Mathematics for Georgia schools, volume III [1962]

for Georgia Schools
VOLUME 3
I

D EPA RTMEN T OF EDue A T ION

CLAUDE PURCELL, SUPERINTENDENT

ATLANTA,

GEORGIA
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:dv-<!CAJ l,y. 7),"v;s/c; .. o{ Lf. sit', ~t/~d

MATHEMATICS

FOR

GEORGIA SCHOOLS

VOLUME III

l

STATE DEPARTMENT OF EDUCATION DIVISION OF INSTRUCTION CURRICULUM DEVELOPMENT

Claude Purcell

State Superintendent of Schools

Atlanta, Georgia

1962

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

Members

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

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

Ui

FOREWORD

TO: Georgia Teachers of Mathematics

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

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

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

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

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

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

DR. CLAUDE PURCELL . State Superintendent of Schools

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

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

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

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

moon-going age.

They never lost sight of the idea that mathematics is a daily necessity for people. They grasped the fact that it is not simply the college-bound who need sound foundations in this subject. Youngsters who plan to work in a filling station, or own a farm, or be a homemaker, will need sound grounding in figures. These are important for the purposes of daily living; income tax, balancing a bank account, figuring a crop return, totaling up the grocery bill, measuring a fence, or building a house. Modern mathematics makes mandatory accurate mathematical knowledge. Whether it is John Glenn figuring a race through space or a young couple figuring their budget, they need mathematical accuracy and understanding.

There is a new interest in math throughout the nation, a ground-swell of concern that students get the foundations of it. Georgia must be in the fore-front of good math teaching and learning. This Guide is a fine addition to the many things that Georgia teachers and administrators have done to improve the schools so that schools can improve our people.

I hope you find it helpful.

Claude Purcell State Superintendent of Schools

tv

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 manipu late 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 indica\es ways in which mathematics programs can be improved. Concrete matei"iaJs, 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 rol,e 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

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, Di rector
vi

MATHEMATICS FOR SENIOR HIGH

Mathematics has had such tremendous growth and applications in so many new fields in recent years that the traditional curriculum of the secondary schools has become inadequate. As a result of this growth new subject matter has been created, and parts of older mathematics have been reorganized and transformed. If the need for people skilled in various branches of mathematics is to be met, curriculum revisions are necessary.

The purpose of this guide is to point out some suggested changes and ways of implementing them. It must be remembered that this is a guide, not a textbook. Members of the committee have made careful study of recommendations of the Committee on Mathematics, of suggested programs of various experimental groups, of curriculum guides from other states, and of new textbooks.

Two programs of study are suggested:

College-Capable

Terminal

I. Elementary Algebra II. Geometry ill. Intermediate Algebra IV. Advanced Algebra V. Probability and Statistics
Analytical Geometry

I. General Mathematics II. Consumer Mathematics

The committee has attempted to make the program for the college-capable flexible enough to meet the needs of all such students. For several reasons, geometry is the suggested second year course in high school mathematics. Most intermediate algebra texts are designed for this sequence since they include many geometric concepts. It is believed that if a student has only two mathematics courses in high school, he should have both algebra and geometry; or if he has three, that algebra should be the last course since too great a lapse of time between high school algebra and college algebra is not good.

The needs of the superior student who will want to take all the mathematics offered are carefully considered. If these students are identified in the elementary grades and a special program provided for them, it should be possible for elementary algebra to be offered to them in the eighth grade. If a superior student has elementary algebra in the ninth grade, topics from intermediate algebra can be included so that by the time he is in the eleventh grade he will be able to study advanced algebra. For all such students a fifth year of mathematics is suggested. It is believed that students who complete the fifth year program in high school will be capable of going directly into a study of college calculus.

The terminal mathematics group is intended to provide both for those students who are not capable of doing college work and for those who plan to attend college but whose backgrounds are not sufficient to allow them to study elementary algebra in first year high school. It is assumed that students in general mathematics, which is to be offered at the freshman level, will be th0se who have failed to gain basic understandings and computational ability in the lower grades. General mathematics is presented on an elementary level; understanding of the structure of the number system is stressed; meaningful drill is recommended. Topics not usually included in such a course are presented in the belief that they will help to arouse the interest 'of those students who do not usually like mathematics.

The course in consumer mathematics should be offered to juniors or seniors only. Since most of these students will, within a very few years be working and managing households, they will have an immediate need for the information on budgeting, taxes, etc., which is presented in this course.

Much of the contemporary mathematics recommended by the Commission and by experimental programs is included in this guide. Principal differences between this and the traditional program are stress on development of concepts, terminology, symbolism, and some new topics The goal of instruction should include both development and understanding of concepts and development of manipulative skills.

TABLE OF CONTENTS

ELEMENTARY ALGEBRA

I. Sets and the Number Line

4

n. Numerals and Variables

5

m. Sentences and Properties of Operations

7

IV. Mathematical Sentences and English Sentences

8

V. The Real Numbers

9

VI. Factors and Exponents

.______________ 18

VIT. Radicals and Roots__________________________________

19

vm. Polynomial and Rational Expressions

21

IX. Solution Sets of Open Sentences

23

X. Graphs of Open Sentences in Two Variables______________________________ 24

XI. Systems of Equations and Inequalities

25

XII. Quadratic Polynomials

26

XIII. Functions _ __

_

26

XIV. Ratio, Proportion, and Variation

27

Voca b u l a r y

.___ __

29

GEOMETRY

I. Introduction

_

II. Points, Lines, Planes, and Angles

III. Triangles

IV. Co ngruences

V. Parallelism

VI. Locus of Points

VIT. Ratio and Proportion

VIII. Similarity

IX. Circles and Spheres

X. Areas and Volumes

XI. Inequalities

XII. Coordinate Geometry

XIII. Construction _

Vocabulary

.

_

34

34 36

37

38

40

43

43

. 43

44

45

.__ 46

52

<________________________________________ 54

INTERMEDIATE ALGEBRA

I. Number Systerns

__ 58

II. Polynomials and Factors
m. RationaI Expressions

.________________________________________ 59 59

IV. Functions: Linear and Quadratic

59

V. Quadratic Equations

62

VI. The Complex Number System

63

VIT. Equations of 1st and 2nd Degree in Two Variables

65

vm. Systems of Equations

67

IX. Exponents and Logarithms

69

X. Permutations and Combinations

70

XI. The Binomial Expansion

70

XII. Sequences and Series

...:-_______________

71

Vocabulary

.___

74

I

I
II

ADVANCED ALGEBRA

I. Sets of Numbers ll. Number Systems
m. Special Properties of the Integers __
IVV.. SFpieeclid~sl Properties of.the Natural Numbers

VI. Polynomial in One Variable as Integral Domain

vVmll..

