.

Examples:

= = If 6 (-3) -18 then -18/ 0
= = If (-5) (-4) 20 then 20/ _~ = = If 2.9 18 then 18/2 9

-3 or -4 or

= = Generalize: If x and yare positive numbers such that ky x then k x/yo = = If ky -x then k -x/yo
= = = If k (-y) -x then k -x/-y, k x/y

Teach substraction of rational numbers by use of examples.

6-4=2. We know this because 2 added 4=6.

3-7= 0 We say "What number added to 7 will give 3?"
7 + 0 = 3 We know from addition of negative numbers that -4 goes

in the box.

6 - (- 3) = D. We say "What number added to - 3 will give 6?"
= - 3 + 0 6. We know from addition of negative numbers that 9 goes

in the box.
= -3 - (+6 D. We say "What number added to 6 will give -3?"
-3 + 0 = -6. We know from addition of negative number that -3

= goes in the box.
- 4 - (- 5) D. We say "What number added to - 5 will give - 4?"

= - - 5 + 0

4. We know from addition of negative numbers that 1

goes in the box.

Generalize these four types of subtraction.

Discuss closure at this point.

Note that the positive integers are closed under addition and multiplication, but they are not closed under subtraction or division.

The teaching of this section will require careful planning. Lead pupils to discover relationships for themselves.
The main objectives are to develop an understanding of the concepts related

83

2. Number phrases a. Numerical phrase b. Open phrase
3. Number sentences a. Symbols for verbs
b. Numerical sentences
c. Open sentences (1) Meaning
(2) Variables in open sentences

to solving equations and to give pupils a mathematical vocabulary which they may use later.
Discuss with pupils the need for equations and the way they are used in many fields.
Give examples of sentences about numbers. Review the meaning of phrase and show that phrases are not sentences. In a sentence about numbers a phrase is called a number phrase.
A number phrase which consists of a numeral, or a numeral which involves other numerals with the signs for operation, is called a numerical phrase. A numerical phrase represents a specific number.
Examples: 7; 3 + 5; 6 4; 2(5 - 2)
Ask what number is represented by x - 8. Students will see that x - 8 may have many values. A number phrase that does not represent a specific number is an open phrase.
Examples: y + 3; x/5; 3x.
Give students opportunities to translate number phrases from words to symbols and from symbols to words.
Review the meaning of sentence. Define a number sentence as a sentence which consists of two number phrases connected by a verb. Be sure students understand the following symbols:
means "is" or "is equal to." oF means "is not" or "is not equal to."
< means "is less than."
<I:: means "is not less than."
> means "is greater than."
:J> means "is not greater than." ~ means "is less than or equal to."
> means "is greater than or equal to."
Give this definition: A numerical sentence is a sentence composed of numerical phrases. A numerical sentence may be true or false.
Examples:
4 + 6 = 5 2. This sentence is true. 5 > 6 + 4. This sentence is false.
Give students opportunities for making numerical sentences and allow them to ask other students to tell if their sentences are true or false.
Ask if this sentence is true or false: x + 7 = 10. To what number does x refer?
= = This sentence may be true or false. If x 3, it is true; if x 6, it is false.
Definition: If a number sentence involves a symbol which can refer to any one of many numbers, the sentence is called an open sentence.
Examples:
x+4=8
y >3
Have students translate these and other similar sentences into words.
Call attention to the fact that x and y in the illustrations used here are called variables.
Definition: A variable is a symbol (It may be a letter, geometric figure, question mark, etc.) used to denote one of a given set of items.
In such sentences as these a variable is a place holder for a numeral which is a name for a number.
Note: If this concept of the term variable is used, it frees students from vague descriptions of variables as a "literal number," "a general number," or "aD unknown quantity."
84

(3) Set of solutions (4) Nature of equations
and inequalities
4. Solution sets of equations
a. Using the addition property

The set of solutions is the set of numbers which make the sentence true.

When we find the entire set of solutions of an open sentence, we say that we have solved the sentence.

If the verb in a number sentence is "=", the sentence is an equation.
If the verb in a number sentence is "<" or ">", the sentence is an inequality.

Have students translate number sentences from words into symbols and from symbols into words. Include statements of equality and inequality.

Consider a statement of inequality and ask if the set of solutions may be determined.
Examples: x - 7 > 5

Ask:How large must the number x be for the statement to be true? Have students try several numbers. Is the inequality true for any value of x greater than 12?

Caution: It is most important that the artificial method called "transposition" not be used in solving equations. A number cannot be "carried" from one side of an equation to the other. Whatever work is done here should be according to sound mathematical principles.

Give students a list of equations and ask them to solve as many as they can by inspection or trial. Include some which are easy to solve and others which are not.

Suggest that there may be a method which would make finding the solution easier.

Proceed as follows: Suppose a and b represent the same number. For example, suppose a is 6 + 2 and b is 5 + 3. Ask: What is a + 4? What is b + 4? If a and b both represent 8. then a + 4 and b + 4 must both be 12.

Then ask: If a = b, does a + 6 = b + 6? If a = b, does a + (-3) = b + (-3)?

This suggests the general principle which is called the addition property. It is stated: If you have two equal numbers and add a third number to each of them, the resulting numbers are equal.

In symbols, if a, b, and c are numbers and a = b, then a + c = b + c and c + a = c + b (by commutative property of addition).

Now apply this property in solving the equation:
x + 6 = 14

Using the addition property, adding the number - 6, we have:

a

b

~~

Ifx+6=14

a+c

~

~

b +c

~

~

then x + 6 + (- 6)

14 + (- 6)

By the associjative property and the fact that 6 + (- 6) = 0 we see that:
(x + 6) + (- 6) = x + (6 + (- 6)) = x ..(x + 6) + (-6) = 14 + (-6)
x = 14 + (- 6)
Since 14 + (- 6) = 8
x=8

Using the addition property, if the equation x + 6 = 14 has a solution, the solution must be 8. However, it must be shown whether or not 8 is the solution set.
Ifx=8 then x + 6 = 8 + 6
so x + 6 = 14

85



c

b. Using the multiplication property

Therefore the solution set is 8; or if S represents the solution set, S = {8}.
Notice in the process of solving the equation two things were done:
We proved a uniqueness statement, showing that there is only one possible solution.
We proved an existence statement, showing that this number is a solution.
(Note: This procedure in the use of the addition property is given here in detail, as found in SMSG, Vol. II, Part 1., Preliminary EditionJ
Ask students if they can discover a subtraction property. Teachers may want to spend time on this idea. Most textbooks use what is called "subtraction axioms." However, the addition property may be used instead.
= Consider examples like these: 3 + ( -4) -1 = = also 3 -4 -1. Then 3 + ( -4) 3-4
6 + ( -2 ) = 4
= and 6 -2 4 then 6 + ( -2 ) = 6 - 2 = -4 + ( -2 ) -6 = and ~ 4 -2 -6
+ then. -4 (-2) = -4 -2
= Generalize: If a and b are numbers, then a - b a + (-b).
Using this principal solve:
= x - 8 10 = = Write the equation as x + (-8) 10 (x + (-8 + 8 10 + 8 by the ad
dition property, adding 8.
x + ( (-8) + 8) = 18 by the associative property of addition.
x = 18 since (-8) + 8 = 0 = = = If x 18, then x - 8 18 - 8 10 so 18 is a solution.
Proceed as follows:

Suppose a and b represent the same number. If we multiply a by 6 and b by 6, could 6a and 6b be two different numbers if a and b are the same?

Give several examples showing this relationship. Then, state the multiplication property: If you have two equal numbers and multiply them by a third num ber, the results are equal.
= = = If a, band c are numbers and a b, then ac bc or ca cb (by the com
mutative property of multiplication.)

Ask: Can you discover a division property of "equals." Time may be given to this development. However, its use is unnecessary, because the multiplication property using reciprocals will be sufficient.
= + Consider the solution of: 3x 2 11 = + 3x 11 (-2)

(By the addition property, adding -2).
= Then 3x 9

ab

= ,.-"--.. ,.-"--..
If 3x 9

c

a

r.A-....,

,.-"--..

then 1/3 (3x)

1/3 9

(By the associative propertity of multiplication)
= = = 1/3 (3x) (1/3 3)x 1 x x = Since 1/3 9 3
= x {3}

86

= If 3x + 2 11 has a solution, then it must be 3 (a uniqueness statement).

If x = 3 then 3x = 3-3 = 9

and 3x + or 3x + 2

2==119

+

2

Therefore 3 is the solution set (an existence statement).
= Therefore S {3}

(Note: This procedure in the use of the multiplication property is found in SMSG, Vol. II, Part I, Preliminary Edition)

Students should find the solution sets of various equations using these properties.

87

SEVENTH AND EIGHTH GRADE VOCABULARY

acute angle acute triangle adjacent angles alternate interior angles alternate exterior angles altitude angle analyze approximate number arc area ascending order assessed value associative property or law a-,rerage (mean) axis
bank balance bar graph barometer base (in geometry) base (in a positional number system) ba3e (in percentage) beneficiary betweenness binary binary operation bin2ry system bisect bisector board foot braces broken line budget
cancelled check capacity carrying charge cash balance central angle check (bank) check stub checking account chord circle circle graph circumference circumscribe closed curve closure coefficient commission commutative property or law compare compass complementary angle composite number compound interest computation

concentric circles cone congruent triangles consecutive converse coordinate coordinate axes corresponding angles counting number cubic curve cylinder
data date of maturity decagon decimal fraction decimal notation decimal place decimal point decimal system deduction degree denominator denominate number dependent depo3it slip depreciation descending order diagonal diameter digit dimension directed numbers discount distributive property or law divided bar graph dividend divisibility dodecagon dodecahedron down payment duodecimal
edge (cube) element employee employer empty set endorse equation equiangular equidistant equilaterial equivalent equivalent fractions even number exact number excise tax

exemption expanded notation expanded numerals expenditures exponent
face (geometry) face (promissory note) face value factor factorization finite formula fraction frequency distribution
gram gross profit gross sales
half-line heptagon hexagon hexahedron histogram hypotenuse
icosahedron identity element inequality indirect measurement infinite inscribed installment price insurance integer interest interest formula inverse isosceles triangles
kilogram kilometer
lateral area line line segment linear measurement list price liter
margin mean median meter metric system mill mode mortgage multiple

88

natural numbers negative negative numbers net price nonagon notation number sentence number phrase
obtuse obtuse angle obtuse triangle octagon octahedron odd number open sentence
paraliel parallelogram pentagon per cent percentage period perimeter pi pictograph place holder place value plane polygon

power precision premium principal prime number prism profit proportion pyramid quadrilateral radius rate ratio rational number ray reciprocal reconciliation rectangle regular polygon rhombus right angle
savings account scale scale drawing scalene triangle sector segment semicircle semiannually

set signature card significant digit similar polygons solution set space sphere square square root statistics stock straight angle supplementary angles symmetry
tangent trapezoid transversal triangle
unique
variable vertex vertical vertical angle
watt withdrawal
zero

89

SEVENTH AND EIGHTH GRADE

Films And Filmstrips
Films

Title:

State Catalog Number

Angles

4254

Areas

.____________________________________________________________________ 4356

Circle, The

3185

Geometry and You

.

2116

Language of qraphs

.. .

.

2975

Language of Mathematics

.

.________________________ 2230

Meaning of Percentage _. .

._. .

.

...

5003

Meaning of Pi

.

.

. 2231

Measurement

2047

Measuring Simple Areas

4256

Origin of Mathematics

.

3256

Percentage

.

. ..

2231

Per Cent in Everyday Life

.__.

.

.__. ._____________________ 2296

Polygons .

..

.

.

.

._____________ 3183

Principals of Scale Drawing (Coronet Films)

Pythagorean Theorem

.

.

.. .______ 3189

Quadrilaterals . ._. . ..

.

.

. 4467

Ratio and Proportion ._..__.__________

4456

Story of Our Number System

2603

Story of Weights and Measures

.

.

2460

The Metric System (Coronet Films)

Filmstrips

SVE - Plane Geometry Series

Title: Areas Basic Triangles Introduction to Circles

.

.._.

.

.

.

.

.

.

.__.

Number: A-541-8 A-541--3 . A-541-10

Quadrilaterals _. . ._.__._.

..

.__________________________ A-541-10

SVE - Understanding and Using Numbers

Addition and Subtraction of Decimals

.

.

A-538-2

Advancing in Linear Measurements

A-538-6

Advancing in Quantity Measurements

A-538-7

Buying and Selling Application of Per Cent

. .___________________ A-539-2

Changing Fractions to Decimals and Decimals to Fractions

Commision, Meaning and Application

.

._____________ A-538-5 A-539--3

Division of Deoimals __. .... ..

. ---._._.._.

.

. A-538-4

Federal Taxes __.__.

.

.___________ A-539-7

Insurance .

.

A-539-5

Interest - Borrowing and Investing

.

.

A-539--4

Meaning and Reading of Decimals

.

A-538-1

Meaning and Understanding. of Per Cent, Percentange

A-539-1

Multiplication of Decimals

.

A-538-3

State and Local Taxes . .

. ..

A-539-6

Curriculum Films - Decimals and Percentage

Adding and Subtracting' Decimals

.

Comparing Decimals

.

._______________________ 213 . 212

Decimals and Common Fractions

211

introduction to Decimals

210

Introduction to Percentage

Multiplying Decimals

.

Problems in Percentage

Using Percentage

216

.

. 214

218

._. . ._________________________ 217

Others:

A Study of Measurement Photo and Sound Production (6 35-mm filmstrips)
90

EQUIPMENT AND CONCRETE AIDS
Abacus Almanac
Area board (to show relationship of square inches to square feet) Blocks, cubic foot, dissectable Blocks, one inch cubes of various colors Calipers (micrometer) Calipers (vernier) Carpenter's folding ruler; T-square Chalkboard drawing sets Charts, place value, deoimal and common fractions Clothespins Coat hangers Colored chalk crayons Compass chalkboard, metal and wooden Compass, magnetic Counting frames and grouping aids, demonstrational Counting frames and grouping aids, student use Cube, hollow, 10 inches on a side Disks, approximately 12 inches in diameter
cut from 3/8 inch plywood and divided to show halves, thirds sixths, eights, and twelfths Draftman's T-square, and triangle Facsimiles of blank checks Flannel board Fraction devices, common, decimal and percentage Fractional wheels Geometric figures, plane Geometl'ic solid models Globe Graph chart (rectangular coordinates) Hundred and hundredth boards Maps Meterstick Measuring devices - cup, pint quart, etc. Measures, Metric Mea:mr,ing disk for determining "pi" Micrometer and a steel rod, approximately a half inch in diameter Model of a cubic yard Number games Odometer Paper punch Paper stapler Parallel ruler (chalkboard size) Place value board Plane figures, ruler sets, chalkboard, stencils Plane figures, ruler sets, one for each student Polygons, pattern dial for drawing, one for each student Protractor, chalkboard, complete circle Protractor, chalkboard, semi-circle Protractor, one for each student Rulers, (English and metric scale) Rulers, decimal (to tenths -and hundredths of a foot) Scales, balance, English system Speedometer, having a dial that can be set at zero. Straight edge, chalkboard Sun dial Tape measures, linen or paper; metric and English; steel decimalized Thermometer, Centigrade Thermometer, Fahrenheit Timetables (bus, plane, train) Wall chart (metric measure) Yardsticks
91

ELEMENTARY ALGEBRA

F

UNDERSTANDINGS TO BE DEVELOPED IN ELEMENTARY ALGEBRA
The basic understanding to be developed in this course is of the algebraic structure of the real number system and its use as a basis for the techniques of algebra. The traditional understandings are developed in this course outline within the framework of the real number system. Concepts not usually introduced in the first course in algebra include inequalities and functions. Pupils should understand the relationship between arithmetic and algebra - that the properties which allow algebraic manipulation are just the familiar properties they have always used in arithmetic, that introduction of variables transforms arithmetic into algebra. They should begin to develop an understanding of the importance of mathematical definitions and the nature of proof. They should, by the end of this course, be able to use the language of mathematics.
95

CONTENT

TEACHING SUGGESTIONS

I. Sets And The Number Line

The concept of "set" has been described as one of the great unifying and simplifying ideas of all mathematics. It is as such that it is introduced here. Emphasis is on the use of the language of sets rather than a formal course in set theory. Hence, begin with only those terms which will be used immediately, bringing in the ideas of "intersection" and "union" of sets later in the course when they have some direct application to the concepts being developed.

A. Sets

A set is a collection of elements with a common characteristic. This characteristic may be only the fact that they are listed together.
{l, 2, 3, 4, 5}

1. Sub-set and proper sub-set

How would one describe this set? Given any element, can one tell whether or not it belongs to the given set? For example, six is not a member of the set consisting of the first five counting numbers. A set is well defined if it is possible to tell whether or not any element belongs to the set.
Suppose set A = {l, 2, 3, 4, 5} and set B = {2, 3, 5}. Set B is a sub-set of set A.
if every element of B is an element of A. From this definition it is evident that A is a sub-set of itself. If there is some element of A which is not an element of B, then B is a proper sub-set of A.

2. Symbols

Usually braces { } are.used if the members of a set are listed. If one wishes to talk about the set as a whole, it is customary to name the set with a capital letter.

3. Null set

Examples can be given to explain to the student the meaning of the null set. The symbol used is <p. Braces are never used with this symbol, since the null set has no elements. Students should not confuse the null set with the set {O}. This'is a set consisting of one element, the number zero.

4. Finite and infinite sets

A set is said to be finite if it is the null set or if the elements of the set can be counted, with the counting coming to an end. Any other set is called infinite.

B. Number line

Another concept which is of use throughout the course is the number line. It is a device for picturing many of the ideas about numbers and operations on them.

1. One-to-one correspondence
a. Successor 2. Graph of a set of num-
bers

o

1

Choose two points on a line and assign to them the coordinates "zero" and "one," then have students assign to other points numbers with which they are familiar. It is suggested by authorities that negative numbers be left for later development unless students have already been introduced to them. Practice in applying properties of operations with familiar numbers, then developing from those properties an extension to include the negative numbers is believed to be helpful in avoiding misconceptions which students frequently have.

The idea of one-to one correspondence between the points of the number line and the real numbers, and the definition of coordinate of a point should be developed. The emphasis here is on the fact that coordinate is the number which is associated with a point on the line.

Introducing the idea of successor of a natural number should lead to the pupil'S realization that there is no last number and to the formulating of a rule fOf finding the successor of a number.