Relations and Functions Circular Functions

IX. Polynomial Functions, Equations, and Graphs X. Exponential Function XI. Logarithmic Function

XU. Trigonometric Functions of the Genera) Angle XIU. Trigonometry Qf Triangles XIV. System~ of CamElex Numbers __ XV. Properties of a Group

Vocabulary .----------------------

. _. .

78 79 80 8812
82 63 86 87 88 89 89 91 _ 92 _ 93
94

MATHEMATICS DEPAR'rMENT EQUlPMENT -----------------.---------- 95

BOOKS FOR STUDENTS

_

96

PROBABILITY AND STATISTICS

I. Introduction

_

102

mD..

Statistics -------------------Probability and I~ l\elation to Stati$tics

. _ 10' -;------ 1()~

Vocabulary ------------.-c--------------------------------------.

103

ANALYTIC GEOMETRY

I. Introduction

-.----------------------------------------,-,,,

108

mll..

The Straight Line --------.---------------The Conic Sections

__. __.__ 109 111

IV. Parametric Equations and Loci

_

112

Vocabulary

114

BOOKS FOR ADVANCED STUDENTS

114

GENERAL MATHEMATICS

I. Numeration

ll. Operations with W):lOle Numbers ---
m. Operations with Fractions
IV. Relations

V. Measurement

...

.__.

VI. Graphs

Vll. Geometry

.

vm. Problem Solving ------------------.

.

Vocabulary

118
1~1
.___ 125 131
.___________ 133 134
.____________ 135
. ., J36 . 137

CONSUMER MATHEMATICS

I. Statistics ---

.-..-

.

mll.. BCureddgiettSinygst-e-m--s----.--------

.______ }A2

. '__

:._

U4
144

IV. Taxes --------_______ V. Insurance VI. Investments -- _
Vocabulary -_~

145 _ lA6
__'_ 147
. 149

REFERENCES FOR TEACHERS

150

GLOSSARY --------________

_ _ 152

ELEMENTARY ALGEBRA

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.

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.
{I, 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 {I, 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 for 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 {I, 3, 5, 7}

Graph
o 123 4 5 6 7 8

{all numbers between 0 and 5}
4

3. Addition and multiplication on the number line

{all numbers between 0 and 5 inclusive}

o 1 2 3 456
(0 and 5 included)

Define closure for addition and multiplication of the natural numbers. 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
r---'~
o 123 456

Multiplication 2 X 3
3 X3
,---A---,~
o 123 ~ 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 parent~eses
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.

5

Associative property Commutative property

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.

b. Corresponding properties for multiplication

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

c. Distributive property

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
-on
48

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.)

552

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

B. Variables 1. Definition

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.

11

You might point out to students that sometimes the pattern or form of a prob-

lem 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.

2. Domain

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.
The set of numbers which a variable can represent is called the domain of the variable. In the case above the domain is ,the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

6

III. Sentences And Properties Of Operations

A. Open sentences 1. Definition

Review truth or falsity of sentences. Then introduce a sentence containing a variable.
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 seITtence?
An open sentence is a sentence 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" =1= means "is not" or "is not equal to"
< means "is less than"
<t 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 <t and> and :t> 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(I) = (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'

~Hl- 5(4) = 20and~= ~(l) = 3r31 = 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
2 5

X

(1)

_3
-2

'5

r151 X l15J

. 10 6

5(2) 3(2)

L(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+b:JI:b+a

7

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 SlimS 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

II

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 rectangle.

(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 not

the "width" but the number of units in the width. Care should be taken to state precisely what number the variable represents.

I

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

I

problem.

Example: We wish to cut a rope which is 28 feet long into two pieces so that one piece will be three feet longer than the other. How long should the shorter piece be?

8

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 student 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

o12 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.

Real Numbers

I

I
Rational Numbers

I
Irrational Numbers

I

I
Integers

I
Rational Numbers

which are not

I

integers

'I

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, -yS is a real number which is a negative
irrational number.

1. Order on the number line

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.

a. Properties of order b. Comparison property c. Transitive property

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 urc real numbers and if a<b and b<c, then a<c.

9

~,..-----------------------

2. Opposites

When pomts 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
"
-3
~

--->

Positive direction

-5 -4 -3 -2 -1 o 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 direction. This places us at -3 on the number line. The second number is positive so now move six units in the positive direction. We see that we have arrived at 3
on the number line, so we can write (-3) + 6 = 3."

After considerable practi<:e of this sort, the student should be ready to formulate a precise definition. He should make use of the idea that the dis-

10

2. Addition property of opposites
3. Addition property of equality
4. Proof

tance of any point from 0 is the absolute value of the coordinate of the point 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 ab-
solute 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, out 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) = O.
You might state this property in words: "The sum of a number and its opposite is 0." If two numbers x and yare such that x + y = 0 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 ad-

.dition rr..ight be used are:

.

5/16 + 28 + (-5/16)

+ (-112) + 7 (-2) + (-3/2) + 2

x = x + [(-x) +3]
n + (n+2) + (-n) + 1 + (-3) =0

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 = -%1 If there is a number x lor which this is true (if the solution set is not the null set) then ]I: + 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 + 0 = (-2) + (-3/5) (Property of opposites)

x = (-2) + (-3/5) (Pi-operty of 0)

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 meaningless manipulative techniques which were learned are based on the fund-

11

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 cait 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) (c<>mmutative 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.

Il l

Instead of the number 2 let's use the variable x to stand for anyone of the

set of real numbers. Then we have:

II

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 bea 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)

12

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)] 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 = -6 (-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 Ibj
2. If one of the numbers a and b is non-negative and the other is negative, then ab = -(Ial Ibj)
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.
Ail studentsshouid 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
13

2. For all real numbers a and b, -ab = -Cab) = (-a) (b) = (a) (-b)

3. Multiplicative inverse

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 - 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.

4. Property of equality

= = 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.

5. Use of properties

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 + (-10)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

6. Equivalent sentences

Stress the fact that operations which involve properties that hold for all real

numbers yield equivalent sentences. Sentences are equivalent if their so-

lution sets are the same. Later equations will be solved using operations which

! I,

7. Theorems

do not yield equivalent sentences. Some useful theorems which might be presented for the superior studen~

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 non-zero real numbers a and b;' 1/ '/b = 1/.

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 solution
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

14

D. Use of distributive property 1. Indicated sums Indicated products Simplifying phrases 2. Several appli,cations
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). U you let [x + (-7)] ccrrespond 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) = x2 - 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:

U 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.

LI [I

a --a-+-'--c------cb

b+ c

n

~ I

b+c

b

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)
-x <-4
4<x

Definition of absolute value Addition property of opposites Addition property of order Associative property of addition Addition property of opposites Addition property of 0 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.

15

3. Multiplication property of order
F. Subtraction and division for real numbers 1. Subtraction
a. Definition
b. Use of definition 2. Division

1. H z = x + y and y is a positive number, then x < z. 2. H 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) U 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 - 27 = 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. H 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 subtraction is related to addition, the definiton of divison could probably be guessed by the students.
16

a. Definition
b. Simplest name for a number
3. Simplifying expressions

For real numbers a and b (b =F 0), "a divided by b" shall mean "a multiplied by the multiplicative inverse (reciprocal) of b."

In symbols,

~=
b

a' t1'

(b =F 0)

The numeral'/b is 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 cO!,lrse.

Theorems which may be proved include:
1. For b =F 0, a = cb if and only. if 'Ib = C

2. For any real numbers a, b, c, d, if b =F 0 and d =F 0 then

ac

ac

b . -d-= bd

3. For any non-zero real number
a, ~ = 1
a
4. For real numbers a and b -a_ a_-a
b- -b- 1)

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 tocror -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
7P

(p =F 0)

(2) 3a2b 5aby

(a =F 0, b =F 0, y =F 0)

(3) 3v - 3 2lY - 1)

(Domain of y must exclude 1)

(4) (3x - 5) - (4 - 2x)

8
(5) (x - 1) (2x - 3 + x)

(Domain of x?)

4x - 4

(6) r - 3 1 r - 7 1
l 8) L 2)

+ (7) xy y
x+l

xy - Y (Domain of x?) x-I

(8) -!. +..!. +..!.23 4

17

(9) 5
3
-1-
"2
(10) ..!.. +...!..
3 -I 1_ 1
"3 .-

VI. Facton And Exponents

A. Factor

You may introduce this idea J>y asking what numbers divide into a certain number exactly, then formulate a precise definition for factor as being a more compact term.

1. Definition of factor and proper factor

The integer m is a factor of the integer n if mq= n, where q is an integer. U the integer q does not equal 1 of n, we say that m is a proper factor of n.

Tests fo.r 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 givl'ln 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 prope~ factors.

The Sieve or Eratosthenes discussed eat;lier 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 2/. to be Il factor of 12, since (3/.) (16)= 12-

Prime factors of many numbers can be found by inspection but if they . canil.ot be found by insp'ection 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 lactors of a number is:

432 2 216 2 108 2 54 2
27 3
93 33 1

432=2x2x2x2x3x3x3

18
II

5. Use of prime factorization to find least common mul t i p Ie
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 n.Imber, 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 tPe 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
v
n factors
The "a" which indicates that number to be used as a factor several times, is called the base. The "n" denotes how many times the factor "a" has been used. "n" is called the expor.ent. The number aD is called the n'b power of a, or sometimes aD 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:
an, an = a m +n

am - n if a = 0 and m > n

an>
a"

an~--Inif a = 0 and n > m

am -aD = 1 if a = 0 and m n

Then by trying to shorterr- 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 -D = l/a"

and we can then write the definition of division as am = am- n an

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 - vb. 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:

19

Cube roots nth roots
B. Irrational numbers
C. Simplification of radicals

2. Restriction on domain of Vx

3. ya

Fa

~b

vb

4. Rational approximation of irrational numben

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
.y -8
After examples using perfect squares have been given, introduce a number like v2 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 y3 Y5?
Does this equal viS? Let's examine
(y3 y'5)2 (v3 y'5)2 = (y'3)2 (y'5)2 = 3.5 = 15 Then v3 v5 must be a square root of 15. Since 15 has only one positive square
root, we can conclude that
v'3 yi5" = ,/15
Drill in simplifying radicals in this way should include discussions of the dqmain of the variable if there is one.
vi ";'Y = Yxy. if x >0 and Y.2: o.
vx~ 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 va'is a real number if a and b are real
v'b . numbers and if a > 0 and b > 0 we know that

= y'a _ va (1)
Vb -: Vb

valVbl'
V'b lVbJ

Emphasize to the student that mding a square root of an irrational number

means findiIig 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
..J2 is between those numbers.
Now by squaring 1.1, 1.2, 1.3, ... 1.9, we can refine this estimate to

1.4 < y2< 1.5 and similarly,

20

1.41 < ~< 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 tOe gether should convince the student that we are getting closer and closer to -)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' + y<; x' + y'; + c' x", 4x" + 3x

2. Degree ofa 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 po"Iynomials

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 th~' 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 admis-

sible

I

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 equations 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 teals are allowed,
x2 - 2 can be factored into (x - -)2) (x + -)2) 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 property used is: If a and b are real numbers such that ab = 0, then either a = 0 or b = O.

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.

21

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
1)T=b(f

(2) ib) -- 1 (3) ~ +~ = a+c
b b -bSimplify:

ab + ab2

a

ab2

Simplify:
x2 + 2x + 1
x2 - 1
x+1
x- 1

1- b
1+b

ab(1 + b)

1- b

'7 a(1 - b) (1 +. b) 1 + b

ab(1 + b) (1 - b) a(1- b) (1 + b) (1 + b)

b[a(1 + b)(1 - b)] (1 + b) [a(1 + b) (1 _ b , By Property (1)

b (1 + b)

If b #- - 1 Property (2) a#-O

x2 + 2x + 1
x2 - 1

x- 1
x+1

(x + 1) (x + 1) (x - 1) (x 1) (x + 1) (x + 1)

, By Property (1)

Simplify:

7

5

----:3=-=6,-'-a=2b- + 24b3

. (x + 1) (x + 1) (x - 1) . (x + 1) (x + 1) (x - 1) Property (1) = 1 By Property (2)

7

5

2232a2b + 233 b3

7

2b2

5

2232a2b 2b2 +. 233b3

3a2 3a2 Property (1)

14b2 2332a2b3 -

+

--;2~33=:2:--a1=2b5'""a"'23'-

Simplify: _ _x x+y

14b2 + 15a2 Property (3)

72a~b3

a#-O and b #- 0

y_ _ x

0 x - y _ _y _

x-y-x+y 'x-y x-y

By use of Property (1)

~ x + y

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

zz

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 eJ<actly 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 rea) 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 rilUltiplied by -131 the sentence +x
obtained would not have been equivalent, since its truth set is {2, - 2}.
In fractional equations specify what values are restricted for the vari~ble. In an example like 1 _ 1 indicate x # 0 and x # 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 + (-Jx 3)2 = 12 X+3= 1
x = -2

If x = -2 then vx + 3 = V -2 + 3
solution set.