Introduce now some uses of the number line. If graphing is done in one dimension, then two-dimensional graphing will be more easily understood.

set {l, 3, 5, 7}

Graph
o 1 2 3 4 5 678

{all numbers between 0 and 5}
96

3. Addition and multiplication on the number line

{all numbers between 0 and 5 inclusive}

o1 23 4 5 6
(0 and 5 included)

Define closure for addition and multiplication of the natural number~. The fact that the set of natural numbers is not closed under division will help to give meaning to the closure property.
Addition 2 + 3 2+3
,-----'~
o 123 456

Multiplication 2 X 3
3 X3
,------'-r--"----...
o1 2 3 ~ 5 6

II. Numerals And Variables

A. Numbers and numerals

The purpose of this section is to distinguish between numbers themselves and the names for them and to introduce the idea of a phrase.
The names for numbers, as distinguished from the numbers themselves, are called numerals. Numbers have a common name, which is the one most often used in referring to the number. Two numerals, for example, which represent
the same number are the indicated sum "6 + 2" and the indicated product
"4 X 2." The statement that these two numerals represent the same number
= + can be written more briefly as "6 2 4 X 2." An equals sign standing be-
tween two numerals indicates that the numerals represent the same number.
Ideas of indicated sum and indicated product are very useful, especially in discussing the distributive property. They should be stressed here to counteract
the tendency to regard "6 + 2" not as another name for the number eight but
as a command to add two and six.

1. Use of parentheses
2. Order of operations
3. Numerical phrase
4. Numerical sentence
a. Properties of addition and multiplication Addition Binary operation

Sometimes parentheses are needed to enclose a numeral in order to clarify the fact that it really is a numeral. The use of parentheses could be compared to the use of marks of punctuation in English. Emphasis should be on their use to avoid ambiguity, not on manipulation with them.
Suppose we have an expression like "2 X 5 + 3." It this a numeral? If students
say "yes" they probably not agree as to what number it represents. Stress the fact that a numeral must represent a definite number, which is the reason that some agreement must be made as to the order in which the operations will be performed.
Any numeral given by an expression which involves other numerals together with the signs for operations is called a numerical phrase. The emphasis here should be on the fact that a phrase is itself a numeral. The use of the word numerical is to distinguish this phrase from the "open" phrase which is introduced later.
Numerical phrases when combined to make statements about numbers are called numerical sentences. A numerical sentence may be either true or false.
= For example, 2(5 + 4) 6 X 3 can be shown to be a true sentence, and it can = easily be verified that (3 + 1) (6 + 2) 30 is false.
Numerical sentences can be used to review the properties of the operations on the numbers. Emphasis here should be on structure. Usefulness might be pointed out, but the main interest is in these properties as the foundation on which the whole subject of algebra is built.
Re-emphasis of the property of closure should be made.
Addition is always performed on only two numbers at one time. Ask students to try to add the numbers 4, 7, and 3 simultaneously.

97

Associative property Commutative property b. Corresponding properties for multiplication c. Distributive property
B. Variables 1. Definition
2. Domain

When 4 + 7 + 3 is written, it means either (4 + 7) + 3 or 4 + (7 + 3). This is considered to be one of the fundamental properties of addition and is written (4 + 7) + 3 = 4 + (7 + 3).
= 3 + 5 5 + 3. It might be well to point out that this property does not hold
for some operations.

Closure, associative, and commutative properties have the same form in multiplication that they do in addition.

Checking several examples will lead to discovery that the distributive property holds:
24+54=74
25(40 + 3) = 25(40) + 25(3)

Demonstrate that in some cases one form is easier to use than the other.

15(7) + 15(3) = 15(7 + 3) favors indicated product.

150(1/2 + 1/ 3) = 150(1/2) + 150(1/ 3) favors indicated sum.
You might point out that the distributive property is used in long multiplication.

24 X23
-----n 48
552

We take 24(20 + 3) as 24(20) + 24(3), or 480 + 72 (The zero at the end of 480 is understood in our form.)

Thus the distributive property is the foundation of the standard technique for long multiplication.

A variable is a letter used to denote one of a given set of numbers. In computation involving a variable, the variable is a numeral which represents a definite though unspecified member of some particular set of numbers.

You might point out to students that sometimes the pattern or form of a problem is more interesting than the answer. Take, for example, the number puzzle:

"Take a whole number no larger than 10. Multiply by 3, add 12, divide by 3, and subtract 4. Did you get the number you started with?" This might seem to be ten separate problems, but by studying the pattern we can show that it is only one. Suppose we choose the number three:

Arithmetic
3 9 21 7 3

Pattern
3 3(3)
3(3) + 12 3(3) + 12
3
3(3) + 12 - 4
3

When we study the final statement in our pattern we can see that, since
12 = 3(4) we can use the distributive property and write 3(3) + 12 as 3(3) +
3(4) and finally as 3(3 + 4), Following the rest of the indicated computation, we can see that our final numerical phrase is a numeral for the number 3.

Because it can now be shown that the pattern does not depend upon the number chosen, but would be the same for any of the ten numbers, we could write:

3n + 12 _ 4 - n

3

-

where n represents anyone of the specified set of numbers.
Im The set of numbers which a variable can represent is called the domain of e
variable. In the case above the domain is the set {1, 2, 3, 4, 5, 6, 7, 8, 9, .

98

III. Sentences And Properties Of Operations

A. Open sentences

Review truth or falsity of sentences. Then introduce a sentence containing a variable.

1. Definition

For example, x + 5 = 7. Is this a true sentence? If we knew the domain of x, could we tell whether or not this is a true sentence?
An open sentence is a s'entence involving one or more variables, and the question of whether it is a true sentence is left open until there is enough information given to decide.

2. Solution set

The set of numbers in the domain of the variable which makes an open sentence a true sentence is called the solution set or truth set of the sentence.

You can draw the graph of solution sets just as you previously drew graphs of sets of numbers.

3. Equations and inequalities

Though you do not call them by these names, you can introduce both equations and inequalities here.
x + 5 =7
x +5>7

4. Symbols

Be sure that the student understands the following:

= means "is" or "is equal to"
=F means "is not" or "is not equal to"
< means "is less than"
<I: means "is not less than"
> means "is greater than"
::t> means "is not greater than"
< means "is less than or equal to"

> means "is greater than or equal to"

B. Properties of addition and multiplication 1. Identity elements
2. Multiplication property of zero
3. Uses of property of one
4. Commutative, associative, and distributive properties

Students should be led to recognize the equivalence of <I: and> and ;j> and <.
By using open sentences and considering their solution sets, students can discover some additional properties of addition and multiplication.
For every number a, a + 0 = 0 + a = a (0 is the identity element for addition)
For every number a, a(l) = (l)a = a. (l is the identity element for multiplication.)
This is also called the property of 1.
For every number a, a(O) = 0

Suppose we wish to find the common name for the numeral 5/ 6 + 3/ s

~il __ 5(4)= 20and1-=~(I) = 3f3i = 3(3) =~

6 L4J - 6(4) 24 8 8

8L3J 8(3) 24

Then~+~=~+~=~ 6 8 24 24 24

Its usefulness for simplifying complex fractions may also be shown.

2

2

2

"3 2
5

:3 2X

(1)

3
-2

P5i
X LI5J =

10
"6

5

5

5(2) 3(2)

~(I) 3

5
:3

The introduction of variables enables you now to state previously mentioned properties of the operations in more general terms.

For all numbers a, b, and c

a+bxb+a

99

= ab

ba

a + (b + c) = (a + b) + c

a(bc) = (ab)c
a(b + c) = ab + ac

You can now show how these properties may be used to change the form of certain open phrases.

By the use of the properties of closure, commutativity, and associativity (3xy) (2x) can be written as 6x2y.

By the use of the distributive property certain indicated sums can be written as indicated products and vice versa.
x(y + 3) = xy + 3x 5x + 5y = 5(x + y) 3a + 5a = (3 + 5)a = 8a

Use of associative and commutative properties for addition together with the distributive property enables you to write an open phrase like:
3x + 2y + 7x + 6y in simpler form 3x + 2y + 7x + 6y = (3x + 7x) + (2y + 6y)
= (3 + 7)x + (2 + 6)y
= lOx + 8y

Use of this approach makes clear to the student why some terms can be added and others cannot.

IV. Mathematical Sentences And English Sentences
The purpose of this unit is to help develop some ability in writing open sentences for word problems.
Begin with phrases and give practice in changing mathematical phrases into English phrases and English phrases into mathematical phrases.
In work with sentences, practice should be given in writing open sentences for equations and for inequalities.
Sometimes statement of a problem in several parts will help the student to see the steps which are involved in forming an open phrase or sentence.
For example, choose a variable for the number of feet in the width of a rec tangle.
(a) Write an open phrase for the length of the rectangle, if the length is three feet less than four times the width.
(b) Write an open phrase for the perimeter of the rectangle described in part (a).
(c) Write an open phrase for the area of the rectangle described in part (a).
We should emphasize the fact that a variable represents a number. "x" is nol the "width" but the number of units in the width. Care should be taken to state precisely what number the variable represents.
In forming open sentences it is sometimes easier to see what the open sentence should be if the student guesses a number for the quantity asked for in the problem.
Example: We wish to cut a rope which is 28 feet long into two pieces so thal one piece will be three feet longer than the other. How long should the shorter piece be?

100

Suppose we guess 14 feet as the length of the shorter piece. Then the longer
piece is (14 + 3) feet long. Since the whole length is 28 feet, we have the sen-
tence
14 + (14 + 3) = 28.

We can show this to be a false sentence, but it points out the pattern which we need for our open sentence. Note that the question in the problem points out our variable. We can now say:
If the shorter piece is n feet long, then the longer piece is (n + 3) feet
long, and the sentence is
= n + (n + 3) 28.

v. The Real Numbers

The stud,:nt has been discovering and using properties of operations on a set of numbers which might be called the numbers of arithmetic. With this background he should now be ready to give names to numbers assigned to points on the left of O. This gives him the total set of real numbers with which to work.

A. Extension of the number line to include negatives

54

3

2 -1 o 1 2 3 4

It is assumed that the student is aware of the need for negative numbers (profit and loss, temperatures, gains and losses on a football field). As before, use the interval from 0 to 1 as the unit of measure and locate points equally spaced along the line to the left of O.

Use the raised dash to indicate the coordinate of a point, as " -I", to be read "negative one," reserving the word "minus" to indicate subtraction. At this stage the student is not to think that something has been done to the number 1 to get the number -1, but rather that .-1 is the name of the number which is assigned to the point located one unit to the left of 0 on the number line.

In order to be sure that a student has a true picture of the number system with which he is now working, it might be advisable to present a diagram showing relationship of the various sets of numbers.

1. Order on the number line a. Properties of order b. Comparison property c. Transitive property

Real Numbers

I

I
Rational Numbers

I
Irrational Numbers

I

I
Integers

I
Rational Numbers which are not

integers

I

I

I

Negative 0 Positive

Integers

Integers

A common misunderstanding is that some numbers on the line are real and others irrational. Until he has become accustomed to their use, a student should be encouraged to say that -4 is a real number which is a rational
number and a negative integer, -";3 is a real number which is a negative irrational number.

Clarify the meaning of the symbol < as applied to the entire number line. Make the agreement that it shall have the meaning, "to the left of," just as it did with the numbers of arithmetic.

By the use of numerical examples, lead the students to develop the properties of order.

If a and b are real numbers, then exactly one of the following is true:

a<b,

b<a

If a, b, c are real numbers and if a<b and b<c, then a<c.

101

2. Opposites

When points to the left of 0 on the number line were labeled, you used successive unit lengths to the left of O. You could think, however, of pairing off points at equal distances from and on opposite sides of O.

a. Opposite numbers
3. Absolute value a. Definition
B. Properties of addition 1. Definition

-2

-1

o

1

2

Since the two numbers in each pair are on opposite sides of zero, it seems natural to call them opposites. The opposite of a non-zero real number is the other real number which is at an equal distance from zero.

Let us use the symbol "-" to mean "is the opposite of." Then you can show that "is the opposite of" a positive number is the same as "negative" and make the agreement that you will not use both symbols, "-,, and "-" with the same number, but will use only "-". The symbol -(-a) would then be read "is the opposite of negative a" and it would follow from the definition that -(-a) = a.

It is believed that this approach would simplify the concept of negative numbers. Signed numbers would not have to be mentioned. No signs would be attached to the positive numbers. They are still exactly the familiar numbers of arithmetic. The symbol -a would mean the "opposite of a," and whether it is positive or negative would depend on the numerical value of a.

The operation of taking the absolute value of a real number is used in extending the definition of addition to include the negatives and should be introduced here.

The absolute value of a non-zero real number is the greater of that number and its opposite. The absolute value of 0 is O.

Careful study of the definition should enable students to discover for themselves that the absolute value of a non-zero number is always positive, and that the distance between a real number and 0 on the number line is the absolute value of that number.

It can then be shown that the absolute value of a positive number is the number itself, and the absolute value of a negative number is its opposite.

You should define now addition for the positive and negative numbers, so that addition of the positive numbers will not be changed, and the fundamental properties will be preserved.

First consider some examples using gains and losses to suggest how addition involving negative numbers should be defined. The number line can also be used to show this.

Negative direction

<--- --->

6 r--_""':3------" -----....,

Positive direction

~

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

Suppose students have the idea of profit and loss to arrive at the sum of

(-3) and 6. You might show them that they can get the same result on.the number line if they make this agreement: "Begin at zero on the number line.

Since the first number is negative move three units in the negative directioJl. This places us at -3 on the number line. The second number is po.sitive ~

now move six units in the positive direction. We see that we have arrIved at

= on the number line, so we can write (-3) + 6 3."

.

After considerable practice of this sort, the student should be reahdY_ formulate a precise definition. He should make use of the idea that t e

102

2. Addition property of opposites
3. Addition property of equality
.. Proof

ta~ce of any point from D is the absolute value of the coordinate of the pomt to develop his definition. It should take something of the following form:

1. If a and b are both positive, their sum is the sum of their absolute values. 2. If a and b are both negative, their sum is the opposite of the sum of their
absolute values.

3. If one of a and b is positive and the other negative, then the absolute value of their sum is the difference of their absolute values. The sum is positive if the number having the larger absolute value is positive, or negative if the number having the larger absolute value is negative.

Proof that the properties of addition are preserved when the negative numbers are included is too advanced for this age level, but use of numerous examples will convince students that this is true.

A useful property of addition which is new for real numbers is:
For every real number a,
+ a (-a) = D.

You might state this property in words: "The sum of a number and its op-
= posite is D." If two numbers x and yare such that x + y D then y is
said to be the additive inverse of x. Therefore (-a) is the additive inverse
of a and vice versa.

Some examples in which this property together with other properties of addition might be used are:
5/16 + 28 + (-5/16)

(-1/2) + 7 + (-2) + (-3/2) + 2

x = x + [(-x) +3]

n + (n+2) + (-n) + 1 + (-3) =D

For any real numbers a, b, c, if a = b, then a + c = b +c. This property has a direct application in finding the truth sets of open sentences. For
example:
= What is the solution set of the open sentence x + 3/5 -2? If there is a
number x for which this is true (if the solution set is not the null set) then x + 3/5 and -2 represent the same number. Point out to students that they have not found the solution set until they have actually checked the number in the sentence to prove that it actually does make the sentence true. Students should justify each step of the work that they do by writing beside it the property they have used.

Example:

= x + 3/5 -2
= (x+3/5) + (-3/5) (-2) + (-3/5) (Property of equality)

= x + [3/5 + (-3/5)] (-2) + (-3/5) (Associative property for addition)

= x + D (-2) + (-3/5) (Property of opposites)

= + x (-2) (-3/5) (Property of D)

= - x

(1-21 + 1-3/51) (Definition of addition)

x = -(2 + 3/ 5) (Definition of absolute value) x = -2 3/ 5 It might be wise at this point to define equation, and root.

It is believed by some authorities that ninth grade students are mature enough to understand formal proof and to prove some simple theorems. One of the principal reasons for this approach to algebra is to show that algebra, like geometry, is a logical structure and that the seemingly mean ingless manipulative techniques which were learned are based on the fund-

103

amental properties of numbers and can be proved to be logical consequences of these properties.
The method of instruction of this idea will determine whether or not the student will accept it. It probably will be better to start with a numerical example and then use that as a model in developing the abstract proof.
The idea might be developed in this way;
We know that (-2) is one additive inverse for the number 2. Do you think that there is another one? Of course, our experience with the number system makes us think that there is not, but suppose we wanted to prove this fact to someone who doesn't think as we do. Let's see if we can use the properties that we have accepted for our number such that 2 plus that number equals O.
Suppose we choose n as variable representing the additive inverse of 2. We can form the open sentence 2 + n = O. What we are really asking ourselves is "Is the solution set of this open sentence just {-2} or does it include other numbers?" We can now find the solution set of this sentence just as we did previously.
2 + n = 0 (Definition of additive inverse)
(2+n) + (-2) = 0 + (-2) (Property of equality)
(-2) + (2+n) = 0 + (-2) (OJmmutative property)
(-2 + 2) + n = 0 + (-2) (Associative property)
o + n = 0 + -2 (Property of opposites)
n = -2 (Property of 0)
Suppose we take our work now to this non-believer and show it to him. We have proved what we believed to be true for we have given as a reason for each step in our work one of the fundamental properties of our number system. But suppose our friend now says, "You've proved that 2 only has one additive inverse, but how about 5, 8, -2.7?" Of course, we can use the same method to find that there is just one additive inverse for each of these numbers, but then he might ask about others. What we really need to do is to show that each number has just one additive inverse. Let's go back and study our problem using the number 2 to see if we can find a way to prove this.
Instead of the number 2 let's use the variable x to stand for anyone of the set of real numbers. Then we have:
x+n=O (x+n) + (-x) =0 + (-x) (-x) + (x + n) = 0 + (-x) [(-x) + x] + n = 0 + (-x)
o + n = 0 + (-x) n = -x
We have now shown that for any real number x, the unique additive inverse is -x. Now, if anyone asks a question about a particular additive inverse, we can show him our problem which applies for all numbers.
Statements of new properties, which can be shown to follow from previously established properties are sometimes called "theorems." An argument by which a theorem is shown to be a consequence of other properties is called a proof of the theorem.
If students understand the proof and wish to try some individually, they might be given the following:
, 1. For any real numbers a and b
+ = + -(a b) -(-a) (-b)
104

C. Properties of multiplication
1. Definition
2. Commutative property Associative property Property of zero Property of one Distributive property

2. For all numbers x, y, and z (-x) + [y + (-z)J = y + [-(x + z)J
3. For any real numbers a, band c if a + c = b +c, then a =b
You might begin by reviewing the properties which are true for the numbers of arithmetic, since you want to be sure, when you define multiplication for the real numbers, that these properties are preserved.
If a, band c are any numbers of arithmetic, then ab = ba (ab) c = a (be)
a(l) =(1) a = a
= a (0) 0
a (b+c) = ab + ac
When you develop a definition for multiplication of real numbers, you must be sure that it agrees with the products which you already have for nonnegative real numbers.
Consider some possible products
(3)(2), (2)(0), (0)(0), (2)(-3), (-2)(-3), (-2)(0)
Since the first three involve no negative numbers, they are already determined. What will the remaining three products have to be in order to preserve properties of multiplication? If the multiplication property for zero is to be preserved, then (-2)(0) must equal O.
You can get the next product from:
o = (2)(0)
o = 2 [3 + (-3)J o = 2 (3) + 2(-3) o = 6 + 2(-3)
Now from the property of opposites you also know that 6 + (-6) = 0, and since it has been proven that every number has a unique additive inverse, it must follow that 2(-3) = 6.
You can show what the last product must be in this way:
o = (-2)(0)
o = (-2) [3 + (-3)J = + o (-2) (3) (-2) (-3) = + o ~ (-2) (-3) = ... (-2) (-3) 6
If the definition is formulated in terms of absolute value:
1. If a and b are both negative or both non-negative, then ab = lal Ibl
2. If one of the numbers a and b is non-negative and the other is negative, then ab = -(Ial Ibl)
The properties of multiplication for the real numbers can be proved using the properties for the numbers of arithmetic and the definition of multiplication for the real numbers. Superior students might be interested in doing this, but it probably should not be an assignment for the entire class.
All students should have practice in applying these properties in problem situations.
Other useful properties which are a direct consequence of these properties are the following:
1. For every real number a,
= (-1) a -a
105

3. Multiplicative inverse 4. Property of equality 5. Use of properties
6. Equivalent sentences 7. Theorems

2. For all real numbers a and b, -ab :::: -(ab) :::: (-a) (b) :::: (a) (-b)
If c and d are real numbers such that cd :::: 1, then d is called a multiplicative inverse of c.
This idea can be developed through the use of examples:
What is the multiplicative inverse of I? of -I? of 2? of O? Students can be led to observe that for any number a, (a =1= 0), the multiplicative inverse is 1/a. The alternate name of reciprocal should be given, and the idea that 0 has no reciprocal should be stressed. Students should observe that this fact follows directly from the definition of multiplicative inverse.
Another important property is the multiplication property of equality. For any real numbers a, b, and c, if a :::: b, then ac :::: bc.
lt is important from the standpoint of structure that students observe the parallel development of properties of addition and multiplication. For each property of addition there is a corresponding property for multiplication.
The properties of addition and multiplication can ve used in the solution of simple equations.
As an example, solve the equation
5x + 8 :::: 2x + (-10)
This equation is equivalent to
(5x + 8) + [(-2x) + 8)J :::: [2x + (-lO)J + [(-2x) + (-8)],
that is, to
[5x + (-2x)J + [8 + (-8)J :::: [2x + (-2x)) + [(-10) + (-8)]
and to
3x -18
This last sentence is equivalent to
(1/ 3) (3x) :::: (1/ 3) (-18), and to
x:::: -6
Stress the fact that operations which involve properties that hold for all real numbers yield equivalent sentences. Sentences are equivalent if their s0lution sets are the same. Later equations will be solved using operations which do not yield equivalent sentences.
Some useful theor.ems which might be presented for the superior student! to prove and for other students to apply are:
1. The number 0 has no reciprocal. (The proof of this theorem is given in SMSG, First Course in Algebra, as an example of indirect proof.).
2. The reciprocal of a positive number is positive, and the reciprocal of a negative number is negative.
3. The reciprocal of the reciprocal of a non-zero real number a is a itself.
4. For any nonzero real numbers a and b, 1/a '/b :::: 1/ ab.
5. For real numbers a and b, ab :::: 0 if and only if a :::: 0 or b =:0.
This last theorem should be stressed since it is the basis of solving quadratic equations by factoring. With this theorem you can determine the solutioJ
set of an equation such as (x + 3) (x + 8) :::: 0
Theorem 5 tells that this equation is equivalent to
x + 3 :::: 0 or x + 8 :::: 0
and one can see that the solution set is
106

D. Use of distributive property 1. Indicated sums Indicated products Simplifying phrases 2. Several applications
E. Properties of order 1. Property of opposites 2. Addition property of order

{ -3, -8}

The distributive property is sometimes described as that property which serves to connect addition and multiplication since it is the only property which includes both operations.

Its use has already been shown in writing indicated sums as products, indicated products as sums, and in simplifying certain phases. This use should now be extended to include the negative numbers.

Students should be led to see that the simple form
= a(b+c) ab +ac

can be used for examples which appear to be much more complex. For example, take the indicated product [x + (-7)] (x + 3). If you let [x + (-7)] correspond to the a in the statement of the distributive property and (x + 3) correspond to (b + c), you have

[x + (-7)] (x - 3) = [x + (-7)] x + [x + (-7)]3
= x2 + (-7)x + 3x + (-21)
= x2 + [(-7) + 3]x + (-21)
= x 2 - 4x - 21

Here the distributive property has been applied several times in one example. Students should be able to point out where it was used in changing this indicated product to an indicated sum.

Properties which have already been introduced, comparison and transitive, should be reviewed. Additional properties can then be introduced and their usefulness illustrated.

A property which connects the order relation with the operation of taking opposites is:

If a and b are real numbers and if
a < b, then -b < -a.
If a, b, and c are real numbers and if a < b, then a + c < b + c.

This property can be illustrated before the formal definition is given by use of the number line.

[C~
-I

a

a+c

b

b+c

c is positive

c is negative

Students should apply these properties in finding solution sets of sentences.

Example:
(-x) + 4 < (-3) + 1-31 (-x) + 4 < -3 + 3 (-x) + 4 < 0 ( (-x) + 4 ) + (-4) < 0 X (-4) (-x) + (4 + (-4 < 0 + (-4) (-x) + 0 < 0 + (-4)

Definition of absolute value Addition property of opposites Addition property of order Associative property of addition Addition property of opposites Addition property of 0

-x <-4 4<x

Order property of opposites

.'. The solution set of this sentence is the set of all real numbers greater than 4.

Theorems which change from order to equality and from equality to order are a consequence of the addition property of order.

107

3. Multiplication property of order
F. Subtraction and division for real numbers 1. Subtraction
a. Definition
b. Use of definition 2. Division

= 1. If z x + y and y is a positive number, then x < z.
2. If x and z are two numbers such that x < z, then there is a positive
= real number y such that z x + y.
Students can discover for themselves the multiplication property of order.
Take the true sentences on the left and insert the two numbers on the right to make true sentences:
1. 5 < 8 and ( ) (2) < ( ) (2) 2. -9 < 6 and ( ) (5) < ( ) (5)
3. 2 < 3 and ( ) (-4) < ( ) (-4) 4. -1 < 8 and ( ) (-3) < ( ) (-3) If a, b, and c are real numbers and if a < b, then
ac < bc, if c is positive,
bc < ac, if c is negative.
In this section define subtraction and division of real numbers in terms of the related operations, addition and multiplication.
One might begin his discussion of subtraction with the way that change is made when he make a purchase at a store. When he buys something costing 27 cents and gives the cashier one dollar, he does not subtract 27 from
= = 100 but adds to 27 the amount necessary to equal 100. The question
"100 - Z7 what?" means the same as "27 + what 100?" Remember
that we solved the equation
27 + x = 100
by adding the opposite of 27 to find
= x 100 + (-27).
We can see then that "100 - 27" and "100 + (-27)" are names for the same
number.
To subtract the real number b from the real number a add the opposite of b to a. Thus, for real numbers a and b,
= a - b + a (-b).
Changing the signs of terms within parentheses preceded by a minus sign follows immediately from this definition.
The sign introduced here, which is the same as that used to indicate "opposite" and "negative" should be read "minus" when subtraction is indicated.
Point out that subtraction is not commutative or associative. If an expression like
5 - 3 - 4 is given
agree that it means
(5 - 3) - 4.
Suppose one gives the problem:
Simplify (3x - 4) - (5x - 7)
= (3x - 4) - (5x - 7) (3x - 4) + [(-5x) + 7] = [3x + (-5x)] + [(-4) + 7] = -2x + 3
Since division is related to multiplication in the same way that sUbtrac:: is related to addition, the definiton of divison could probably be gue by the students.
108

a. Definition
b. Simplest name for a number
3. Simplifying expressions

For real numbers a and b (b =1= 0), "a divided by b" shall mean "a multiplied by the multiplicative inverse (reciprocal) of b."
In symbols,
= ~b a' .-.bl ' (b =1= 0)
The numeral aIbis called a fraction with a the numerator and b the denominator.
Since the number 0 does not have a multiplicative inverse, the definition automatically excludes division by zero. This fact should be stressed throughout the remainder of the course.

Theorems which may be proved include:
= 1. For b =1= 0, a cb = if and only if alb C

2. For any real numbers a, b, c, d,

if b =1= 0 and d =1= 0 then

ba



c -(f-=

bacd

3. For any non-zero real number a, ~ = 1 a
4. For real numbers a and b -a_ a_-a
b- -b - j )

The above theorems are quite useful in simplifying certain numerals. Common names for numbers were discussed previously as being the simplest names for numbers. It might be well at this point to define what is meant by the simplest name for a number.

The simplest name for a number:

Should have no indicated operations which can be perforemd.

Should in any indicated division have no common factors in the nu-

merator and denominator.

a

-a a

Should have the form -Tin preference to (for -d

Should have at most one indicated division.

The definitions of subtraction and division together with the theorems which have been listed can be used to simplify the following types of problems:

(1) 35p ''fr)

(p =1= 0)

(2) 3a2b 5aby

(a =1= 0, b =1= 0, Y =1= 0)

(3) 3v - 3 .
2~y - 1)

(Domain of y must exclude 1)

(4) (3x - 5) - (4 - 2x)
8
+ (5) (x - 1) (2x - 3 x)
4x - 4
(6) r - 3 1 r - 7 1
l8J L2J

(Domain of x?)

(7) xy+y xy-y (Domain of x?) x+1 x-I

(8) .l:. + ~ +..!.23 4

109

(9) 5
3'
-1-
'2
(10) J:.-. +~ 34 11 "3 4

VI. Factors And Exponents

A. Factor
1. Definition of factor and proper factor

You may introduce this idea by asking what numbers divide into a certain number exactly, then formulate a precise definition for factor as being a more compact term.
= The integer m is a factor of the integer n if mq n, where q is an integer.
If the integer q does not equal 1 of n, we say that m is a proper factor of n.

Tests for divisibility will need to be discovered by the students or presented to them so that they can test to see if a number is a factor without actually dividing.

2. Tests for divisibility

(1) Two is a factor of a given number if the number ends in 2, 4, 6, 8, or 0

(2) Three is a factor if the sum of the digits is divisible by three.

(3) Four is a factor if the last two digits are divisible by four.

(4) Five is a factor if the number ends in 5 or O.

(5) Six is a factor if 2 and 3 are factors.

(6) There is no simpl~ test for divisibility by seven.

(7) Eight is a factor if the last three digits are divisible by eight.

(8) Nine is a factor if the sum of digits is divisible by nine.

(9) Ten is a factor if the number ends in O.

3. Prime number

A prime number is a positive integer greater than 1 which has no proper factors.

The Sieve or Eratosthenes discussed earlier in this volume is a good method for finding primes. (Also in SMSG, First Course in Algebra)

4. Prime factorization

Stress the fact that when you speak of factoring an integer you mean over the positive integers. For example, if the possible factors are not
specified, you might consider 3/4 to be a factor of 12, since (3/4) (16) = 12.
Prime factors of ~any numbers can be found by inspection but if they cannot be found' by inspection a system for factoring is needed.
A suggested approach is:

(1) Find by testing for divisibility the smallest prime factor. Divide by this number.

(2) Take the quotient from (1) and repeat the process.

(3) Continue this process remembering that the largest prime you must try is less than or equal to the square root of the quotient obtained from above.

An orderly way of writing the factors of a number is:

432 2 216 2
108 2
54 2 27 3
93 33 1

432 2 x 2 x 2 x 2 x 3 x 3 x 3

110

5. Use of prime factorization to find least common multiple
B. Exponents 1. Definition
2. Operations with exponents

The least common multiple of two numbers is easier to find if one first writes the numbers as the product of prime factors.
Example: What is the least common multiple of 288 and 432?
288 2 x 2 x 2 x 2 x 2 x 3 x 3 and
432 2 x 2 x 2 x 2 x 3 x 3 x 3
For the least common multiple use each factor the greatest number of times it occurs in either number, so the L. C. M. is
2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 or 864.
This principle is useful in finding the least common denominator when adding and subtracting fractions.
The form given above for writing the prime factors of a number will lead to the definition of exponent.
a= axax ... xa n f;ctors
The "a" which indicates that number to be used as a factor several times, fs called the base. The "n" denotes how many times the factor "a" has been used. "n" is called the exponent. The number an is called the nth power of a, or sometimes an is called the power.
One should define multiplication and division first in terms of positive exponents, giving examples to show why these operations are defined in this way:
am an = am+n

an

am - n if a =/=-0 and m > n

1.

an---

-
In

I

f

a

=/=-0

and

n

>

m

= am

.

-an = 1 If a =/=-0 and m n

Then by trying to shorten the definition of division to one statement, you can introduce the idea of negative and zero exponents.
It will be necessary to make the following agreements:
= aO = 1 and a -n I/an
and we can then write the definition of division as

VII. Radicals And Roots

A. Roots

If b is a positive real number, and a2 = b, then a is a square root of b.
If a is a square root so is -a. The positive square root of b is denoted by Vb
and is usually called "the" square root of b. The negative square root of b is
indicated by - y'b. The square root of 0 is 0, and in this case there is just one root.

The symbol V is called the radical sign. An expression which consists of a
phrase with a radical sign over it is called a radical.

It might be well to introduce this idea with a discussion of inverse operations (subtraction is the inverse of addition, division is the inverse of multiplication).

The process of squaring a number has been studied:

111

Cube roots nth roots
B. Irrational numbers
C. Simplification of radicals

If x = 15, then what is the value of x2?

Now consider the same kind of question in the opposite direction:
= What is the solution of x2 225?

The student should observe that there are two numbers whose squares are 225.

Mter practice with squares and square roots, students should be able to formulate a definition for cube roots and finally for nth roots. It should be made clear to them that negative numbers are admission for cube roots or for any odd index.

3
yI -8

5~
V -32

After examples using perfect squares have been given, introduce a number
like yl2 and ask what it equals. Squaring the answers given should indicate to the students that there is no rational number whose square is 2. If students have been accustomed to proving simple theorems, they will be able to un derstand the proof for this statement. Emphasize the fact that there are
many irrational numbers and ask for other examples.

Properties of operations on radicals can be discovered through the use of examples. Is there a simpler expression for yl3 yl5?
Does this equal yl15? Let's examine

Then v3 yl5 must be a square root of 15. Since 15 has only one positive square
root, we can conclude that

2. Restriction on domain of yx
3. yla
v'b
4. Rational approximation of irrational numbers

Drill in simplifying radicals in this way should include discussions of the domain of the variable if there is one.
ylx -Jy = ylxy if x >0 and y,> O.
ylx~ domain of x is all real numbers. In this case be sure to indicate that a > 0 and b > O.

Show also how the property of one is used to simplify an expression having a radical in the denominator. Since yla'is a real number if a and b are real
ylb

numbers and if a > 0 and b > 0 we know that

yla _ yla (1) Vb - ylb

yla[ylbTVtb LylbJ

Emphasize to the student that finding a square root of an irrational number means finding a rational approximation to that number. At least one example should be shown using the iteration method even if the familiar algorithm is used later.
For example:
= We know that 12 1 and 22 4 so
v'2 is between those numbers.
Now by squaring 1.1, 1.2, 1.3, ... 1.9, we can refine this estimate to
1.4 < yl2< 1.5 and similarlY,

112

1.41 < f i < 1.42
We can continue this process, getting more accurate rational approximations to y'2.
Use of the number line to show that the approximations are closer to-
gether should convince the student that we are getting closer and closer to .,j"2.

VIII. Polynomial And Rational Expressions

A. Polynomial expressions

1. Definition

A phrase formed from integers and variables, with no indicated operations other than addition, subtraction, multiplication, or taking opposites, is called a polynomial over the integers.
Examples: x + yC; x" + yO; + c" x"; 4x' + 3x

2. Degree of a polynomial

When a polynomial in one variable is written as a sum of monomials, the degree is the highest power of the variable in any monomial.
+ Example: 4x5 3x4 + 3x2 is a polynomial of degree five.

3. Operations on polynomials

Emphasize that when polynomials are added, subtracted, multiplied or divided, you are using the same properties that were used for the real numbers. Students may realize that the set of all polynomials is an algebraic system in which the elements are not numbers. They may be led to see that this system is like the system of integers since it is not closed under division. However, this should not be forced on them.

Emphasis in this section should be on structure. Mechanical manipulation of symbols is to be avoided. If the student shows signs of this, it is important to go back to the real numbers. (If you assign a value to the variable, a polynomial represents a real number.)

4. Factoring of polynomials
a. What factors are admissible
b. Use of distributive property
c. Common monomial factor
d. Difference of two squares
e. Perfect squares f. Trinomials of second de
gree g. Solving quadratic equa-
tions by factoring

Mention the similarity between factoring polynomials and factoring integers.
Since the student is working with polynomials over the integers, when a polynomial is factored the factors should be polynomials over the integers. For example, if polynomials over the reaIs are allowed,
x2 - 2 can be factored into (x - .,ji) (x + .,j2) but it cannot be factored
with polynomials over the integers.
Stress use of the distributive law in factoring, particularly in the case of a common monomial factor.
Through inspection of these, both in factored and simplified form, the student should discover for himself ways of dealing with them. It might be pointed 'out that the distributive property is still the basis for his work,
but that in an expression like x2 + 8x + 16, it is not always possible to use
a direct application of this law. Some tricks or devices are needed.

This topic gives motivation for factoring. Emphasize again that the pro-

= = perty used is: If a and b are real numbers such that ab = 0, then either

a

or b 0.

B. Rational expressions

A rational expression is a phrase which involves real numbers and variables with, at most, the operations of addition, subtraction, multiplication, division, and taking opposites.
It is obvious from this definition that a polynomial is a rational expression. Similarity between rational numbers and rational expressions will also be noted. Remembering that for each value of its variable a rational expression is a real number one can see the same properties will hold for both systems.

113

1. Properties which hold for both systems
2. Properties applied to rational expressions

The following properties will be if importance in work with rational expressions, such as, if a, b, c, and d are rational expressions, then
(1) a c ac
1lT=1ld

(2) b __ i)-

1

t3) ~ + ~ = a+c b b -b-
Simplify:

ab + ab2

a

ab2

Simplify:
x2 + 2x + 1
x2 - 1
x+1
x-I
Simplify:

1- b
T+l>

ab(l + b)

1- b

+ = a(l - b) (1 b) 1 + b

ab(l + b) (1 - b)
a(l - b) (1 + b) (1 + b)

(1

+b[ab()1

+
[a(l

b+)(l

b)

b)] (1 _

b)]

,By Property (1)

(1 +b b)

If b #- - 1 Property (2) a#-O

x2 + 2x + 1
x2 - 1

x-I
X+T

+ + (x 1) (x 1) (x - 1) (x - 1) (x + 1) (x + 1) (x + 1) (x + 1) (x - 1)
. (x + 1) (x + 1) (x - 1)
1 By Property (2)

, By Property (1) Property (1)

+ 7
--:3:C-6::a-=2bc-

5 24b3

7

5

+ 2232a 2b

233 b3

+. 7

2b2

2232a2b 2b2

5

3a2

233b3 3a2 Property (1)

~=:-1=4~b_2 +

15a2

233 2a 2b 3

2332a 2b3

14b2 + 15a2 Property (3)

72a~b3

a#-O and b #- 0

Simplify:

x

y

x

+ x-y _~y~_ x y

x+y-x-y=x+y x-y

x-y

x+y

By use of Property (1)

x(x - y)

y(x + y)

(x + y) (x - y) - (x + y) (x _ y)

_ x(x - y) - y(x + y)

-

+ (x y) (x --y)--

By use of Property (3)

_ x2 - xy - xy _ y2
+ - (x y) (x - y)

_ x2 - 2xy _ y2

-

x2 _ y2 .