--vI = 1. Therefore, - 2 is the

Solve ..JX + x = 2
y!X+x-x=2-x
vx= 2 - x

23

D. AbsolUite value sentences

(-JX}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 - lxl = 1 x - 1 =Ixl (x - 1)2 = (lxl)2
+ x2 - 2x 1 = x2
2x - 1= 0
z=%

Checking reveals that lh is not a soiution to the equation; therefore its solution set is the null set. '

x. Graphs Of Open Sentences In Two Variables

A. Extension from the real num ber 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 r.eal humber plane would be even more helpful. Let's see how numbers can be associated with points of the piane.

I III

:f

r



-3

-~

--12


0


1


2

L
3 45

C

B. Ordered pairs of numbers associated with points

Consider a point A. H 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 dumber 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 th~ 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 the point (b, a) except when a and b are equal. Be sure that students understalld 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 nmnbers.

24

C. Ordered pairs as members of truth sets for linear equations in two unknowns
D. Graphs of sentences containIng absolute values
E. Graphs of Inequalities
F. Writing open sentences from graphs
G. Graphs of more than one sentenee 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=0 2x+y-1=0

A. Systems of equations as com.pound 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.

25

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 """' 0. 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 = lhx2 = (5) y lh(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 % (x + 3)2, and to verify it by = = sketching the graph. They should note that the graph of y 1;2 x2 and y
lh (x + 3)2 have exactly the same shape. Introduce later an equation of the
= form y 1h (x - 3)2 + 2 and have students observe the shape of the graph
and the location of the origin.

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.

B. Standard form of a quadratic polynomial

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. Factoring 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 r eal 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 root!; of equations, or it may be given as a formula which can be d~rived, 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 ass.ociation of numbers is called a function. The given set is called the domain of the function, and the set of assigned num'bers is called the range of the function.
Give examples to show that the rule of a function may be given by a table, a diagram, a graph, a verbal description, or by an expression in one variable.
26

I~ I i('=z79i""?"i ! 9 I

B. Notation
c. Graphs of functions
D. Linear func:tlona 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. U 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 "f of x." The number f(x) is called the value of fat x.

In some cases several equations may be used to describe one function.

= { f(X)

> X, X
-X,

X~0

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.

U 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) 'Is 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) ia 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. U 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 xn. 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 p. 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

11 a and b are real numbers and b ~ 0 the ration of a to b is ;b., and this can also be written "a : b."

B. Proportion

A proportion is an equation whose members are ratios. Give student& practice in forming proportions from verbal problems.

Some properties of proportions which may be proved are:
= = (1) The proportion .;b I d is equivalent to ad be
(2) ;, = ;., is equivalent to ;. = b/
= (3) ;b = ;d is equivalent to +b;b +;. = (4) .fb = ;. is equivalent to ;b
~

27-

C. Variation 1. 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 = % gt2 where s varies directly as the square of t.
V = 4/ 3 "-1'3 where V varies directly as the cube of r.
Students should have oPIJ0rtunity to describe and work with variations of all these types.

28

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

29

GEOMETRY

,

"'i

UNDERSTANDINGS TO BE DEVElOPED IN GEOMETRY

The geometry studied in the tenth grade is closely related to the algebra that is studied immediately before and after, and it should be taught in such a way that these relations are evident. Therefore, an understanding of a broader concept of plane geometry and .solid geometry should be developed.
The concept of the various branches of mathematics as a unified whole is approached in the use of algebraic concepts to develop basic postulates and the use of algebraic skills in proofs. The introduction of coordinate geometry allows this notion of unification to be extended into the second course in algebra.
The content of the course should be presented ,in a manner which will guide the student in developing an understanding of the nature of logic, nature of proof, and the development of skill in applying various types of reasoning both in mathematical and non-mathematical situations.

33

CONTENT

TEACHING SUGGESTIONS

I. Introduction

11

A. Kinds of problems

Give several examples showing that the answers to some problems are obvious

while others require reasoning in arithmetical or algebraic processes. H a line

1. Problems having obv~ous answers

segment 8 inches long is divided into two parts and one of the parts is 5 inches, how long is the other part? H the sum of the measures of two angles of a triangle is equal to the measure of the third angle, the triangle is a right tri-

angle.

2. Problems requiring reasoning

Discuss these types of reasoning: analogous circular non-sequitur. deductive inductive indirect

3. Optical illusions

In how many ways does the box open?

B. Undefined terms and unproved statements
1. Undefined terms a. point b. line c. plane d. between
2. Unproved statements

How many blocks are there?
Definitions are just agreements that we make to allow us to substitute a word, phrase, or symbol for other phrases. A logical treatment of any branch of mathematics must start with undefined terms. Then other words are defined in terms of these undefined terms.
Example: H point, line, and between are undefined then we define a line segment as that part of a line contained between and including two distinct points on a line.
It is impossible to prove every statement since every proof is based upon a hypothesis that is assumed. Therefore geometry begins with some unproved statements that must be assumed as a basis for that structure.

II. Points, Lines, Planes, And Angles

A. Points

Call attention to the fact that a geometric point is not a dot. The dot is representative of the point. Use capital letters to name points.

34

B. Lines
1. Perpendicular lines
2. Parallel lines
3. Perpendicular bisector of a segment
4. Distance frC?m a point to a straight line
5. A line perpendicular to a plane
6. Oblique lines

Give some examples to illustrate these statements:
Collinear points are points that are contained in the same straight line. Coplanar points are points that are contained in the same plane. Every straight line contains at least two distinct points. Space contains at least four non-coplanar points. Every line is a set of points
Give some examples to illustrate these statements:
Given any two distinct points, there is one and only one straight line containing them.
Given two distinct straight lines which intersect, the two lines contain exactly one common point.
Point out that the intersection of two sets of points composing two straight lines may be the empty set.
A straight line segment has only one midpoint.

7. Line oblique to a plane

8. Intersecting lines

9. Sub-sets of lines

11

intersects
A

12 if
B

and

ouly

if

I 1n12

=F

'"

'-0

0-'

~

AB

a. Straight line b. Line segment c. Ray
d. Half-line e. Unequal segmentf C. Planes

A straight line extends infinitely far in both directions.