If x#- y, x#- -y

x #- y x#- -y

114

.....

IX. Solution Sets Of Open Sentences

A. Linear equations and inequalites in one unknown

The purpose of this unit is to help the studene discover some ways of solv. ing equations and inequalities. It should include considerable drill, but the drill must be meaningful.

1. Equivalent sentences

Certain operations when applied to the members of a sentence yield other sentences with exactly the same solution set as the original. If two sentences have the same solution set, they are said to be equivalent. The procedure for solving a sentence consists of performing permissible operations on a sentence to give an equivalent sentence whose solution set is obvious.

2. Operations which yield equivalent sentences

Two operations which yield equivalent sentences are: (1) addition of a real number to both members, and

(2) multiplication of both members by a non-zero real number.

In the case of inequalities, with (2) specify that order is unchanged if the multiplier is a positive number and is reversed if the number is negative.

After students have solved many of them, they will become convinced that these operations are sufficient for dealing with linear equations and inequalities.

B. Equations involving factored expressions

Extend the idea of solving a quadratic equation by factoring to include polynomials of degree higher than two.
These problems should be presented in factored form:

(x + 1) (x + 3) (2x - 3) 3x + 2) = 0

In solving an equation of the form (3 + x) (x2 + 1) = 5 (3 + x) one should be careful not to multiply by some expressions which is not a real number. For example, if we multiplied by 1 and if x = - 3, then
x +3
we have multiplied by a meaningless symbol. In this case one should solve it in this way:
= (3 + x) (x2 + 1) 5 (3 + x) = (3 + x) (x2 + 1) - 5 (3 + x) 5(3 + x) -5(3 + x) = (3 + x) (x2 + 1) - 5(3 + x) 0
+ + (3 x) [(x2 1) - 5] = 0
= (3 + x) (x - 2) (x + 2) 0

C. Fractiorial equations

and

the

truth

set

is {-3,

2,

-2}.

If

we

had

multiplied

by

_ 3

_+l_

1
x

the sentence

obtained would not have been equivalent, since its truth set is {2, - 2}.

In fractional equations specify what values are restricted for the variable. In an example like 1 _ 1 indicate x 7"= 0 and x 7"= 1. Solve these equations
x-l-x
= in the familiar way. When one multiplies by x (1 - x) he is using the proper-
ties of real numbers and 1 is not a real number if x O.
x

If it is necessary to square both members of an equation, the resulting equation may have a larger solution set than the original.

Solve

VX+3= 1
= (v"x + 3)2 12
x +3= 1
x = -2

= = If x -2 then yx + 3 y -2 + 3
solution set.

--vI = 1 Therefore, - 2 is the

Solve ..JX + x = 2
V'x+x-x=2-x
yx= 2 - x

115

D. AbsolUite value sentences

(vfx)2 = (2 - x)2
= + x 4 - 4x x2 = o + 4 - 5x x2 = o (x - 4) (x - 1)
x 4 or
x =1

Checking here indicates that 4 is not a solution of the original equation but 1 is a solution.

Absolute value sentences are solved in the same that those containing square roots are solved.

Solve

x - Ixl = 1
x-I =Ixl

(x - 1)2 = (IXJ)2
x2 - 2x + 1 = x2
2x - 1 = 0
x = 1,2

Checking reveals that Yz is not a solution to the equation; therefore its solution set is the null set.

x. Graphs Of Open Sentences In Two Variables

A. Extension from .the real number line to the real number plane

Students have had practice in drawing graphs on the number line. Graphing should now be extended to a set of coordinate axes in the plane. It might be done in this way:

The number line has been useful in determining relations among the real

numbers. Perhaps a real number plane would be even more helpful.

Let's see how numbers can be associated with points of the plane.

:f

I



-3 -2 -1

--12


0


1


2

L
3 45

C

B. Ordered pairs of numbers associated with points

Consider a point A. If it is on the original number line, say four units to the right of 0, then one can associate with it the number 4. Suppose however, that it is not on the line, but is above the number 4. It seems then that one might associate with this point not one number, but two. List the number on the real number line first and say that the coordinates of this point are (4, 3). Suppose you had another point P which is located two units below the number 4. How shall one represent it? Since you used the number 3 to indicate that the point was 3 units above the line, it seems natural to use the number -2 to indicate that the point is two units below the line.
Locating a number of points such as Band C on the diagram and drawing the vertical lines will show the need for a line of reference and the usual set of coordinate axes is introduced.
Stress the fact that order is important. The point (a, b) is not the same as th: point (b, a) except when a and b are equal. Be sure that students understan that the point (a, 0) is on the x-axis and the point (0, b) is on the y-axis. For every pair of numbers there is a point and for every point a pair of 'numbers.

116

C. Ordered pairs as members of truth sets for linear equations in two unknowns
D. Graphs of sentences containdng absolute values
E. Graphs of inequalities
F. Writing open sentences from graphs
G. Graphs of more than one sentence on the same set of axes

Take an open sentence like
2y - 3x + 6 = 0
Make the agreement that when the two variables are x and y, x will always be considered first. Ask students to assign the values 2 and 0 to the variables (2 to x and 0 to y by the above agreement). Is the sentence now a true sentence? Given any pair of numbers, students will find that the sentence is either true or false. The sentence then divides the set of all pairs of numbers into two sub-sets: those which make the sentence true and those which make it false. Plotting several points on coordinate axes will enable the students to see that the graph is a straight line.
= = = The graph of a sentence like Ix- 31 = 2 is really the graph of two line for
Ix - 31 2 is equivalent to two sentences x - 3 2 or x 5 if x - 3 > 0
= and -x + 3 = 2 or x = 1 if x - 3 < O. The graph then consists of two vertical
lines, one at x 1 and the other at x = 5.
Given an inequality, 2y - 3x + 6> 0, or 2y - 3x + 6 < 0, to draw the graph, first draw the graph of 2y - 3x + 6 = O. If the symbol > is used, the graph is the region above the line, if < is used the graph is the region below the
line. The line is not included unless the symbol.< or > is used.
This might be included as an optional topic for better students.
Graphing two open sentences on the same set of axes will serve as an introduction to the next unit on systems of equations. Graphing can help to develop the idea of slope before a definition is given. Graphs of y = 2x + 3;
y = 2x - 3; y = 2x; y = -lhx will give the student some idea of the part
played by the coefficient of x when a linear equation is written in this form.

XI. Systems of Equations and Inequalities

Discussion of systems of equations and inequalities follows directly from the section on graphing. Take a system of equations:
x+2y-5=O 2x+y-l=O

A. Systems of equations as compound sentences

Graph these equations on the same set of axes. If the two are considered as a compound sentence, then they could be stated as x + 2y - 5 = 0 and 2x + y - 1 = 0, or they could be stated as x + 2y - 5 = 0 or 2x + y -1 = O. If the first statement is used, then the solution set is the set of all ordered pairs belonging to the solution sets of both the equations, in this case {I, 3}. The sentences containing "and" are of most interest.

B. Finding the truth sets of systems of equations

The solution sets of systems of equations are found in the usual way. Stress the properties which allow the various steps in solving systems of equations by these three methods.

Methods: Graphing Addition SUbstitution

Call attention to the fact that some systems have no solution (null set as solution set) and that some have many solutions (the set of all pairs of real numbers). Students should recognize the graphs - parallel lines in the first case and the same line in the latter.

C. System of inequalities
D. SYstems of an equation and an inequality

Use only the method of graphing for these. Graph both inequalities on the same set of axes. The solution set is the set of coordinates of points in the region which is shaded twice or the intersection of the two sets which are solutions of the separate inequalities.
Use the method of graphing. Graph the inequality and the equation on the same set of axes. The solution set is the set of coordinates of all points on that part of the line which lies within the shaded region of the inequality.

117

XII. Quadratic Polynomials

A. Graphs of quadratic polynomials

Quadratic polynomials were first studied in unit VIII. The method of solving by factoring was given then. In graphing quadratic polynomials emphasis is on examining the polynomial in standard form to see what information can be learned about the graph without actually plotting it.
Exery quadratic polynomial can be written in the form Ax2 + Bx + C
where A, B, and C are real numbers and A 7'= O. The graph of the polynomial
Ax2 + Bx + C is the graph of the open sentence
y = Ax2 + Bx + C.

A drawing can be made of the graph of a quadratic polynomial by locating some of the points of the graph.

Students will profit from sketching the graphs of many related quadratic polynomials and observing how they are related. A suggested list might include:

(1) Y x2 (2) Y _x2 (3) Y 2x2 (4) Y V2x2
= (5) y V2(x - 3)2

Point out that in each of the first four cases the vertex is at the origin. The graph opens upward if the coefficient of x2 is positive and downward if the
coefficient is negative. In the fifth case, the vertex is located on the x-axis
= three units to the right of the origin. Students should be able to guess the
location of the vertex of the graph of y V2 (x + 3)2, and to verify it by
= = sketching the graph. They should note that the graph of y V2 x2 and y
1h (x + 3)2 have exactly the same shape. Introduce later an equation of the
= form y 1/2(x - 3)2 + 2 and have students observe the shape of the graph
and the location of the origin.

B. Standard form of a quadratic polynomial

Generalization should follow: If a quadratic polynomial is written in the form
y = a(x - h)2 + k then the point (h, k) is the vertex of the graph, and if a is
positive the graph opens upward, if a is negative it opens downward. It has
= the same shape as the graph of y ax2.
A quadratic polynomial in the form y= a(x - + h)2 k is said to be in standard
form. The method of completing the square is used to change a quadratic polynomial to the standard form. This form is useful both for determining information about the graph and for solving a quadratic equation.

C. Quadratic equations 1. Faotoring 2. Graphing 3. Completing the square 4. General equation

The traditional approach is used here. Emphasis should be placed on the idea of equivalent equations derived from the original equation by certain permissible operations. Stress the fact that a quadratic equation has two roots, that it may not have roots in the system of real numbers but it will have roots in a larger system to be studied later.
The quadratic formula may be developed and then used in finding roots of equations, or it may be given as a formula which can be derived, leaving its deriv~tion as a special assignment for superior students.

XIII. Functions
A. Definition

This unit presents another basic idea in mathematics - the idea of _a function. This material should definitely be included if one is working with a class of superior students but might be omitted with an everage class.
Given a set of numbers and a rule which assigns to each number of this set exactly one number, the resulting association of numbers is called a function. The given set is called the domain of the function, and the set of assigned nUID" bers is called the range of the function.
Give examples to show that the rule of a function may be given by a tab~e,
a diagram, a graph, a verbal description, or by an expression in one variab e.

118

B. Notation C. Graphs of functions
D. Linear functions E. Quadratic functions

The work on functions will be devoted almost entirely to the last type, but students should recognize that this is not the only kind of function.

Use letters of the alphabet as names of the function. If f is a function and x is a number in the domain of f, then f(x) represents the number which f assigns to x. The symbol f(x) should not be used to represent the function. The symbol f(x) should be read elf of x." The number f(x) is called the value of f at x.

In some cases several equations may be used to describe one function.

X, x> 0

f(x)

{ -x, x::::: 0

In this case the domain is derived into two parts, and a different rule is used for each part.
One way to represent a function is by means of a graph. The graph of the
= function f is the graph of the truth set of the equation y f(x). It will consist
of all the ordered pairs (x, f(x) ) for which x is in the domain of f.

If the domain of a function is not explicitly stated, it is assumed to be all the numbers for which the rule has meaning. For example, if the rule of the function f is given by the equation f(x) = 11x then the number 0 must be excluded from the domain.
Students should be able to recognize whether or not a graph represents a function. Since each x in the domain must have assigned to it one and only one y, any vertical line must intersect the graph in one and only one point.

A function whose graph is a straight line (or a portion of a straight line) is called a linear function. Linear functions were dealt with in Unit IX, but here they were treated as linear equations. All linear equations, except those of
= the form x a where a is any real number, are also linear functions. If the
domain of a linear function is not stated, it is assumed to be all real numbers.
A quadratic function is one which is expressed in terms of a quadratic poly-
nomial in one variable, Ax2 + Bx + C. These were treated in Unit XU. In this
section the students' work with functions should extend their knowledge of quadratic equations. They might study the nature of the roots as observed from the graph. They should have practice in graphing functions with a speci-
= fied domain. Graphs of equations in the form x Ay2 + By + C should be
done to demonstrate that if x is a number in the domain, this equation does not define a function.

XIV. Ratio, Proportion, And Variation

This unit is included here because it might be taught from the standpoint of the equations for variation representing functions. It could equally well be taught earlier, introducing the idea of ratio when division is defined, and including the other material when equations are studied.

A. Ratio

If a and b are real numbers and .b # 0 the ration of a to b is Ia b" and this can also be written "a : b."

B. Proportion

A proportion is an equation whose members are ratios. Give students practice in forming proportions from verbal problems.

Some properties of proportions which may be proved are:
= = (1) The proportion Ia b 'I. is equivalent to ad bc = = (2) a/" 'I" is equivalent to ai, b/. = (3) 'Ib = 'I. is equivalent to '+bIb '+1. = (4) .'/b = 'I. is equivalent to 'Ib ,+0
b+"d"

119

C. Variation
l. Direct
2. Inverse 3. Joint 4. Other types

= The number y is said to vary directly as the number x if y kx where k is a
constant.
The number y varies inversely as the number x if xy = k where k is a constant.
= The number y varies jointly as x and z if y kxz where k is a constant.
Other types of variation will include:
= s lf2 gt2 where s varies directly as the square of t.
= V 4/ 3 ,,-r3 where V varies directly as the cube of r.
Students should have opportunity to describe and work with variations of all these types.

120

abscissa absolute value additive inverse approximation associative property axis of symmetry
base binary operation
closure coefficient common name commutative property comparison property compound sentence constant contradiction coordinate coordinate axes counting numbers
deductive reasoning degree difference direct variation distributive property dividend divisible divisor domain
element empty set equal equation equivalent exponent
factor factorization finite set fraction function
graph
horizontal

VOCABULARY
identity elements index indicated product indicated quotient indicated sum indirect proof inductive reasoning inequality infinite set integer intercept inverse operation inverse variation irrational number
least common multiple linear equation linear function
minus monomial multiplicative inverse
natural number negative number non-negative non-zero null set number number line number system numeral
open sentence operation opposites order ordered pair ordinate
parabola phrase plane point polynomial positive number power

prime factorization prime number product proof proper factor property proportion
quadratic quotient
radical range ratio rational expression rational number real number reciprocal reflexive property remainder root rule
sentence set simplest name simplify slope solution set square root standard form subset successor sum system of equations
terms theorem transitive property truth set
unary operation unique
variable vertical
whole number

121

GENERAL MATHEMATICS

UNDERSTANDINGS TO BE DEVELOPED IN GENERAL MATHEMATICS
General mathematics is designed for the pupil who enters the ninth grade. The materials in this course should give the pupil ~ background sufficient to prepare him to take algebra or consumer mathematics at a later date. The review which will be necessary for the pupils in this course should be presented in a more mature language than that used for problems involving the same principles in the earlier grades. Many of these pupils will have a better understanding of the decimal system if they are taught the use of other bases of numeration including duodecimal notation. Geometric constructions in this course should include material of a more advanced leveL Solving equations should be extended in this course to a greater degree of competence than that acquired in the Eighth Grade.
125

CONTENT
I. Numeration
A. Decimal notation
B. Numerals as names for numbers.

TEACHING SUGGESTIONS

Recall that our system is the decimal system, and stress the reason for its name.

The position in which a symbol is written, as well as the symbol itself determines the number for which it stands.

Our system of writing numerals has the base ten. Starting at the one's place, each place to the left is given a value ten times as large as the place before. Display a place value chart which shows this fact.

Give an example of decimal notation and show its meaning.

Example:

4176 = (4 X 1000) + (l X 100) + (7 X 10) + 6, or

= + + + (4 X 103)

(1 X 102)

(7 X 10) 6

The small numerals, 3 and 2 are called exponents, while 10 is called the base.

Remind pupils that by using only ten symbols one can write numbers as large or as small as one wishes.

Show this diagram:

U2

:U:r2:

i=l U2

E-t

<Z
:U:>2
:0:r:
E-t

i=l ~ p:: i=l
Z
~

en Z
~
E-t

en
~
0

:U:r2:
E-t
Z
~
E-t

i=l ~ p::
iz=l
::::r>:

3 5 76 75

C. A number may indicate size or position

Explain that this number is -read: "three thousand, five hundred seventy-six and seventy,five hundredths," or "three thousand five hundred seventy-six point seven five."
The decimal point has no place value. It is used to separate the whole number from the fraction. It is read "point," or "and."
A decimaf fraction is name<;l by its position just as a decimal whole number is. Each place still has the value of ten times the number that succeeds it.
Have pupils become aware of naming the decimal fraction according to the last place occupied.
Also, lead them to rediscover that the decimal fraction ends in "ths" while the whole number does not.
Pupil activities: Practice reading and writing numbers including very large and very small numbers. Bring illustrations of the use of numbers from newspapers, magazines, ~~_c. Make a list of numbers heard in conversation, other classes, TV, etc . during a glve'n period of time.
Extend knowledge of cardinal and ordinal numbers.
Lead pupil to remember that a cardinal number tells how many and that ordinal number tells position.

126

Examples: Ordinal

3 2

1

D. Historical background
E. Numeration bases other than ten 1. Definition of base of a system of numeration
2. Numeration in base 8.
a. Counting in base eight

One may say first step or step one. Used in this way one and first are both ordinal numbers.

Cardinal

DOD



...

3 BLOCKS

Study some of the early systems of numeration such as Egyptian, Roman, Babylonian, Mayan, Greek, and Chinese.
The student should not try to develop any mastery in the use of these but should merely understand how cumbersome these were for computation and thus develop an appreciation for his own "base ten" system.

Let this review help him discover what constitutes a modern number system. Review numeration in bases studied in grades seven and eight.

The base of a system of numeration is the number which is used for the fundamental unit of grouping as:

base 3, ternary system base 4, quartenary system base 5, quinary system base 6, senary system base 7, septenary system base 8, octonary system base 9, nonary system

Explain to pupils that I xxxxxxxx I xxxx means one ei.ght, four units, and
I xxxxxxxxxx I xx means one ten, two units.

The first is read "one four base eight"; the second is read "twelve base ten."

Table of values:

Number
one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen etc.
127

Base Ten
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16 17

Base Eight
1 2 3 4 5 6 7 10
11
12 13 14 15 16 17 20 21 etc.

b. Operations in base eight

When pupils have developed an understanding for counting in base eight, have them make tables for addition and multiplication. They should appear as follows:
Addition Table

1'+ 0 1 2 3
:'
0 0, 1 2 3 1 1 i' 2 3 4
2 2 3 '4 , 5

33456

.

~

~

~

~

5 5 6 7 10

n

n

~

~

~v

~~

7 7 10 11 12

4: 5 ~ 7 45~ 7 5 6 7 10
6 7 0 11
7 10 1 12
LV ~~ 12 13 1 )2 13 14
12 13 14, ,15
13 14 15 16,

Multiplication Table

,
X, 0 1 2 3 4 5 0 1'0 0 0 0 0 0

1 0 1, 2 ~ 4 5
I'
2 0 2 4, 6 10 12

~

n

u

v

~

n I' n

1A

1'"

"-
4 0 4 10 4 20, 24

5 0 5 12 7 24 31

u

v

u ...~ 22 30 36

7 0 7 16 25 34 43

~7 )0
7 4 16 22 25 30 34 36 43 44, 52 52 61,

The diagonal of the table is the dotted line. Lead pupils to become aware of the symmetry; that is, both sides of the table (formed by the diagonal) are alike.

This symmetry in both tables gives evidence of the commutative property in addition and multiplication.
= Tables shows that 6 X 3 3 X 6 etc.

Provide problems in base eight. Use problems in the four fundamental operations.

Example:

Add:

24,10.'

Look in addition table for 6 + 4. It is 12.

26.lgh'

Write 2 in its proper place.

II

52ctght

2 + 2 + 1 (group regrouped) gives 5.

c. Converting from base ten to base eight.

Change from base ten to base eight

8 / 314
8/39 8/4
-0

remainder 2 7 4
128

d. Converting from base eight to base ten.
3. Numeration and operationsin other bases.
F. Natural numbers
G. Whole numbers

Now read up the column of remainders, and your number is 472' I 'hl.

Change 3125.I,hl to base ten.

= + 3125. " h' (3 X 83) (1 = + (3 X 512) (l X 64)

X
+

+ 82) (2 X 8) (2 + 8) + 5

+

5(1)

= 1536 + 64 + 16 + 5
= 1621 1...

When we write a number without the subscript number for its base we mean base 10.

Notice the structure for base eight and base ten is the same.

Pupils should have some knowledge of the binary and duodecimal systems. The binary system is of interest because of its use in the digital computers. The duodecimal system is of interest because advocates of its use argue that twelve as a base offers more factors than ten and therefore would make work with fractions easier.

Many books which give details concerning numeration and operations in other number bases are available.

Let pupils consider the counting numbers. Be sure that they know these numbers are called natural numbers. For interest let pupils try numbers to determine if they are perfect numbers. Recall that a perfect number is one that equals the sum of its factors, not including itself.

Example: The factors for 6 are:

1 and 6 2 and 3
1 + 2 + 3 = 6 (The factor 6 cannot be used for the rule says "not including
itself")

Example: The factors of 12 are

1 and 12 2 and 6 3 and 4
1 + 2 + 3 + 4 =/= 12

12 is not a perfect number.

Be sure that pupils know that whole numbers are the natural numbers and zero. Therefore, the natural numbers and zero are subsets of the set of whole numbers.

Lead pupils to see that there is no largest whole number. Familiarize them with the idea that whole numbers are members of an infinite set. This notation is expressed:

{O, 1, 2, 3, 4 ...}

II. Operations With Whole Numbers

A. Using the number line to
Show meaning of fundamental operations

The use of the number line is one of the most successful techniques in developing an understanding of numbers and operations with numbers.

Use a number line to show the addition of three and five.

<------------> o 1 234 5 678

v
5 units

'-r----'
3 units

The sum is 8.

129

B. Using the number line to show properties under fundamental operations
1. Commutative property of addition and multiplication

Let pupils undo this operation (subtraction, the inverse of addition). Begin at 8 and count back three spaces to show 8 - 3. They will stop on the number line at 5.

Use the number line to show multiplication.

3

3

~-----.

< o 1 2 3 4 5 6 7 8 9 10 11 >

'-v--' '-v--''-v--'
222

Example: Two threes equal three twos or 2 threes equal 3 twos.

This process can be "undone" by division (the inverse of multiplicart:ion).

The number line may be used to help pupils recall that addition is commutative.
On the number line show 5 + 3 and 3 + 5.

3

5

~

<

>

rr 1 2 3 4 5 6 7 8

'---.;---~

5

3

5 + 3 means three units to the right of five which is eight.

3 + 5 means five units to the right of three which is eight.

We may "undo" this operation. Begin at 8 and count off three units to the left of eight. This number is 5. This process is subtraction, which is the inverse of addition.

Subtraction is not commutative in the set of whole numbers. Show on the number line that 8 - 3 .,;: 3 - 8.

Multiplication may be taught using the number line.

3

3

,...--A------,r---"----.,
o 123 4 5 6 7 8

2. Associative propel'lty.

2 threes are equivalent to six; 3 twos are equivalent to six; multiplication is commutative.

This process can be "undone" by showing the number of times three is contained in six and again by finding the number of times 2 is contained in six.

Division is the inverse of multiplication.

In the set of whole numbers 6 ..;- 3 .,;: 3 ..;- 6 so we say division is not commutative.

'The set of whole numbers has the associative property for addition.

Ask pupils to consider these examples:

(1 + 2) + 3 = 6 means 3 + 3 = 6 1 + (2 + 3) = 6 means 1 + 5 = 6

5

Show on number lines:

A

130

Have students work examples to show that the associative property does not hold for subtraction in the set of whole numbers.

Consider this number line
= (3 X 2) X 4 3 X (2 X 4)

6

6

6

6

"

"

A

A

3

3

~

o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

'--y-J '--y-J '--y-J '--y-J
22 22

- - - - - ' ....

'------.,~---'----'

8

8

Multiplication is associative.

By use of examples lead students to discover that division is not associative.
= Does (8 -:- 4) -:- 2 8 -:- (4 -:- 2) ?

3. Distributive property with respect to multiplication ov~r addition

Show 3 7 and 3 5 on the number line. This may be shown first on the number line:
3X7

o

7

14

21

'--v---' '--v---' '-----y---J
7+ 7+721

3 X5

o 5 10 15

= '-v---' '-v---' '-v---' 5 + 5 + 5 15

21 + 15 = 36

o7

14

21

36

4. Closure

Now have pupils discover that 3 sevens and 3 fives can be grouped and made
into 3 X (7 + 5) or 3 twelves.

oI







.







.

I
12













.





1
24











.

j 36

On the number line 3 twelves is 36.

This is the distributive property with respect to multiplication over addition.

Practice: 5 248 means 5 200 + 5 40 + 5. 8

Lead pupils to see that 374 X 132 uses the distributive property repeatedly

(374 lOG) + (374. 30) + (374 2) or

(300 100) + (70 100) + (4 100) + (30030) + (7030) +
+ (4 30) (3002) + (70 2) + (4. 2) or

30000 + 7000 + 400 + 9000 + 210 + 120 + 600 + 140 +8

Help pupils realize that closure involves two considerations: it must be possible to operate on any two elements of a set, and the result must always be an element of the set.

Remind pupils that a single counter example is sufficient to prove that the set is not closed under the operation.

Example: Develop understanding that the set of whole numbers is closed under addition and multiplication.
= 3 + 8 11: The sum of any two whole numbers is a whole number.

131

C. Classifying whole numbers. 1. Number names
2. Even and odd numbers
3. One, the identity element for multiplication
4. Zero
a. Identity element for addition
b. Division by zero is impossible
5. Factors and primes

= = 8 - 3 5, but 2 - 7 ?
You do not have closure for subtraction. -5 is not in the set of whole numbers.
Try examples for multiplication and division. You have closure in the set of whole numbers for multiplication but nat for division.
Any number may be represented by many different symbols and names as
1, 5/ 5, 5 X 1/ 5, 50, one, I. One is the smallest counting number. There is no largest.
Lead pupils to discover that the successor of a number is always found by adding one to the number you already have.
Have pupils select numbers at random. Are these numbers divisible by two? If they are divisible by two they are even numbers. If they are not divisible by two they are odd numbers.
= 10 + 2 5. Ten is an even number.
The successor of 10 is 11. for it is one more than ten.
11 + 2 = 5lh 11 is an odd number because there is a remainder.
Make pupils conscious that an odd number is always obtained when 1 is added to an even number.
Proof: 2 N is an even number because it is divisible by 2.
2N + 1 is an odd number because ,all odd number is always one more than an
even number.
The next natural number is (2N + 1) + 1 so
= (2N + 1) +1 2N + (1 + 1) (Associative Law). and = 2N + 2 2(N + 1) (Distributive Law).
Because 2(N + 1) is divisible by two it must be an even number.
Suggest that pupil multiply several numbers by one. Bring out that any natural number multiplied by one is that number. Therefore one is called identity element for multiplication.
Division is the inverse of mulJtiplication so any number divided by one is that number.
If John had 3 dogs and gave away all three he had left "not any." This is expressed by the symbol O.
o 1 2 etc.
Zero is to the left of one on the number line, one unit away from one. It is the first whole number.
Locate zero on the number line. Add 1 and you get 1. Locate 1 on the number line. Add zero and you get 1. The identity element for addition is zero.
When one multiplies three by zero his product is zero. Since multiplication is
repeated addition the sum of three zeros is 0 + 0 + 0 and this is zero.
Division is the inverse of multiplication.
= 0/ r; 0 means 0 X 5 = 0 = = Bring out that 5/ 0 is impossible, as this would mean 0 X 0 5. Help pupil
realize that division by zero is meaningless and impossible. 0/0 has no meaning.
Pupils should drop the use of the words "multiplier" and "multiplicand" in' favor of the word "factor"; for a factor times a factor gives a product.

132

6. Composite and primes
D. Unique factorization property of natural numbers and its use.

Reteach: A product is even if one of its factors is even; the sum of two even numbers is even; the sum of two odd numbers is even; the sum of an even and an odd number is an odd number; the number zero has every number a factor but is itself a factor on none; the number 1 has just the factor 1; the prime numbers have exactly two different factors, themselves and 1; the composite numbers hav,e more than two different factors.
Suggestion: Let pupil use the "Sieve of Eratosthenes" to find the prime numbers less than 100. (See guide for seventh grade.)
Have pupils select some natural numbers that have exactly two different factors. (This means the number and 1.) Explain that these numbers that have only two factors are primes.
Explain that all the numbers with more than two factors are composites.
Ask pupils if the set of all primes put with the set of all composites, gives the set of natural numbers.
If the pupils say yes, remind them that the number 1 does not appear in either group, then ask if they obtained the set of natural numbers. (Although 1 was a factor it was not one of the original numbers, set of primes or set of composites.)
There are three kinds of natural numbers classified according to factoring; primes, composites, and the number 1.
Let pupil add two primes such as 5 and 7. Ask for others but be careful at this time to skip the number 2.
After several sums are done ask the pupil what he concludes.
After they have assumed that the sum is an even number (if they do - and some will not) have them add 2 and 7, etc. Explain that their guess was false. Only one false example is necessary to make assumption false.
Recall and reteach that the only even prime is 2 and that although 1 has only two factors, itself and 1, it is not prime for the two factors one and one are alike.
/
Review that every composite natural number can be factorsd into primes in only one way, except for order of the factors.
If 2 is a factor of 14, then 14 is a multiple of 2.
If 2 is a factor of 6, then 6 is a multiple of 2.
While 1 is a common factor of any set of numbers, in 6 and 14 we find 2 is a common factor.
Recall that all even numbers are multiples of 2 and that the product of two odd numbers is odd.
The greatest common factor is found by using the unique factorization property.
36: find prime factorization

54: find prime factorization

The greatest common factor is 18.

III. Operations With Fractions

A. Introduction

Man introduced fractions when he began to measure as well as count.

At first Egyptians used only unit fractions, that is fractions whose numerators are 1 such as 1/ 2, 1/ 3, etc. Any other fraction that had to be used was expressed

133

B. Common fractions or rational numbers 1. Meaning of rational numbers
a. Definition of a rational number
b. Definition of a fraction c. Whole number 2. Properties of rational numbers
Simplest form of a fraction

+ in/terms of unit fractions as 5/ 6 = 1/2 1/ 3,
Babylonians usually used fractions with denominators 60, 602, 603, etc., because the base of their notation system was 60.
Romans primarily learned about fractions with denominators of twelve.
Over the years many other notations have been used. The fraction bar came into general use in the 16th Century.

"A number may have several names" is a concept pupils should acquire. Explain that 15, XV, 3 5, etc., are names or numerals for the same number. Other names may be 3/2, 45/ 3, etc.
Another number may be named as 4/ 3, 8/ 6, 12/9, etc.
3/ 2, 45/ 3, 4/ 3, 8/ 6 and 12/9 are rational numbers.
Explain that symbols for natural numbers were used long before symbols for fractions were introduced. Our cumbersome language of improper fraction, common fraction, unit fraction, mixed numbers, etc. is an attempt by man to give a better idea to something a little unusual. Many authors and books now use the term rational numbers to mean this collection of numbers we think of as fractions.

Recall that the definition of a rational number is that if a and b are whole

= = ./ numbers, b # 0, and bx a then x

b.

= Call attention here that the pupil has already learned that division is the
inverse of multiplication and that he is using this when he says x alb, b # 0,
= because b. x a

If a and b are numbers, b # 0, the symbol / b is called a fraction.

Remember that sometimes a/ b (b # 0), is a whole number. This, of course, happens if and only if b is a factor of a.

Remember that the whole number 5 can be written 5/ 1 which shows that 5 is a rational number.

Lead your students to discover that the rational numbers will now have the same properties as whole numbers.

Review properties with pupils. Give illustrations of each using the number line when feasible.

The set of rational numbers is closed with respect to the operations of addition and multiplication. .

The commutative property holds for the operations of addition and multiplication for the rational numbers.

The associative property holds for the operations of addition and multiplication for the rational numbers.

The distributive property holds for the operation of multiplication over addition for the rational numbers.
= Zero is the identity for addition or the additive identity, that is, 0 + a a +
0= a.
One is the identity for multiplication or the multiplicative identity, that is,
= 1 a = a 1 a.

If the numerator and the denominator have no common factors except 1, the fraction is said to be in simplest form.

= = = ';" To write the fraction a/ b in simplest form, find the greatest common factor k
of a and b, where a = kc and b kd, then by property of one, a/ b .c/

134

Reciprocals
5. Equivalent fractions
6. Operations with rational numbers a. Addition
b. Subtraction

Recall definition of a rational number.
Now consider bx = 1
x = 'Ib and
b rIb) = 1
If the product of two numbers is 1, the numbers are called reciprocals or multiplicative inverses of each other .
Example: a is called the reciprocal of 1/a, and 1/a is called the reciprocal of a.
Lead pupils to discover that if a / b "/ a = 1, neither a nor b is zero.
Definition: The reciprocal of the rational number a/ b is the rational number"/ a
Show that 6/ 8 = 15/ 20
= = Show pupil that 6/ 8 becomes 3~2/ 4-2 3/ 4, and that 15/ 20 becomes 3-5/ 4 _5 3/4 ,
and that since each number is equal to 3/ 4, the given numbers are 'equal to each other.
Make student aware that if you divide each term of a fraction by a common non-zero factor, the resulting fraction names the same number as the original fraction.
From this lead the pupil to discover that when he multiplies the numerator and denominator of the fraction by the same number, the new fraction is equivalent to the original fraction.

+ + Take an example: 1/ 3 1/ 3 1/ 3 = + + 1 1/ 3 1 1/3 1 1/3 =
= = + + (l 1 1) 1/3 3. 1/ 3 1

1, and give full proof.

Have pupils discover that the property used here is the distributive property of multiplication over addition.

This shows pupils why only numerators of like fractions are added when like fractions are to be added.

Review method of finding prime factors. Recall that this process is used in finding the Least Common Multiple (LCM) which is the least common denominator (LCD) of the denominators of fractions.

Review addition of unlike fractions. Stress the changing of fraction to equivalent fractions which have the same denominator and stress the use of the distributive property as the reason for adding numerators of like fractions.

Use the number line to increase understanding of addition of fractions.

Restate that subtraction is the inverse of addition. To solve subtraction problems find a common denominator as in addition, and then subtract numerators.

Use a number line to make the meaning clear.

I

I

2

Just as whole numbers can be plotted on a number line have pupil extend his concept to fractional values. For instance, have pupil locate the points whose coordinates are 1fz and 11fz.

Pupil could also locate the points that divide the unit segment into fourths or thirds.

By use of the number line show 1fz - lf4.

135

c. Multiplication

Locate %. Count from % to the left %. The point located is %. %-%=%
Show other examples of subtraction on the number line. Then review the subtraction algorithm. Again stress the meaning of the operation.
Example: One half of three

d. Division of rational numbers

This rectangle represents three units. If each of the crossed section is one half of the original unit, the crossed sections represent 3/ 2,
Written in symbols this means 1/2 of 3 or 1/ 2 X 3.
Through the use of this explanation, help pupils to understand the meaning of the algorithm for multiplication of fractions.

Then 1/2 X 3 = 1/2 X 3/ 1 = 3/ 2 1 X 3 is 3 and 2 X 1 is 2 so the product is 3/ 2, This simple problem gives the foundation for all multiplication.

Example:

1

2

2X1

2.1

1

2 X-5-=2X"5=2T=s

Example:

7

5

15 X T

By inspection determine that 5 is a factor of both the numerator and denominator and that 7 is also a factor of both. This becomes
7 .5 ~
This may be done mentally, or by showing the division of 5 and 7 by using a slanting line, as:

1l - 'f, 3-\1-' Do not call this "cancellation." This is division. You are dividing both the numerator and the denominator by the same number. In the first case it may be seven and in the second it may be five, or vice versa.
Redevelop the idea that a fraction is a name for a rational number just as numeral is a name for a number.
Restate the five types of multiplication developed in other grades:
A fraction multiplied by an integer A mixed number multiplied by a fraction A.n integer multiplied by a fraction A fraction multiplied by a fraction A mixed number multiplied by a mixed number.

If an understanding of these types of multiplication is lacking, refer to the guide for seventh grade and reteach whatever is necessary.

Relate division of fractions to that of natural numbers.
Example: How many halves are there in three? From early work pupils recall that there are two % 's in 1, and generalize that 3 -;- % is 6.