A0

0B

A ray is a set of points containing a given point of a straight line and all of the points in the line on only one side of the given point.
A ray has one and only one end-point The name of the end-point is always the first letter in the name of the ray. Do not confuse ray AB with segment AB.
L..A B . C.~
The point B in the straight line L separates all the other points in the line into two sets of points. Point B does not belong to the set of points on either side of B. The set on either side of B, not including B, is called a half-line.
Give illustrations for the following statements:
Given any three distinct non-collinear points, there is one and only one plane containing them. Given any two distinct points in a plane, the line containing these points is contained in the plane. If a straight line and a plane are distinct and if they intersect, they have one and only one common point.
35

D. Angles 1. Interior of an angle
2. Exterior of an angle

If two distinct planes intersect, their intersection is one and only one straight line.
Define angle as the union of two rays which have the same end-point but do not lie in the same straight line. The set of all points that lie inside the angle is the interior of the angle; and the set of all points that lie outside the angle is the exterior of the angle, as in the picture:

EXTERIOR
"" " EXTERiOR L .............8
"""

"

EXTERiOR

LABC is an angle in a plane P. A point R lies in the interior of LABC if R
~
and A are on the same side of BC and also if Rand C are on the same side of
~
BA.

The exterior of LABC is the set of all points in the plane that do not lie in the interior and do not lie on the angle itself.

The interior of an angle is the intersection of two half-planes.

III

A

L.

!tfirl-+--+-t-+-r+-

./ "
""

3. Kinds of angles
III. Triangles
A. Classification of triangles 1. Sides 2. Angles
B. Sum of the interior angles of a triangle

is the interior of L ABC. In addition to the usual angles studied include these:
dihedral angles right dihedral angles plane angles trihedral angles Use this definition of adjacent angles: Adjacent angles are two angles in the same plane that lie on different sides of a common ray.
Extend this section to include sum of the angles of a polygon.

C. Sum of the extenor angles of a triangle

Extend this section to include sum of the angles of a polygon.

D. Line:; in the triangle 1. Perpendicular bisectors of sides of the triangle

Find the incenter, circumcenter, centroid, and orthocenter of the triangle by constructing these lines of the triangle.

36

2. Bisector of angles of the
triangles 3. Altitudes of the triangle 4. Medians of the triangle

IV. Congruences
A. One-to-one correspondence
B. Basic congruence postulates
C. A.S.A Theorem
D. Uses of congruence 1. Exterior angles 2. Inequalities in triangles 3. Perpendiculars in lines and planes

Even though the unit deals mainly with congruent triangles, do not limit the concept to these figures in the introduction. It may be advisable to require the student to present detailed proofs at this stage; that is not to omit minor steps. As the student progresses he should be allowed and encouraged to omit minor steps when convenient. The student should be encouraged to develop flexibility based on the background of listed postulates and theorems. In the proof the listing of postulates and theorems by numbers should not be allowed.
The congruence of segments and the congruence of angles have been presented in prior units, but no relationship between them has been established. In this unit the concept of congruency of triangles by a one-to-one correspondence should be developed. Stress should be given the manner of notation of congruency of angles, se'gments, and figures. Lead the students to discover that for some triangles or figures there is a unique one-to-one correspondence between vertices, while in others (isosceles and equilateral triangles) there is more than one congruence between them.
Definition. If every pair of corresponding sides is congruent, and every pair of corresponding angles is congruent, then the correspodence is a congruence between the two triangles. The class may obtain the intuitive background for the S. A. S. Postulate by use of ruler and protractor construction. For example,
give four or five construction problems such as (1) Ii. ABC with AB = 5
= = = inches, AC 7 inches, and L BAC 30; ',2) Ii. ABC with BC 7 inches, = = CA 3 inches, and L BCA any number of degrees; or (3) A ABC with
= = AB 5 inches, BC 4 inches, and L BAC = 35. Encourage the students to
compare their triangles. On the basis of the assumption of number one above, the S. A. S. Postulate may be introduced. Basic theorems dealing with the isosceles and equilateral triangles may be introduced at this time or delayed until after the completion of the basic congruence theorems. Definitions of median of triangle and angle bisectors of a triangle should be presented with related problems.
With the proofs of the congruence theorems, sufficient related problems should be worked to allow familiarization with the theorems. It is suggested that the teacher present, if they are not included in the textbook, exercises in which there is not enough information to prove triangles congruent in part Band C of the outline. For such exercises the student should list needed or missing congruent parts and, if there is more than one solution, consider alternate possibilities.
One rewarding feature of the new approach to plane geometry is that with the new definitions and postulates there is no longer any need for superposition as a method for proving triangles congruent.
The material covered in this unit is very similar to that found in the traditional texts. The main difference is in the comparison of two segments by their lengths or two angles by their measure. This advantage is possible with the introduction of real numbers. A review of the priniciples of inequalities should introduce this section.
Perpendiculars in 2-space geometry may be presented by proof of construction of the basic theorems dealing with (1) perpendicular bisector of a given segment, (2) perpendicular to a line at a given point on the line, and (3) perpendicular to a line from a given point not on the line.

37

v. Parallelism

Basic definitions in this section demand sufficient time and appropriate method of presentation to be meaningful to the student. A sizeable demonstration model and student models would be most effective in 3-space geometry. Basic theorems dealing with perpendicular lines and planes may be found in many textbooks.
Two or more lines in the same plane that do not intersect are called parallel lines. Emphasize that the phrase in the Mme plane is necessary in the definition of parallel lines because two lines not in the same plane which do not intersect have a different name. (Skew lines)
Set language may be used in defining parallel lines by answering the following question:
Which figure reprents 11 n 12 + ep? Which figure reprents 11 n 12 + ep?
______ 1\
_______ lz

A. Angles formed b1 parallel lines 1. Alternate interior angles 2. Alternate exterior angles 3. Corresponding angles 4. Consecutive Interior angles 5. Exterior angies on the side of the transversal
B. Way to prove lInes parallel

Ii this true? For every pair of lines 11 and 12 such 11 ::J P and 12 c where P is a plane, 11 1112 if and only if 11 n 12 = 41.
Example: If two lines are cut by a transversal, and if a pair of alternate interior angles are congrument, then the lines are parallel.
LZ

-----+:~~----L,3
If L 2 and La are cut by transversal and L1 and LX=< L Y then L 2 II La.
Indirect proof: Let L1 be the transversal cutting L2 and La' intersecting them. at Rand S. Suppose that a pair of alternate interior angles are congruent. There are two possibilities: (1) L2 and LB intersect at point P
or
(2) L:! II La
38

If L2 and L3 intersect at point P we have a figure like this:

p

C. Parallels in space

D. Statement, converse, inverse,

contra-positive

'