136

c. Decimal fractions as an ex-
tension of rational numbers 1. Decimal notation
2. Operations with decimals a. Addition
b. Subtraction

Through enough applications pupils generalize that division by '/.. is equivalent to multiplication by u.

Lead student to see that division is defined in terms of multiplication, that to do a division'/ b we look for a number c such that c X b = a. If in / b, a is 1
1
= and b is lh, we have"""; c.

We look for c such that when multiplied by lh gives 1 (our a). The number multiplied by lf2 to give 1 is 2.
Refer to guides for grades seven and eight and reteach if necessary, understanding of decimals which are developed there.

Stress that every rational number may be changed to a decimal which terminates (repeats zero) or which repeats a pattern of digits.

Use the number line to show decimals equivalent to common fractions.

Review decimal notation of whole numbers and decimals stressing the meaning of place value. Show by expanded notation that the same concept of place value applies in a decimal.
= Example: 4578.2346
4(103) + 5(102) + 7(101) + 8(100) + 2(1/10') + 3(1/102) + 4(1/10') + 6(1/ 10,)
Extend ability of pupils to:

Read and write decimals to millionths.

Express decimal fractions as common fractions and vice versa.

Arrange decimals as to size in ascending or descending order.

Determine pupils' understanding of the basic processes with decimals. Reteach whatever is necessary.

Detailed suggestions are not given here, but suggestions are made which may add understanding to what has been taught previously.

Show regrouping in this manner:
= 4.37 4 + .3 + .07 = +2.68 2 + .6 + .08
6 + .9 + .15 (.15 = .1 + .05)
= 6 + 1.0 + .05 (.9 + .1 1.0)

7.05

Show the operations involved in subtracting the following: 7.32 - 4.78

7.32 means 7 + .3 + .02

4.78 means 4 + .7 + .08
Rewrite 7 mentally as 6 + 1; then in your problerr. you have to regroup 1 and
.3 and you have

6 + 1.3 + .02
4 + .7 + .08

Understand that 1.3 came from regrouping 7 as 6 + 1

6 + 1.2 + .12
4 + .7 + .08 2 + .5 + .04

comes from regrouping 1.3 as 1.2 + .1 then regroup combining .1 and .02

This number is written 2.54

Provide ample problems to deveiop speed and accuracy. Have pupils show the details as above, on several problems, and then allow the regrouping to be done mentally.

137

c. Multiplication

= 100 X 1000 100,000
Show pupils the ease of multiplication using exponents. Rewrite 100 as 102 (The exponent is the same number as the zeros in 100.)

Rewrite 1000 as 103 (The exponent is the same as the zeros in 1000.)

Now the problem is 102 X 103

The exponent is the same number as the number of zeros in the 100 and 1000.

The product is 105 because 100,000 had five zeros in it.
Help pupils to draw the conclusion that they could add the exponents 2 and 3 and obtain 5.

Show that placement of decimal point in multiplication makes use of multiplication of this type.

In work in multiplication students multiplied as this example indicates:

111
100 X -1000 = 100000

Have them rewrite the problem using exponents:

1

1

1

102 X 103 = 105

Change a common fraction to a decimal fraction. For example:

= 1/10 .1

Note: The divisor has one zero and the quotient has onl decimal place

Call attention to the fraction 1/100' Show that it means 1/10' or .Ol.
1/100 has been rewritten as a decimal fraction. There are two zeros in divisor and there are two decimal places in the quotient.

Thus 1/10" means write 1 and count over to the left five places beginning with the one written; then insert the decimal point.

This process may be used as an additional explanation in multiplication and other operations.

Example:

.37 X .026

37 X 1/100 X 26 X 1/1000

(37 X 26) X 1/10' X 1/10"
(37 X 26) X (1/105)
Your answer or solution set is the product of 37 X 26 with five decimal places pointed off.

. Division

750,000 --;- 2500 means
= (75 X 104) --;- (25 X 102) = 3 X 102 300
This application is made with decimal fractions also.

Example:

.00008 --;- .004
= (8 X 1/10.) --;- (4 X 1/10') = 2 X 1/10' .02
Example:

.00008 --;- .00004
(8 X 1/10.) --;- (4 X 1/10,) = 2 X 1/100 = 2

I 138
II

IV. Relations
A. Equivalence
1. Equivalent equations
2. Basic principles 3. Number sentence
4. Root B. Inequality
1. Solution sets for inequalities
2. Ordering inequalities

Students may question:
.008 ..;- .00002
(8 X 1/ lOa) ..;- (2 X 1/10,)
4 X 1/10"'"
4 X I/IO~
Your better pupils should know that a negative exponent is possible. It will be studied in a more advanced unit of work or a more advanced Course.
Show that the work can be done by finding a common denominator.
.008 ..;- .00002 means 8. 2 1000 -:- 10000 Your common denominator is 10,000 From the property of 1 you may multiply
10800 by 1100 and have
~ -'- _2_ _ 40 10000 . 10000-
Redevelop the idea that the rational numbers (perhaps thought of as fractions), may be constructed as ordered pairs of numbers.
2/ 3,6/9,9/ 12
(2, 3), 6, 9), (8, 12) ... all equal
These illustrate a set containing an infinite number of different equal fractions and this "set" is called an equivalence class.
Equivalent equations are sentences that have the same solution set.
x + 4 = 14, x + 2 = 12
The solution set here is in the set of positive integers, but have students try non-positive integers as well.
Both members of an equation may be multiplied by the same number, or divided by any number except zero, and equivalent equations result.
Number sentences may be regarded as statements of facts about the members of a number system. They embrace equations and inequalities.
As in the case of verbal statements, number sentences mayor may not be true. Consider:
False 3+4=6
3+4~7
Remind pupils that members of the solution set are called roots of the equation. The number that is a root makes the equation a true statement when it re-
= -nkC!}.~iable in'the equation. In 3x 9, 3 is a root.
Approach inequality with a reminder t~at an inequality expresses the fact that one number is either greater-than-ot less than another number.
Provide practice exercises in which symbols are used as 3 < 5.
Consider a statement of inequality and ask if the set of solutions may be determined.
Example: n - 7 > 5
Help pupil discover that when he adds a number to each of a pair of numbers or multiplies a pair of numbers by the same positive number, the corresponding

139

C. Order of decimals
D. Ratio 1. Meaning 2. Other meanings
E. Proportion 1. Property I.
2. Property II 3. Scale drawing F. Per cent

sums or products will be in the same order as the original pair of numbers.

4>2

8>4

Review methods for finding whether two fractions represent the same number. If they are not equal, determine which is the larger.

012345678910
10 10 10 10 10 10 10 10 10 10 10
> < 6/ 10 occurs to the right of 5/ 10 so
6/10 5/ 10 or 5/10 6/ 10
If fractions are not decimal fractions but common fractions, convert into decimal form and compare.

Example:

2/ 5 and 4/ 11
2th/ 5us=0.400.40> 0.36

4/ 11 <= 0.36

You may find a common denominator for these two fractions; convert each to a decimal and then compare.

> 22/ 55 = .40 .40

20/ 55 """ .36 .36

Review Seventh and Eighth grade guide on ratio.

Develop the concept that ratios are ordered pairs. For example, 4/ 5 is not the same as 5/ 4 ,
As an added explanation, recall that a ratio is any element in the set of ordered pairs of the form alb, where a and b are integers and b ~ O.

Stress the concept that two numbers may be compared by subtraction or by division.

Thus: If the numbers are 12 and 3, you may say that the first is 9 more than the second, or you may say that the first is 4 times the second. Suggest that this be expressed by saying the ratio of the two numbers is 12 to 3 or 12/ 3,
Review definition of a proportion as a statement of equality of two ratios.

A brief review of Property I and II for proportion is suggested here.

Property I. if a/" = e/ d b ~ 0 and d ~ 0, prove ad = bc

= If a /" = e / d, then the property of 1, a /" X d / d = e / d X b / b or,

ad/ ...

eb/ db

By the commutative property for multiplication you can show that bd db; thus ad =cb by the basic principle of division.

By the commutative property again cb = bc so ad = bc.

Develop Property II the same way. If ad = bc then alb = e/ d.
= If ad bc
'0/ d = be/ d Dividing both members by d

a = b X e/ d

Property of 1

Dividing both members by b

= , / b

e/ d

Property of 1

Show how scale drawing uses the concepts of ratio and proportion: SCale drawings of floor plans may furnish interesting exercises for students.

There should be a review of per cent as given in Seventh and Eighth grade section of the guide.

140

Present per cent now as a ratio method.

Example: Find 12% of 60

12 N
100 =60
= lOON 720

N = 7.2

Allow pupils to experiment with other types of per cent to determine that they can also be worked as ratio.

Example: 5 is what % of 25

5

N

25"= 100

25N 500

N 20

Ask pupils what this 20 is Be sure they understand that it is the value of N

and that N per cent.

is

not

20%.

While

5

is

20%

of 25,

it

takes

the

ratio

N/ 100

to

give

the

= = The percent N/ 100 20/ 100, and because this is 20/ 100 means 20% for
a ratio in which the second term of 100 can be changed to a decimal number or a per cent.

Students should work various problems in per cent as found in every day living and business transactions (interest, commissions, discounts).

V. Measurement
A. Measuring things that are continous
B. Subdivision and measurement
C. Metric system
D. Linear measure

Emphasize that all measures are comparisons and all measures are approximate and involve some error.

Ask questions in a manner that will show pupils that measurements may be found by counting or by measuring.

If two geometric continuous figures or sets are both made up of parts such that every part of one can be matched to a part of the same size in the other, then the two continuous figures or sets have the same size.

A geometric continuous figure or set may be subdivided.

Show a line-segment contained in a part of a line as

A

B

C

<--------------->

How do the segments AB and 'Be compare?

AB may not be equal to BC but may be approximately equal to (_).

Sizes of continuous quantities can be found by measuring. Emphasize that the unit of measure must be of the same nature as the thing to be measured and secondly, it must be possible to move the unit of measure.

Offer information on the origin of the metric system and the way in which it gained world-wide acceptance.

Explore the basic conveniences of the metric system.

Familiarize pupils with Greek and Latin prefixes used in the metric system.

Display tables showing metric values and ask pupils to work problems involving conversion.

All measurement of line segments is called linear measurement.

141

E. Area F. Volume
G. Location on a sphere
H. Other measures 1. Temperature 2. Weights
3. Time

The history of linear units of measure is interesting material for reports (yard, hand, span, cubit, fathom, pace, and the like). Develop the concept that area is the number of square units in the interior of a closed geometric figure. Develop experimentally, the meaning of volume and the concept of cubit units.
Have pupils contrast inch, square inch, and cubic inch so that they understand the difference between linear, square, and cubic measures. Lead pupils to derive area and volume formulas for solving problems that are within their experience and understanding.
Develop formulas for the solids given in VII, E.
Develop the concept of azmuth, longitude, and latitude.
Have pupils construct a simple clinometer and use it to measure the angle of elevation of several objects.
Demonstrate finding latitude using a sextant, if one is available.
Have pupils aware of number line readings when they read a thermometer.
Review the units on weight in the English system of measures.
Differences between the ideas of mass and weight are of great importance in the physical sciences. A pupil's weight varies according to his distance from the center of earth but the mass of his body remains the same.
Develop understanding of time zones. Let pupils work travel problems where there is a need to calculate time zones.

VI. Graphs
A. Purposes of the various types of graphs
B. Histogram and frequency polygon

Graphs should be treated as an important part of statistics.

Each kind of graph has a definite purpose for which it is particularly suited.

Pictograph - shows thi'ough use of interesting symbols the subject of the graph.

Bar - vertical, horizontal, or divided - pictures ratios.

Broken line - best shows how rapidly a set of numerical data is changing, increasing or decreasing.

Circle - best shows parts of a whole or how each group is related to the whole set.

Point out that histograms are special kinds of bar graphs; they make graphic presentation of frequency distribution.

No. Yrs. car used
5 4 3 2 1

Frequency Distribution Tally
I II +++t III IIII

Frequency
1 2 5 3 4

142

C. Summarizing data
VII. Geometry
A. Points and lines
B. Angles
C. Complementary and supplementary angles
D. Plane and solid geometric figures Triangle Quadrilateral Circle Other polygons Rectangular solids Cube Prism Cylinder

c

5

A V\

4

3

"'" ~

1\
\D

2 1

""- ""-.E

o

~

1

2

3

4

5

NUMBER OF YEARS

Join midpoints of a histogram by connecting these points successively by line segments. The resulting graph is a frequency polygon.
illustrate statistical averages - mean, mode, and median.

Example:

= Sum of items
Number of items

mean

Median represents the middle one of a set of numbers arranged in order. Mode is the item which occurs more than any other one.

For some pupils, geometry may be a new word. Develop its origin (geo - earth; metry - measures).
See and review any and all parts of geometry, grades 7 and 8. The extent and depth of coverage will be determined by the ability and needs of the students.
Develop the concept that lines have many points.
Provide practice in drawing and identifying lines according to position and relation to other lines.
Position: vertical, horizontal, oblique
Relation: parallel, perpendicular, intersecting
Define angles as a set of points on two rays with a common end point which is called the vertex. The rays are called sides. Angles are also defined in terms of their measure.
Give pupils extensive practice in constructing, measuring, and labelling angles.
Review the concept that the sum of two complementary angles equals 90 degrees and that the sum of two supplementary angles equals 180 degrees.
Encourage pupils to develop an awareness of these designs in nature, in indus try, and in other places.
Have pupils learn to identify these figures by name, especially the family of quadrilaterals and triangles (scalene, isosceles, equilateral, right, acute, obtuse).
Provide practice for drawing and constructing these figures. Review meaning of perimeter.
Use graph paper to explain the number of square units contained in the area of a rectangle.
Review understandings about the circle, its construction and its parts. By a series of measurements of circular objects, lead pupils to discover for themselves the relation of circumference to diameter.

143

Cone Sphere E. Similar triangles
F. Indirect measurement 1. 3-4-5 Relationship (Pythagorean rule) 2. Square root

Help pupils to discover the formula for area of a circle. Cut a circle into pie shaped pieces. Rearrange the pieces to obtain a figure which approximates a rectangle whose height is the radius of the circle and whose base is half the circumference.
Encourage pupils to develop an awareness of these designs in nature, in industhe triangles are similar.
Continue this procedure using other kinds of triangles that are different in size. Help pupils conclude that if triangles are equiangular, they are similar.
Use similar triangles in indirect measurement to determine the missing term by use of ratio and proportion.
Eighth grade guide on geometry gives detailed directions for proving the Pythagorean property.
Check to be sure the concept of square root is understood. Then ask pupils to find square roots by each of the following:
(a) An algorism of the approximation method
(b) Prime factor technique
(c) Table of square roots

VIII. Problem Solving

Lead pupils to think of symbols of operation as "operators" used to indicate a particular pattern of mathematical computation.

A. Vocabulary

Add

Sub.

+

Mul.
X, ab, aob

Div. a ,(b "'" 0)
b

Because of the wide differences in reading abilities, plan exercises for developing a vocabulary necessary for problem solving. Use techniques for unblocking words by configuration clues, picture clues, structural analysis, and the like.

Give exercises in vocabulary as (a) find meanings, (b) match words with objects, and (c) classify and identify words and concepts,

Use teaching aids and concrete devices.

Make drawings or diagrams of the problem (number line, any form of grouping, or graphs).

B. Mental computation

Have pupils think through a problem then work it orally, giving a reasonable estimate for his solution.

Provide practice in solving problems orally. This helps pupils to concentrate by focusing his entire attention on the problem.

C. Steps in problem solving

Suggest pupils follow a definite procedure in problem solving as:

a. Read the problem and list information given b. Determine what is to be found c. Estimate the result d. If necessary, draw a sketch, or diagram e. Solve problem using formulas or logical reasoning f. Check answers

144

GENERAL MATHEMATICS VOCABULARY

approximate number ascending order associative property average (mean)
betweenness binary operation binary system
chord circle circle graph circumference circumscribe closed curve closure coefficient commutative property complementary angle composite number concentric circles cone congruent triangles consecutive natural numbers constant coordinate corresponding angles counting number cube cylinder
data decagon decimal notation decimal system degree descending order diagonal digit

dimension directed number distributive property divisibility duodecimal
edge element empty set equation equiangular equidistant equilateral equivalent equivalent fractions even number exact number exponent
face (geometry) factor factorization finite formula fraction frequency distribution
histogram hypotenuse
identity element inequality indirect measurement infinite inscribed integer irrational inverse
lateral area line

median mode multiple
natural number negative negative integer number sentence number phrase
odd number open sentence
parallel perimeter pi . pictograph place holder place' value power precision prime number proportion
quadrant quadrilateral
rate ratio rational number reciprocal
significant digits similar polygons solution set square root statistics supplementary angles
variable variation
zeto

145

VOLUME II
BOOKS FOR PUPILS
Adler, Irving. The Magic House of Numbers. John Day. 1957. Adler, Irving. The New Mathematics. John Day. 1960. Adler, Irving.tim~ in Your Life. John Day. 1955. Adler, Irving; Hess, Lowell, and Fehr, Howard. The Giant Golden Book of Mathematics.
Golden Press Co. 1960. Anderson, Raymond. Romping Through Mathematics. Alfred A Knof, Inc. 1947. Bakst, Aaron. Mathematics. Its Magic and Mastery. D. Van Nostrand Co., Inc. 1952. Bakst, Aaron. Mathematical Puzzles and Pastimes. D. Van Nostrand Company, Inc. 1954. Bell, Eric I., Men of Mathematics. Simon andSchuster, Inc. 1937. Bendick, Jeanne. How Much and How Many. McGraw-Hill. 1947. Freeman, Mae and Freeman, Ira. Fun With Astronomy. Random House, Inc. 1953. Freeman, Mae .and Freeman, Ira. Fun With Figures. Random House, Inc. 1946. Galt, Thomas. Seven Days from Sunday. The Thomas Y. Crowell Company. 1956. Gardner, Martin. Mathematical Puzzles. Thomas Y. Crowell Company. 1961. General Moters. Department of Public Relations. Precision: A Measure of Progress.
General Motors Corp. 1952. Hogben, Lancelot. Mathematics for the Million. W. W. Norton and Company, Inc. 1937. Hogben, Lancelot. The Wonderful World of Mathematics. Garden City Books. 1955. Kenyon, Raymond. I Can Learn About Calculators and Computers. Harper and Brothers.
1961. Larson, Harold D. Enrichment Program for Arithmetic. (A series of pamphlets for each
grade.) Row-Peterson. 1956. Meyer, Jerome A. Fun With Mathematics. World Publishing Co. 1952. Molt-Smith, Geofrey. Mathematical Puzzlers for Beginners and Enthusiasts. Dover Pub
lications, Inc. 1954. Reid, Constance. From Zero to Infinity. Thomas Y. Crowell Co. 1955. Ruchlis, Hy and Englehart, Jack. The Story of Mathematics. Harvey House Publishers.
1958. Sanford, Vera. A Short History of Mathematics. Houghton Miflin Company. Smith, David Eugene and Ginsberg, Jekuthial. Numbers and Numerals. The National
Council of Teachers of Mathematics. 1937. Smith, George O. Mathematics. The Language of Science. G. P. Putnam's Sons. 1961. Sticker, Henry. How to Calculate Quickly. Dover Publications, Inc. 1956.
146

REFERENCES FOR TEACHERS A. Periodicals

Mathematics Student Journal. National Council of Teachers of Mathematics, 1201 Sixteenth Street N W

ton 6, D. C.

.

, . " Washing.

School Science and Mathematics. Central Association of Science and Mathematics Teachers, Oak Park Illin .

The Arithmetic Teacher. National Council of Teachers of Mathematics, 1201 Sixteenth Street, N. W' W o~.

6, D. C.

., ashmgton

The Mathematics Teacher. National Council of Teachers of Mathematics, 1201 Sixteenth Street N W W h'

6, D. C.

.

' . ., as lDgton

B. Books
Adler, Irving, The Giant Golden Book of Mathematics. Golden Press Co., 1960. Adler, Irving S. The New Mathematics, John Day Company. 1958. Banks, J. Houston. Learning and Teaching Arithmetic. Allyn and Bacon, Inc. 1959. Bell, E. T. Men of Mathematics. Simon and Schuster. 1937. Bell, E. T. The Queen and Servant of Sciences. McGraw-Hill. 1951. Brandes, Louis G. A Collection, of CrossNumber Puzzles. Teacher's Edition. J. Weston Walch Company. 1957. Brueckner, L. J., Grossnickle, F. E. and Reckzeh, John. Developing Mathematical Understandings in the Upper
Grades. John C. Winston Company. 1957. Clark, J. Rand Eads, Laura K. Guiding Arithmetic Learning. World Book Company. 1954. Clarke, John R, Jimge, Charlotte W., Mooser, Harold E., and Smith, Rolland R. Teaching Arithmetic in Grades One
to Eight. World Book Company. 1957. Dantzig, Tobias. Number, The Language of Science. McMillan. 1954. DeVault, M. Vere. Improving Mathematics Programs. Charles E. Merrill Books, Inc. 1961. Dumas, Enoch. How to Meet Individual Differences in Teaching Arithmetic. Fearor Publishers. 1957. Eves, Howard. An Introduction to the History of Mathematics. Rinehart and Company, Inc. 1955. Fliegler, Louis A. Curriculum Planning for the Gifted. Prentice-Hall, Inc. 1961. Gamow, G. One, Two, Three Infinity Viking Press. 1958
General Motors. Precision - A Measure of Progress. General Motors, Department of Public Relations, Educational Service.
Grossnickle, F. E. and Brueckner, L. J. Discovering Meanings in Arithmetic. John C. Winston Company. 1959. Grossnickle, F. E. and Brueckner, L. J. Principles of Mathematics. John C. Winston. 1959. Grundlach, D. H. Basic Mathematics for Elementary Teashers. Educational Research Council of Greater Cleveland,
1961. Hogben, Lancelot. The Wonderful World of Mathematics. Garden City Books. 1955. James, G. and James, R. C. Mathematics Dictionary. Nostrand Company, Inc. 1959. Larsen, H. D. Enrichment Program for Arithmetic. Peterson and Co. 1956. McSwain, E. T. and Cooke, Ralph J. Understanding and Teaching Arithmetic in the Elementary School Henry Holt
and Co. 1958. Meserve, Bruce and Sobel, Max. Mathematics for Secondary School Teachers. Prentice-Hall. 1962. Mueller, F. J. Arithmetic: Its Structure and Concepts. Prentice-Hall. 1956. Morton, R. L. Helping Children Learn Arithmetic. Silver Burdett Co. 1960. National Council of Teachers of Mathematics. Insights into Modern Mathematics. Twentythird Yearbook. The Coun-
cil. 1957. National Council of Teachers of Mathematics. Evaluation. Twenty-sixth Yearbook. 1961. National Council of Teachers of Mathematics. Instruction in Arithmetic Twentyfifth Yearbook. The Council. 1960. National Council of Teachers of Mathematics. The Growth of Mathematical Ideas. Twentyfourth Yearbook. The _
Council. 1959. Russell, David. Children Learn to Read. Ginn and Co. 1949. Sawyer, W. W. Prelude to Mathematics. Penguin Books. 1957. Spitzer, Herbert. 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 Arithmetic. Holt Rinehart and Winston. 1960.

147

C. Articles, Pamphlets, and Bulletins

A Guide to the Use and Procurement of Teaching Aids for Mathematics. Prepared by Emil Berger and D. A. Johnson. Publishers: National Council of Teachers of Mathematics.
Cili~k, John R. et al Notes for the Arithmetic Teacher. Yonkerson-Hudson: World Book Company.

Dean, Richard A. "Defining Basic Concepts of Mathematics." The Arithmetic Teacher, Vll:8. March 1960.

De~igning the Mathematics Classroom. National Council of Teachers of Mathematics.

Education for the Talented in Mathematics and Science. By K. Brown and P. Johnson. Bulletin 1952 No. 15. 34 p. ; Superintendent of Documents. U.S. Government.

Faulk, Charles J. and Landry, Thomas R. "An Approach to Problem Solving." Arithmetic Teacher. 4:157160; April, 1961.

Free and Inexpensive Aids for the Teaching of Mathematics by Kenneth E. Brown. Circular So. 348. 7 p. Department of Health, Education and Welfare. U.S.Government.

Gesling, Martha M. "The Demon of Arithmetic - Reading Word Problems." A monograph from Elementary Teachers. Evanston: Row Peterson and Company. 1954. 7 p.

Ginn Games for Arithmetic, Grades 3-8.' Bulletin. Atlanta: Ginn, 1959.

Hamilton, E. W. "Number Systems, Fad or Foundation?" The Arithmetic Teacher. VTII:5, May, 1961.

Hooten, Joseph R. "The Modern Language, of Elementary Mathematics." The Resourceful Teacher. Silver Burdett.

Johnson, Donovan A. and Glen, William H. Exploring Mathematics on your Own Series. St. Louis: Webster Publish, .' ingCompany, 1960; ;

Jones, Emily. "Decimals versus Vu~ar Fractions." The Arithmetic Teacher. 4:184-188; April, 1960.

Mueller, Frances J. "On the Fractions as a Numeral." The Arithmetic Teacher. 8:5; May, 1961.

Research Service Department. Helping Teachers Teach Arithmetic. New York: Silver Burdett Co., 1954.

Selected Bibliography of Current Articles n Mathematics Education by Kenneth E. Brown. Circular No. 346 Department of Health, Education and Welfare. Washington, D. C.

Selected Bibliography of References ,and Enrichment Material for the Teaching of Mathematics by Kenneth Brown. Circular No. 347 7 p.Department of Health Education and Welfare.

The Elementary and Junior Hig'h' School Mathematics Library by Clarence Ethel Hardgrove. Publishers: National

Council of fTeachers of Mathematics.'

,

Thorpe, Cleata B. "These Problem Solving Perplexities." The Arithmetic Teacher, 4:152-156; April, 1961.

Un~el;,Ester R. "Children Are Naturals at Solving Word Problems." The Arithmetic Teacher. 4:161-163; April,1961.

Urbanek, Joseph J. (ed) Mathematical Teaching Aids. Chicago School Journal, Vol. 35:3-6. Chicago Teachers College, 1955.

Van Engen, Henry. "Rate Pairs, Fractions, and Rational Numbers." The Arithmetic Teacher, 7:8, December, 1960.

Vincent,J. G. and Hunnicutt, C. W. What Does Research Say About Arithmetic. Washington: National Education Association, 1958, 4 p.

Woodward, Edith J. and McLennan,. Roderick C. Elementary Concepts of Sets. New York: Holt Rinehart and Winston, Inc., 1959.

D. Courses of Study
Denver Public Schools. The Mathematics Program of the Denver Public Schools: Grade Six. Denver, 1960. High School Mathematics~ Units IIV. University of Illinois Press, 1960. School Mathematics, Study Group. Mathematics for Junior High School, Volumes I and II. Stanford University:
School Mathematics Study Group, 1961.
University' of Maryland Mathematics Project. Mathematics for the Junior High School, Second Book Part 1. Maryland: College of Education.

148

GLOSSARY FOR TEACHERS

ABSCISSA. U the ordered pair of numbers (a, b) are the coordinates of a point of a graph, the number a is

the abscissa.

.

ABSOLUTE ERROR. One -half the smallest marked interval on the scale being used.

= = AIaBISOLU-aT.EOVnAtLhUe En.uTmhbeerablsinoelutaebsvoalluutee

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
a in the set. The :;ymbol for the identity is

set of usually

numbers that has 0; in the complex

the following numbers it is

p0ro+peOrti,y:anId

+
in

a a for all 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.

ALGORITHM. (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. U 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 MEANS. 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 arid multiplication are stated as (a + b) + c a + (b + c)
= and (a X b) 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.

149

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 more than the base of ten literally means "ten and one". Twenty means two tens or two of the base.
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 l.
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 Y B.
CHECK. To verify the correctness of an answer or solution. It is not to be confused with "prove".
CmCLE. 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.
cmCULAR 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.
CmCUMFERENCE. 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 pictures, 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' -l.
COMPOSITE NUMBER. A counting number which is divisible by a smaller counting numbers different from l.
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 differ 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. {t, 2, 3, 4, ... j
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, etc. In measurement of time it represents a period of ten years.
150

DECIMAL EXPANSION. A digit. for every decimal place.

DEDUCTIVE 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 aooO.

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.

DIFFERENCE. The answer or result of a subtraction. Thus, 8 - 5 is referred to as a difference, not as a remainder.

DIGIT. Anyone 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".)

DIHEDRAL 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.
= DffiECT 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) (3X10) + (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 con-
= = tained in another. For any real numbers a and b, b o 0, "a divided by b" means a multiplied by the reciprocal of
b. Also, a -;- b c, if and only if a b c.

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 '" 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 band 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.

151

EXPONENT. In the expression aD the number n is called an exponent. If n is a positive integer it indicates how many times a is used as a factor.
aD=axax ... xa
n fa~tors Under other conditions exponents can include zero, negative integers, ratio~al 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 m q = 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. The 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. If 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. If a sphere is divided into two parts by a plane through its ,center, each half is called a hemi-

= sphere.
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'+2xy+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.

152

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 ",,", >, <, 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. If 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. Two or more lines that pass through a single point in space are called intersecting lines.
n INTERSECTION OF SETS. If A and B are sets, the in tersection of A and B, denoted by A 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. If 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 f-'.
INVERSE VARIATION. The number y is said to vary inversely as the number x if xy = k where k in a constant.
mRATIONAL 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 if 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 UJ: 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 Z 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 %.S.
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:

1 [ a,a2

bb2,

c1
C2

d1 d2

J aa ba Ca da
a4 b4 C4 d4

153

MEAN. 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 intro-
duc~d.
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.
= M1JLTIPLICATIVE INVERSE. The multiplicative inverse of a non-zero number a is the number b such that
ab 1. It is usually designated by '/. or a-I. MUTUALLY DISJOINT SETS. Two 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.
= NFACTORIAL. The expression "n!" is read "n factorial". n! n(n -1) (n -2) ... 2.1.
NULL SET. A set containing no elements. It is sometimes called an empty set. The symbol for the null set is '" 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 groups whose sizes are powers of 10. That is, ones (10), tens (10'), 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, 16/2, 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" a" ... , am) such that a, is the first number, a, is the second number, a, is the third number, ... , and am 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 the pair. 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.
154

PARTIAL PRODUCT. Is used in elementary arithmetic with regard to the written algorithm of multo r .

Each uct s.

digit

in

the

multiplier

produces

one

partial

product.

The

final

product

is

then

the

sum

of

the

parItl~aIlcaptrlodn-.

PARTIAL QUOTIENT. In long division, any of the trial quotients that must be added to obtain the

It

quotient.

comp e e

PERIMETER. The sum of the measures of the sides of a polygon. The measure of the outer boundary of a poIygon.

PERIOD. The number of digits set off by a comma in an integer or the integral part of a mixed decimal In

repeating decimal the period is the sequence of digits that repeats.

.a

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 domain of f. The smallest positive number p for which this holds is called the period of the function.
PHRASE. 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,xn - 1 + ... + an_'X + an 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 oF O.
PYRAMID. A pyramid is a polyhedron, one of whose faces is a polygon 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.

155

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. If a and b are whole numbers with b not zero, the number represented by the fraction 01. 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
of all points C for which it is true that B is between A and C. The point A is called the end-point of AB.
RECIPROCAL. Multiplicative inverse.
RECIPROCAL FUNCTION. Pairs of functions in the set of real numbers whose product is 1
Example: (Sin </ (Csc </ = l.
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 P' in the line AB if P, P', and AB all lie in the same plane with P and P' on opposite sides of AB and if the perpendicular distances PO and P'O to the point 0 in AB are equal.
REFLEXIVE PROPERTY. If 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 mul-
= tiplication, 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.
RIGHT 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.
RIGHT 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 magnitude 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.
156

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 a number 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 (PIP2) is the num ber m such that m pair (Xl' Yl) and P2 is the ordered pair (x2, Y2)' SOLUTION SET. The truth set of an equation or a system of equations.

y~
X9.

--xy,,

where PI is the ordered

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 fro m a sample.

= = = SUBTRACTION. To subtract the real number b from th e 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+l.

n

l : 1: ' SUMMATION NOTATION. The symbol ak. 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
= l : ak a.+a.+a.+a.

k=2

00

I : When the summation includes infinitely many terms it is written

a.. 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. If 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. If 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 ha ve the transitive property.
TRIANGLE. If A, n, and C are three non-collinear points in a given plane, the set of all points in the seg-
ments having A, B, C as their end-points is called a triangle.

UNEQUAL. Not equal, symbolized by =1=.

157

------_-...._-

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 x.S.

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.

r VECTOR. In physical science, a quantity having magnitude and direction. In mathematics a vector is a matrix

of one row or one column as (al

bl

cl )

or

I

al a2

1 I

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.

158

=-:FLOW ~CHART

======'=K===='=1= ==-=='= = =--=2======-1==3========\= ~~~4==

5- .I

6

NUMERATION

7

..I

8

I I 111 I ELEME9NTARY

GENE9 RAL .,

10

ALGEBRA

MATHEMATICS GEOMETRY

MEDIINATTEER ADVA12NCED ALGEBRA ALGEBRA

I 11 or 12 Consumer Mathematics

HISTORY

History of Roman numerals

Appreciation of decimal 1\YStem through historical development of other number

Historical development of numeration, Hindu-Arabic, retaught and extended through re-

Historical development of algebra

Research to show how historical background helps pupils discover what constitutes the modern num-

Appreciation of development in geometry from ancient to present

Continuation of historical development of algebra

History of efforts to solve equations of degree greater than two

Historical developments of taxation, investment and credit system

systems

search

ber system

time

-------------------~c-ar-d-i-na-l-s-----l-R=~e~a-di-n~g-a-n~d~-w~r~it-in-g---C--ou-n-t-in-g--ex-p-e-r-ie-n-c-e-s-e-x-te-n-d-e-d~---------C-o-u-n-ti-n-g-t-h-ro-u-g-h--3-00--b-y-1-'s-,-2-'s-,-3-'s-,-4-'s-, 1---C-o-u-n-tin-g-b-y--6-'s-, -7-'s-, -B-.'s-, -9-'s-, -a-n-d-b-a-c-k----Or-d-i~-a~l~s~th~r~o~ug~..u~~rh.u~n~d~-re--d-th-----------t-Cro~nc~ep~t ~an~d ~us~e ;ofrolrd;in~al ~an~d~ca~rd~i- rC~o~mp~ar~e ~de~ci--~S~ym~bo~lis~m=ex-- ~-~Irr~at~ion~a-l --lK~n~ow~le~dg~e ~o-f -,~I~ntr~od~uc~e -r~Pu~re~~im~ag-in1a--~P~o~la-r-f-or-m--o~f f----------------

SYMBOLS FOR

through 12 to 99

through 200 by 1's, 10's, 5's, 2's;

5's; through 1000 by 10's and 100's; ward from 100 to 1

Reading and wr ing beyond LXXX

nal numbers extended

mals and

tended to in- numbers

Roman numer- symbols

ry number

complex

NUMBERS AND

Ordinals

Counting -by 2's, s:;, through 20 by 4's; through 30 by 3's;

backwards from 100 by l's, 2's, 5's, Ordinals through fiftieth

Understanding L C, D, M, and their Reading and writing Roman numerals Roman sys-

elude Baby-

als and other

used rar- system, the

numbers

COUNTING

through

10's to 99

through 1000 by 100's

lO's

Reading and writing through LXXX and combinations

with facility

.

terns of nu- Ionian and

symbols main- ticularly in symbol i

1. Hindu-Arabic

fifth

Reading and writing Ordinals through fifteenth

Ordinals through thirtieth

recognizing D, C, M

Counting and OI!erating in base 8 Comparison of Roman numerals with meration

Egyptian

tained

geometry Complex num-

2. Roman

Reading and

number names

Reading and writing Roman numerals Reading number names through 100

Base 5 introduced

and base 5

Arabic system

Computation Computation

Bases a, 12 and

ber

3. Others

writing

through 10

through Xll

Writing number names through fifty

Extend operations in other bases

with bases

with bases

others reviewed

numbers 1-5 Ordinals through tenth

Reading and writing through XXX

through 9

through 12

RECOGNITION OF
CARDINALITY OF SETS (Groups)

Sets (Groups) to 3
Vocabulary introduced by using objects More than and less than

Sets (Groups) to 10 One to one matching Members of a set Vocabulary extended to understanding of more than, greater than, less than, and others

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 ~ ~

Union of sets (U) Intersection Cnl Subsets

Extend understa ding of sets

Extend understanding of sets

Extend under- Strengthen

standing of

the concept

sets

of sets

Set notation including finite and infinite sets

Interpretation of sentences expressed 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

BASES FOR NUMER
AT ION 1. Decimal
Notation (Base 10) and Place value 2. Notation and

Place value of ones and tens Reading and writing two digit numerals

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 wl'iting 4 digit numerals (perhaps 5 and 6 digit)
Understanding use of 10 as the base of our system of numeration
Zero in 1's, 10's and 100's place Regrouping numbers through 100

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 p!aces Reading and w1i ng decimals to hundredths Place value thrc !ugh millions Place value in ase 5 and 8

Numerals to 12 places Base 5 and base 8 reviewed Place value in bases other than 10 Reading and writing decimals to
millionths