T is a point in L2 on the opposite side of L, from P. Then L TRS is an exterior angle of triangle RSP, and L RSP is one of the opposite interior angles.
... mLTRS>mLRCP
But we know by hypothesis that two alternate interior angles are congruent. From a previous theorem we know that both pairs of alternate interior angles are congruent.
.', mLTRS=mLRSP
Since statement (1) leads to a contradiction and is false then statement (2) is true,
Prior to this time we have used the direct method of proof in proving theorems and exercises. This method consists of putting together definitions, assumptions, theorems, and the hypothesis in logical steps which lead to the conclusion.
Another method of proof that we often use is the indirect method, Points to remember when proving a theorem by the indirect method:
(1) Enumerate the conclusion and all the other possibilities,
(2) Prove that all the other possibilities lead to a contradiction of previously learned truths.
(3) The conclusioB is therefore true.
Include the following theorems:
(1) If a plane intersects two parallel planes, then it intersects them in two parallel lines.
(2) If a line is perpendicular to one of two parallel planes, it is perpendicular to the other.
(3) Two planes perpendicular to the same line are parallel.
(4) If two planes are each parallel to a third plane, they are parallel to each other.
(5) Two lines perpendicular to the same plane are parallel.
(6) A plane perpendicular to one of two parallel lines is perpendicular to the other.
(7) Two parallel planes are every where equidistant.
The extensive use of the connectives "and," "or," "not," "if ... then," "only if," etc., demands investigation of certain rules of logic. As advanced statements in mathematics are built from combinations of simpler statements, it is essential that the student understands when equivalency exists between these four implications. A non-mathematical situation for class investigation might be: Statement: If he is on the football team, then he plays football. Converse: If he plays football, then he is on the football team. Inverse: If he is not on the football team, then he does not play football. Contrapositive: If he does not play football, then he is not on the football team.

39

E. Theroems and proofs
VI. Locus Of Points
A. Circle locus B. Properties of a 10cM C. Other loci
D. Project

It should be observed that the statement and' its contrapositive are always equivalent, and that the converse and the inverse are always equivalent.
U the original statement and its converse are both true, then all are equivalent. It might be desirable in later examples to introduce the term theorem and examine the above implications in terms of several theorems that were studied in the eighth grade informal geometry.
An excellent comparison of direct proof and indirect proof in terms of the above section may be found in such books as Fundamentals of Freshman Mathematics by Allendoerfer and Oakley. A review of the direct proofs used in the unit on real numbers and the indirect proof used in the unit on radicals in first year algebra would serve to remind the student that these techniques are not new to him in formal course content.
A locus of points is the set of all points, and only those points, that satisfy certain conditions.
Example: Find the locus of all points in a plane at a given distance r from
the given point o.
In a plane, the locus of all points at a given distance r from a given point 0 is a circle with the given point 0 as the center and the given distance r as radius.
The locus includes all the points that satisfy the given condition.
The locus excludes those points which do not satisfy the given condition.
In the circle locus above all the points on the circle are at the given distance from the given point and those points that are at the given distance from the given point are on the circle.
Give examples of the following:
Find the locus of points:
(1) At a given distance from a given line. (2) Equidistant from two parallel lines. (3) Equidistant from two points. (4) At a given distance from a given circle. (5) Equidistant from the sides of an angle. . (6) Equidistant from two concentric circles.
Make models representing loci.

Find the locus of points equidistant from two concentric circles.
40

E. Intersection of loci

Find the locus of points at a given diatance from a given circle.

I
LS:EX
GIVEN :SIDE
I I

Find the locus of the vertex of an isosceles triangle having a given base.

~ _-"\

J

... ,,'

Find the locus of the vertex of a triangle having a given c;ide and a given median to that side.

Other models of the following problems may be made:
Find the locus of the vertex of a triangle having a given side and a given altitude. Find the locus of the vertex of the right angle of a right triangle having a given segment as hypotenuse. Find the locus of points at a given distance from a given point and also at a given distance from a given line. There are five possibilities. Draw them.

,... /,-
/

I

,

, .p J

-- \ "

./

, '" ---~---

I ./

\

,\

/I

...... _ /

41

F. Loci in space

-- --- -,.~..... , /

.. - _*,L...._. _......~ _y _

-----r-- 1

\

, _- I .p I

/ - .....

, .p I

-~ .....

"

\

/ X,

'y

, . /

(

~

.... "

4\

.p I

-~

I

~- - --,llN--

_/

x, Y, Z, and W represent the points that satisfy both conditions.
Consider the locus of all points equidistant from two fixed points in two dimensions and then three dimensions.
Examples of loci in space.
(1) The locus of points at a given distance from a given point is a sphere with the point as center and the given distance as radius.

(2) The loc~ of points at a given distance from a given line is a circular cylindrical surface whose axis is the given line and whose radius is the given distance.
(3) The locus of points equidistant from two parallel plal1es is a plane paral leI to them and midway between them.
(4) The locus of points at a given distance from a given plane is a pair of parallel planes, one on each side of the given plane, parallel to it and at the given distance from it.
" ~,LOC6UI~V5E'>I_~ , ~-,'l..Oeu~ ~--~ 42

G. Conic Sections 1. Circle 2. Ellipse 3. Parabola 4. Hyperbola

VII. Ratio And Proportion

A. Meaning of ratio

For all real number x and y =F 0, the quotient of x divided by y is the ratio of x to y.

B. Meaning of proportion

A proportion is a statement that two ratios are equal.

C. Terms of a proportion

D. Fundamental properties of a proportion

VIII. Similarity
A. Definition
B. Ways to prove triangles similar
C. Uses of slmilar triangles D. Areas of SlmiLar triangles E. Similarities in right triangles,
The Pythagorean Theorem

If the measures of the corresponding angles of two polygons are congruent and the measures of the corresponding sides are proportional, the correspondence is a similarity and the polygons are said to be similar.
It would be desirable to extend the concept of similarity to polygons based on the definition of a polygonal region being the union of a definite number of coplanar triangular regions.
The treatment of these sections should follow the conventional pattern. Appropriate comparison should be made between the definition of a similarity and of a congruence.
The basic theorem and its corollaries dealing with similarities in right triangles should be .presented with emphasis on the geometric mean. This will allow for an algebraic proof of the Pythagorean Theorem. Traditional area proofs will appear in the next unit.

A~B D
= (1) x + y c
(2) ~ =~r x =~

-L=~or ac

y

=~ c

(3)

~+~=
cc

c

+ (4) b 2 a2 = c2

IX. Circles And Spheres

The first part of this unit deals with the common properties of circles and spheres relative to the intersection with lines and planes.

A. Basic definitions

Through the use of concrete illustrations or dissectible models the definitions for circle, sphere, radius, center, diameter, concentric, chord, and secant should be developed. The analogy between 2-space and 3-space geometry should be emphasized. It is important that the student obtain a clear distinction between interior, exterior, and circumference of a circle, and the interior, exterior, and surface of a sphere.

43

B. Tangent lines, fundamental theorem of circles
C. Tangent planes, fundamental theorem of spheres
D. Arcs of circles 1. Central angle 2. Inscribed angels 3. Angles formed by chords, tangents, and secants

Basic theorems and corollaries dealing with tangents, chords, congruent circles, tangent circles, and concentric circles should be presented in the conventional treatment of this section.
The level of ability of the class will determine whether or not you will use formal proof of this theorem. The basic theorem and its corollaries dealing with a plane tangent to a sphere should be presented.
The second part of this unit deals with measure of circular arcs and related properties of angles and arcs, chords, secants, and tangents. Traditional treatment of this material is appropriate. The last suggested proof of the Pythagorean Theorem mentioned in the preceding unit may be presented following the last section.

x. Areas And Volumes
The study of the circumference and area of a circle should be approached by means of inscribed regular polygons. This section should include areas of polygons, ratios of areas of two polygons, Hero's formula for the area of a triangle, areas of similar polygons, areas of irregular polygons, area by coordinate geometry, areas by the offset method, area of a circle, area of a sector of a circle, and area of a ring and an ellipse.

A. Area of circles and sectors 1. Polygons 2. Circles and regular polygons 3. Circumference of a circle, number pi
4. Area of a circle

Basic definitions concerning polygons should be developed.
Material in this section should be limited to theorems sufficient to provide adequate background for the sections to follow. Constructions involving regular polygons and inscribed and circumscribed circles should be presented.
Definition. Circumference of a circle is the limit of the perimeters of the inscribed regular polygons. This definition and the one in the following section are not the same as those in the glossary. These definitions are designed for theorem proofs. The theorem dealing with the ratio of the circumference of a circle and its diameter should be presented. With an elementary introduction to the concept of limits, a process of computation of pi may be presented. By use of a unit circle with an inscribed and circumscribed hexagon, comparison of their perimeters with the circumference of the circle in light of the above definition may be made in terms of inequalities. It is not intended that a formal treatment of limits be given, but that the student develop only a basic intuitive concept.
Definition. Area of a circle is the limit of the areas of the inscribed regular polygons.
Informal treatment of this material is recommended. Beginning with the formula for the area of an "inscribed n-gon An = lh ap where a is the apothem and p is the perimeter of n-gon, find out what limits these quantities approach as n becomes very large.
1. An-+ A of the circie.
2. a -+ r as a is always slightly less than r.
3. P-+ C with C = 2,..r.
= = Then the formula An lhap takes the form of A lhr 211'r = lI'r2

5. Length of arcs Areas of sectors
B. Solids
1. Prisms 2. Pyramids 3. Cylinders and cones 4. Spheres

Basic theorems dealing with length of arcs and sector areas should be presented at this time.
Cavalieri's Principle may be presented as a postulate or given only informal treatment. Material in this section should be limited to theorems on surface area and volume and their respective formulas. The intuitive concept of limits will be required again in this section. Various proofs of above theorems may be found in many textbooks of solid geometry. These proofs should be presented with definite series of statements (or suggestions) for procedure of proof.
44

XI. Inequalities
A. Fundamentals 1. Greater than relationships
2. The sets of positive numbers, negative numbers, and zero.
3. Basic inequality axioms
4. Uniqueness of order
5. Betweenness 6. Transitivity of order 7. Addition 8. Multiplication 9. Subtraction 10. Division 11. Powers and roots B. Absolute value
C. Inequalities in the triangle

If x and yare any two real numbers represented by points on a horizontal number scale directed to the right, then x > y if and only if the point representing x lies to the right of the point representing y.

~---<~.....-_-

o----+-_o---

-6 -5 -4 -3 -2 -1 0 1 2 3> -1

o----+-_o----+-_~
3456 -2> -4

x > y if and only if x - y is a positive number.

-4 - (-5) = 1

The negative of a number a is defined as the number -a such that
+ a (-a) = 0.

If a is a real number then one and only one of the following statements is true:
(1) a is the unique member zero of set 0; (2) a is a member of the set P of positive numbers; (3) -a is a member of the Set P.
If a and b are real numbers, then one and only one of the following relationships holds:
a = b, a > b, or b > a. If a and b are numbers of the set P of positive numbers, then the sum a + b
and the product ab are members of the set P.
For every x and y one and only one of the following relations holds:
x < y, x = y, x > y.
The square of any real number other than zero is a positive number. Of course
= 02 0 ... the theorem follows: Any real number a satisfies the inequality
a2 > 0. The sign on equality holds if and only if a = O.
For all real numbers a, b, and c the expression <lb is between a and c" means
that either a < b < c or a > b > e is true.
If a > band b > c, then a > c.
If a > band c > d, then a + c > b + d'-If a > band c is any real number then a + c > b + c.
If a> band c > 0, then ac> bc. If a> band c < then ac < bc. If a> b >
and c > d > 0, then ac > bd. Multiplying by a negative number reverses the order.
If a > band c > d, then a - d > b - c. If a > b, and c is any real number, then a - c > b - c.
If a > b > 0 and c > d > 0, then &;d> b; ,.
if a > b > 0, if m and n are positive integers and if a f, and b {- denote positive nth roots, then a W> b~.
The absolute value of a real number a, symbolized lal, is defined as a if a is positive or zero, and as -a if a is negative.
121 = 2 101 = 0 1-21 = -(-2) = 2
For any real number x, if x is less than zero, then Ixl -x. If x is not less
than zero, then Ixl = x.
Show similarities between equalities and inequalities.
Give examples for the following basic inequality theorems:
(1) The exterior L of a 11 is greater than either of the opposite interior angles.
45

D Inequalities in the circle.
1. Unequal central angles, arcs, and chords
2. Unequal chords and their distances from the center

(2) H two sides of a triangle are not congruent then the angles opposite these two sides are not congruent, and the larger angle is opposite the longer side.
(3) H two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.
(4) The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
(5) H two sides of one triangle are congruent respectively to two sides of a second triangle, and the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.
(6) H two sides of one triangle are congruent respectively to two sides of a second triangle, and the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.

XII. Coordinate Geometry

A. Introduction.

Allow students to give brief reports on the life and works of Rene Descartes. The approach presented in this unit is important to the student who plans to take the second course in algebra. The unit on quadratic functions pre-supposes student mastery of this unit.

! IIII

B. Coordinate systems in a plane

1. Review of ordered pairs and plotting

Review use of the Cartesian plane studied in the first course in algebra; such as x-coordinate, y-coordinate, ordered pairs, quadrants, plotting of points, arbitrary labeling and placement of axis. The three dimensional coordinate system and the plotting of ordered triples should be introduced.