Extend understanding of place value in the decimal system and other bases

Emphasize "base" and "position" by use of exponents

Logarithms to different bases and method of conversion

Maintain skills developed in our decimal notation

Place Value for

Other Bases

VARIABLES

Use of 0 in mathematical sentences

Use of 0 and N in mathematical sentences

Use O, A, and N in mathematical sentences

Use of D and N P, equations;
in problem solvlig

Use mathematical sentences in problem solving

Use in mathe matical sentences and formulas

Use of variables extended Application made in number sentences

Variable de fined in set language Domain of the variable

Usage reviewed and extended in equations

Quadratic variables Equations with three linear variables

Sine x and cos x as vari ables in real number system

--------------------~----------~----------------~----------------------------------1----------------------------------r--------------------------------- --------------t----------------r,u~s;.e~in~ar~e:a~f~o;r:m~u~la:s~.i~in~:p~la:c:e~va~l~u:e1-1E~x~p~o~n~e~n+.ts~---~~U~s~in-g~b-a-se-s---~~P~o-ffi~.t~i-ve-,--n-e-g-.+-R=-ev-i~e-w---an-d~-e-x--I------~---~-N~e-ga-t~iv-e-,----~--F-u-r-th_e__r _i_n__---1------------

EX PONENTS

of the decimal ana other number bases

used to emphasize role otphfoesibtaidoseene1 mai nna1d Zseyrsoteemxponents

other than ten

ative integral exponents Zneerontsexpo-
Multiplying and dividillg

tension

fractional, irrational exponents

vestigation with irrational number
as exponent

SERIES AND SEQUENCES
PROBABILITY AND STATISTICS

Simple averages

Mean, median, and mode

Use of mean, median, and mode Frequency distribution

Histogram and frequency polygon

Meanings, sum of series, sum of sequences, sum 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 and interpretation of data. Percentile in terms of normal curve

FLOW CHART

K

2

3

4

I

I I I I I I EL~MlNTARY IMATHE~ATicsl GEO~~TRY INTE~~DIATE ADvl~cED c~n:~.,::,- 1
sO11 PERATIONS=====6===========1======a===== ~LG=EB~RA=9==G=EN=ER=AL=====~~====AL=GE=BR~A ~==~dALbGE=BR=A ~~~M~ath~ ema~tlc- :a

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

M stery of 81 basic combination facts;
Afldcitns gin1v,ol2v, inagndze3rodigit numbers with addends with internal zeros and rerouping
Su tracting 2 and 3 digit numbers with mternal zeros and regrouping

Adding 4 digit numbers, regrouping to lO's, lOO's, and 1000's
Adding 7, 1 digit numbers; 5 and 6, 2 and 3 digit numbers; and 5 four digit numbers
Subtracting with regrouping

Addition of 7 digit numbers Subtraction with regrouping

Commutation with high degree of mast- Whole numbers,

ery

decimal and

common frac-

tions

Use of

Alg braic ex-

pr operties prlssions

in solving

problems

Application of commutative; associative, and distributive properties

Addition and sub- Vector addition of

-

traction with com- complex numbers

plex numbers

MULTIPLICATION AND Readiness

Regrouping of

Adding twos and dou- Discovering and mastering basic multi- Developing multiplication and division Multiplicat ion with 4 digit multipliers Use of 4 digit multipliers;

Whole numbers, Use of Alg1 braic ex-

Application of

Multiplication and Multiplication and

DIVISION

2's and 3's to form other groups

bles Separating twos and
doubles

plication and division facts with 2's, facts with 6's, 7's, B's, 9's

using zero in different places

's, 4's, 5's, and lO's M ltiplying by 1 digit multipliers

Multiplying by 2 and 3 digit multiplier Dividing by 1 digit numbers with and

T2 edstisgiftordidviivsiosrisbilit)l1continued

3 digit dividers Tests for divisibility extended Casting out 9's

decimal and common fractions

properties prEssions in solving problems

properties to solving problems

division of complex numbers, 12 =-1

division of complex numbers in polar form

Di idmg by 1 digit divisors

without remainders

Rounding divisors to get first quotient Distributive law

Rationalization of DeMoivre's t heo-

Understanding what is meant by aver- figures

denominators

rem

age

Powers and r oots of

_F_R_A_C_T--IO__N_S--(C_o_m__m_o_n~)-+--I-n-tr_o_d_u_c_e---th-e~~U~n'd-er-s~t~a~n'd~in~g~--+-'M~ea~n~i~n~g~o~f~~a~n~d~-o~f~t-~~~~~---~~~~--------------------~~FI~in~nt~rdo~idn~ug~~cpe~arf~atcs~t~oor~fin~wg~h-ol~e__a_n_d~o-f;-g=r_o_u_p_sT-E~q=u~iv=a~le=n=t~fr=a=c~t~io-n~1~-----------------t~M~u~lt~ip=~ly:i:ng:~a~n~d~d7i.v:i~d7i.n~g~fr=a~c:t~io~n~s~----t-Fr~a~c=t7io~n~s~i:n-:si:~m-41~E;,:xt~e:n~d~=us=e=TC~-o~n-tp7le-x-:f-ra-c------7-~P~r-a-ct~i-ce~i~n--~--------------+-~R~a~t7io-n-a~l--ex-p-r-e-s-s~io-n-s7-co~m~pl~ex~n~um~b~e~rs~-----------

idea of one-half and writing .!.
of a whole 2
Understanding
and writing !.
of a whole 4

an object ;; ;- 2' ' 4' 5' 6' a of a whole and of a group

. . . 1 & 1 U e of symbols for fractions and use of
Familiarity With: 2 4 erms numerator and denominator

of a group and of an object

U

~eriotfindgemciomnaely

point in reading expressions

and

Introduce use of deci-

mal point

Simplifying fractions

Use of fundamental operations in deal

ing with money

-

Relationship between the cent sign and

decimal point

Adding and subtracting like fractions

Addition, subtractiop, multiplication, and division of factions
Adding and subtracting tenths and hundredths with regrouping
EDsetciimmaatliomn ixofedrensuuml~sbers

Short cuts of multiplying and dividing by lO's, lOO's, and 1000's
Estimation and r ounding off Use of reciprocals Distributive law
Multiplying and dividing decimals

plest form Division of fractions by use of Principal of One; by com-
mon denominator method

of fractions

tio s Rat onal expressio11s Pro erties of a fie d

computation using fractions

Complex algebraic fractions

PROPERTIES

Understanding of commutative, associa- Use of proper terminology tive, and distributive properties

Extend use of Laws of Arithmetic

FielII properties

Commutative, associative, and distributive

Properties of field, integral domain, group

FACTORING

U e of Factors

Use of Factors

Use of factors in dealing with fractions Use of factors in more complex frac Factoring com-

Use of terms factor1 prime, multiple

tions and in getting common denomi posit numbers

nator s

into primes

H. C. F .

L. C. M., Nat ral numbers

H. C. F .

(pr mes)

and L. C. D. Co,rnl!non monomi-

use in

al actor

solving Diff~rences of

problems sqt ares

Per ~ct squares Qua~atic tri-

no nials

Importance of factoring in solving problems

Basic laws reviewed Special cases of factor ization including trinomials reducible to differ ence of squares

Factor theorem, synthetic division

PROPERTIES OF 1 and 0

Di covering generalizations: n+o=n; Understanding of the generalizations in Reviewing general~tions

Extend use of property of 1

~-o=n; n-n=o; n x o=o; ox n=o

reference to 0 and 1

Use of property or f 1 in simplifying

xn=n; n...;...l=n

fractions

n+ O-n; O+n-n nx l=n; lxn=n n-O=n; n...;...l=n

Emphasize If ab- O then

properties a= or b=O;

and use

a + (- a) = 0;
ax~=a

axl-a;..! - a
1

Use of property of
I one in simplifying fractions

Idea that 1 and 0 need not resemble 1 and 0

PROBL~M SOLVING

TYPES OF PROBLEMS Daily activity

One step problem Gradual introduction of abstract terms

Pictorial representations Development of problems with word symbols and number symbols Relating all four processes in simple oral and written problems

Ar.alyzing and solving one step probems using mathematical sentences
Es imation of answers Ct ecking of computation

Analyzing and solving problems involving four fundamental processes
Various types of problems Estimation of answers Checking of computation

Using four fundament al operations Using ratio to sympolize rate and com-
parison Finding averages Using equations re ated to processes
taught

Finding quantities in various situations Three basic per-

Expressing numbers as fractional num- centage prob-

erals

lems

Variables with solution set containing Life situation

one or more digits

pr oblem

Processes with variable extended to in-

clude: a x

b c

Extension

Line~ equations Qua ratic equatio s Sys ems of linear quations

Extension

Proof of statements concerning geometric figures

Application of function to physical processes VVorded problen11s involing one vari-
able, two variables, quadratic variables

Solutions of simult aneous equations using set language for representing solutions Practical problems using trigonometr ic functions

LOGIC

Readiness through step by step reason- Extend step by step reasoning

Analyzing prob- Extension Ded11ctive

Statement, con-

Conjunction

ing

lem solving pro-

re sonmg

verse, inverse,

Disjunction

cedure

and contraposi-

Estimation of

tive

answers

Deductive and

inductive reasoning

-

CONCEPT OF PROOF

Readiness

Using concrete

with concrete materials

materials

Matching and other

E tend procedures to show correctness Using more than one solution

simple procedures

of answers

used to show correct-

Using Laws of Arithmetic as reason. Checking

Using Laws of Arithmetic Using more than one solution Checking

Checking reason- Giving

Dlr~lct and indi-

ableness of

reasons rec proof

solution set

for steps

Extension

Direct and indi Direct and indirect Algebra as a logical

rect proof used proof of selected theorems

systen11 with facts capable of being

ness of answers

in prob-

proved from a basic

Comprehending in-

lem solv-

set of postulates

verse process

ing

Proofs emphasized throughout course

--

\

EQUALITY INEQUALITY

K Readiness
Readiness

:Mathematical sentences
Mathematical sentences

2

3

FLOW

4

I5 I

6

Principles of base ten numeration used in understanding a collection of roins EXperiences with measurement
Opportunity to learn 5 is less than 6, and 4 is greater than 3 Use of number line
to show addition

Understanding names for the

same number Commutative law of addition

and multiplication

Associative law of addition

and multiplication

Use of concrete objects to show

1

2

7=4

Demonstration with counters and number line that some numbers are greater than or less than other numbers

Understanding of commutative and associative laws extended

Skill in making comparisons:

unequal, greater than, less

than Symbols:

< , > , <F

Use in equation and problem solving Use of many names for the same number
Use of terms with proper symbols:
<>

Extension to include deci mals
Use of symbols in prob lem solving

CHART
7
RE LATIONS
Simple number sentences

I

8

I I I 9 ELEMENTARY ALGEBRA

9 GENERAL MATHEMATICS

10 GEOMETRY

I

11 INTERMEDIATE
ALGEBRA

I

12 ADVANCED ALGEBRA

I CO11NSoUrM12ER
MATHEMATICS

Properties of equations

Linear equations :Quadratic equa-
tions Systems of 2
equations with 2 unknowns

Solving number sentences

Equality as iden tity in definitions

Linear equations, quadratic equations
Systems of linear equations Systems of quadratic equations Systems of linear equations with
three variables

Reflective, symmetric and transitive properties Proofs of theorems

Nature of inequalities Numbe sentences with inequali-
ties

Nature of inequalities

Linear inequali-

Number sentences with in- ties

equalities

Properties of in

equalities

Review and extension of in equalities

Exterior, remote interior angles of triangles
Longer side and sum of other two sides

Quadratic inequalities Graphs of quadratic inequali-
ties

Axioms of inequalities Proofs of theorems

FUNCTIONS VARIATION

Graphs of

Graphs of simple compari-

sim.ple com- sons

parisons

Ratio tjlld proportion in number
sente~ces

Introduction of concept: Linear Quadratic

Direct Inverse Joint

Application in formulas and in mathematical sentences

Linear, quadratic functions Exponental function Logarithmic function Direct variation: y=mx+b
where b=O

Idea of inverse function Detailed study of circular polynomial, rational, exponental, logarithmic, trigonometric functions

RATIO AND PROPORTION
PERCENT
UNITS OF MEASURE ESTIMATION

Identifying equal groups Understanding of
!. !. !.
2' 8' 4

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 iS the comparison of one

integ r with another (except 0)

Rate Scale

acraawriantgios

Used in solving per cents Scale drawings

Readiness experiences through classroom activit!~ related to 100%

Classroom experiences: 100%, 50%, 25%

Experiences extended

Use ratio to introduce per cent

Per cent compared with cwnmon and decimal fractions
Meaning of three cases of per cent
Use in problem solving

Pe~orll~~~ by use of ratio and pro-
Use of er cents greater than 100 and ss than 1
Roun off per cent Use of er cents in business

Extend use of per cents

Developmental experiences; clock, calendar, money, inch, foot, yard, weight

Telling time: to hour, and onehalf hour Recognition of coins

Skill in esti mating

Identifying and using standard measures as needed
Estimating using objects in classroom

Identification and use of stand ard units of measurement to include relationships between units
Sums and differences by round ing off
Use of number line marked in units of 100 to find approximate differences

Use of standard units of measure continued
Conversion from one unit to another
Adding and subtracting like units
Use of meter sticks Rounding numbers to 1000s

Conversion from one unit to another Meaning of square measure Rounding large numbers

Meaning of cubic measure Principle of regrouping
in multiplication and division
Estimating in linear, square, and cubic measure

MEASUREMENT
Difference between counting and measuring weight and mass
Advant11ges of metric measure over English system
Accuracy and precision
I
Importa ~ce of precision

Nautical measure, longitude and latitude
Square root Measures of science Continued

Review and reteaching of ratio Properties of proportion

Sine, cosine, tangent Similarity of geometric figures Area of sectors

Extend use of per cents

..

V'2 on number line Scientific notation

Linear, area, volume, degree, radian

-

Estimation as compared with
accur~cy ~d
preCISlOn m
measurement

Trigonometric functions as ratios in right triangles

Paying cash through loans and using carrying charge.

Radians as units for measuring arc, angles Methods of conversion to degrees

Extension of use of metric measure

..

I

FLOW CHART

POINTS, LINES AND PLANES
GEOMETRIC FIGURES

K

Recognition of rectangle, line, circle, square, triangle

Explore S<J.Uare, triangle and rectangle

Cube

Cylinder, sphere

2 Straight line, square corner

I I3 I
l
Identify lines and surfaces

I

4

Parallel lines and right angles

5

6

7

GEOMETRIC CONCEPTS

Using number lines to show re lations
Lines, points, plane, ray, line segment, and end point defined
Perpendicular lines

Study of angles: r ight, acute, obtuse

Points, lines, planes, and their properties

8 Recognizing 1, 2, 3, di
mensional space

Recognizing triangle, circle, scqyulianrdee, r,coannea1 cube, sphere

Identifying nd reproducipg ~ircle, square, triaMle, rectangle, cpbe, sphere, con , cylinder

Making, collecting and using geometric figures Finding area and perimeter of square and rectangle

Defining closed curves, parallelogram, triangle (equilateral and isosceles), rectangle, square, quadrilateral, circle, polygon
Studying area and perimeter Recognizing prism, pyramid and
rectangular solid

Using term: degree in longitude1. latitude problems
Extenaing study of area Finding volume Using term "congruent" Law of Pythagoras

Extension of concepts of 6th grade
Perimeters of the triangle, quadrilateral, pentagon
Areas of the triangle, square, rectangle, parallelogram, circle
Symmetry, similar triangles Measurement of angles

Area and volume Measurement of angles

I I I ELEME'NTARY

GEN~RAL

ALGEBRA

MATHEMATICS

10 GEOMETRY

11 INTERMEDIAT E
ALGEBRA

12 ADVANCED ALGEBRA

J1 or 12 Consumer Mathematics

Points, lines, a~ Ruler, postulate, rays, One-to-one correspon-

Graphs of functions and relations

-

P"'"" ., " I segments, congruent dence between ordered on Cartesian plane

segments; parallel

pairs and points on the Addition and subtraction of geo-

lines; lines and

xy plane.

metric vectors

planes; perpendicu- One-to-one correspondence

lar lines and planes between ordered triples

and points in three-space

II

Application of Rays, angles, triangles, Conic sections

measurement o polygons, quadrilate-

geometric fi-

rals, circles, spheres,

gures

prisms, cones, cylinders

CONSTRUCTION

Construction of simple figures by means of straight edge and protractor
Constructing figures to scale

Extension of simple construc- Bisection of angles and lines

tion of scale drawing

Perpendiculars

Three demensional figures from Angles equal to given angles

drawings Using Compass

.

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

Rmeveinetwal ocfonfsutnrdu~ej: tions Skill in use ofi the straight l edge, compass, and the protrac tor

Inscribed and circumscribed circles of tri angles Inscribed and circumscribed polygons of circles Segments of line in proportion

-A-N-A-L-Y-T-IC-A-L---~~~--------~---------+-------------~-------~--~-------------_,----------------------~---------------------+----------------------+-----------------+----------r----------+.~D~eri~ vat~ion~o~f s-lop-e-i~n ~3~ -dim-e-ns~io~na-l -g-eo-m~et-ry-1~T~an-g-en-ts-to-c-u-rv-e-s;-a-re-a-s -un~d-er-~-----------

GEOMETRY

tercept form

Equation of plane

curves

Introduction of 3-di-

Intensive study of straight line

mensional graph

and circle with proofs of theor-

ems from plane geometry

Extension of study of conics, po-

lar form of equations, paramet-

ric equations, rational forms

GRAPHS AND CHARTS

GRAPHS

Readiness

Observing and manipulating

Reading thermometers Making simple
graphs of daily ex-
periences

Making sim le bar and picture graphs
of classroom experiences j
Drawing simple maps of sc 'ool, neighborho d and class

Making and interpreting simple graphs
Extending ability to draw maps and make and read scale drawinis

Picture graphs Bar graphs Line and circle gr aphs Reading maps

Extension of construction of graphs and maps to include

Construction of graphs: Histograms, bar, broken line,

large numbers and scaling

circle, pictograms Scale drawings

Extend construction of graphs

CHARTS

Readiness

Readiness Observing and
constructing simple charts

Making simple charts Using information on charts Developing awareness of the function of a chart

Making wea ~er charts, nun ber charts of IJasic facts

Making and inter- Reading simple charts and tables preting charts extended

Reading of charts and table!. extended to include interpretation and construction from statistical data

Reading charts

Frequency distribution chart and table

- -------+------------------------ -----------------~----------+-----------~----------------+--------~----~--------------+------------------------1-------------

jNumber line Coordinat e plane Linear and quadratic equations

Inter pretation and constructio of gr aphs

Introduction of 3-di mensional graph Linear equation Quadratic equations in form; x2+y2=r2

Graphs of linear equations, Quadratic equations Solution of systems

Elementary statistics Extension of grade 7 and 8 skills Inequalities in ~ plane

Table of trigonometric functions

Table of Logarithms

Plotting graph of function defined by algebraic equation
Study of graph of exponential and logarithmic function
Writing equation from points on graph
Advantage of quantity and quality buying. Home owner ship versus rent.

===================~======================~==~==========~======================~=======~~==============~==========~================~==========~============d:================~======
II