2. Slope of a Jl()n-vertical line

If the ordered pairs (x" y,) and (x" y.) are two points on a nonvertical line, the slope is define to be y. - y.
X. - x,.

a. m > 0
Z 1

PI = (2, 3) P2 = (5,4)

1 Z 3 .. 5

The

slope

of

PIP2

.
IS

the

number

m

=

Y2- YI
X2 -XI

== 3-5---24= 1

b. m < 0

Pt.
hc~'Yl.l The slope of P I P 2 is the number
46

c. m = 0

ffi=

(-4-6) - 5-(-4)

-10

10

-9-= --9-

d. m is meaningless

The slope of PIP2 is the number

m= 3-3 =~=O

4- 2

2

('1',) P,

bl.,)/l) Pt

PI = (1, 1) P2 = (1,2)

The slope of PIP2 is meaningless.

m=

Y2 - Yl
X 2 -X1

=

(2 (1-

1) 1)

=

1

.

-(0meamngless)

Include the theorem: On a non-vertical line, all segments have the same slope.

Case 1.

PI = (2, 1) P2 = (4,2) Pa = (6, 3)
==- -_ PzR
The slope of P1PZ = RP1
PzR _ 1
RP - 2 1
__ PaS
The slope of PzP3 = =-
SPz
PaS 1
SPz =T
47

= .".The slope of PIP~ the slope of P 2P 3 .
Case 2

= The slope of PtP~ o/:.?

0/2 = 0

The slope of P2P3 = 0/"

0/" = 0

Case 3

p~
,, ()l~) Y3) I I

I

I

Pz.

RI- - - fll;l.Yl.)

s - - - - ---- Pt 11, Yl)

3. Parallel and perpendicular lines

The

slope

of

PtP?

= -RP-3 = RP

--34-

2

__ The slope of P 2P .3

=

=PSP2-S] =

-3 4

.".The slope of P t P 2 = the slope of P 2P3 Show that we not only find the slope of a segment but also the slopes of lines.

The algebraic conditions in terms of slope for two non-vertical lines to be parallel or perpendicular should be presented with related geometric proofs and theorems. Related applications of these definitions may be used; such as giving the vertices of a quadrilateral without plotting to determine if it is a parallelogram.

Include these theorems with proofs:

lz.
The slope of L] is the same as the slope of L2.
:.L1 II L 2
The non-vertical lines are parallel if and only if they have the same slope.
48

4. The distance formula

The slope of Lt is the negative reciprocal of L2 ...L t ..lL2
The non-vertical lines are perpendicular if and only if their slopes are the negative reciprocals of each other.
With the use of the Pythagorean Theorem and definition of absolute vaine derive the distance formula.

5. The mid-point formula
6. Aplication to previous theorems
7. Describing a line by an equation
8. Various forms of equations of a line

The development of the distance formula and the mid-point formula for lines in space should be considered as an optional unit.

Use the picture above and find the mid-point of P t P2

+ P _ (-4 2)

-

2

-3 +4 2

P = -2/ 2, 1/2

P = (-1, '/,)

With the use of the coordinate system, properties of slope, the distance and mid-point formulas present new proofs of theorems previously presented. For example:
The segment between the mid-points of two sides of a triangle is parallel to the third side and half its length.

If the diagonals of a quadrilateral bisect each other and are perpendicular, then the quadrilateral is a rhombus.
Much of algebra is a study of linear equations in x and y, while much of geometry is a study of (straight) lines. Therefore these sections should serve as coordinating units for algebra and geometry. Review of the graphs of simple conditions would lead into consideration of the condition which characterizes a non-vertical line. Point-slope form should be proved:
y - y1 = m(x - Xl)'
Using the definition of y-intercept, point (0, b), translate the point slope form
into the slope intercept form y = mx + b. The point (0. b) is on the line and
the second point may be obtained by assigning an arbitrary value to x and obtaining point (x, y). The second method may be preferred. Given the point (0, b) and considering the slope,

49

9. Intersection of lines

m = Y-Yl
X-Xl

One may count to the right or left (depending on whether positive or negative slope) (x - Xl) units and up (y - Yl) units from the y-intercept.

The theorems and proof of sub-cases related to the general form of the linear
equation (Ax + By + C = 0) should be presented at this time.

The geometric problem of finding the point P of the intersection of two lines is equivalent to the algebraic approach. Emphasis should be placed on the recognition by the student of consistent, inconsistent, and dependent systems of linear equations when written in slope-intercept form based on the compari sons of m and b.

The equations for a circle having its center at the origin and for a circle not having its center at the origin should be derived algebraically. This may be done by use of the distance formula (See SMSG) or by use of the Pythagorean Theorem (see diagram below). The theorem stating that "every circle is the graph of an equation of the form
X2 + y2 + Ax + By + C = 0" and the theorem dealing with the solution sets
of the above equations should be proved (See SMSG).

y PN

,.jill"= y - b

-X ON
a OM
b RN

QR=x-a
= QR2 + PRz Qp2 = + (x - a)Z (y - b)Z r2

y

p
@

0

a

MN

10. The graph of a condition

a. If OA is 3 and 0 is the

y

origin

A )(

A circle with the origin as center and radius 3.
50

b. If OA < 3 and 0 is the
origin

c. x = 0

The interior of the circle with center at the origin and radius 3.

d. y = 0

---------4I--------_'X
The y- axis
y

e. x > 0 and y = 0

The x- axis

)(
,i
i
I

10

A

-+
The ray OA where A = (2, 0) 51

f. x = and y < 0
g. If the coordinates of point A are both positive

Cl
A
-+
The ray OA where A = (0, -2)

In the first quadrant

Consider the following for individual study or for reports: (1). The coordinates of the point dividing a line segment in a given ratio. (2). Another way of writing an equation of a line through two points.

!I

XIII. Construction

In addition to the elementary constructions which are used in other constructions, include the following:

(1) Constructions based on fourth proportionals.

(2) Constructions based on mean proportional.

(3) Division of d line segment into equal parts.

(4) Division of a line segment into segments proportional to given line segments.

(5) Inscribe regular triangles, quadrilaterals, pentagons, and hexagons in circles.

(6) Line parallel to a given line at a given distance.

(7) Line parallel to a given line through a point not on the line.

(8) Nine-point circle.

(9) Tangent to a circle.

(10) Construct triangles when

(a) Two sides and included L are given.

I1

I1

(b) Two LS and the included side are given.

"

(c) Three sides are given.

Ii'

52

III
~I

The class will have covered various aspects of this section during related presentation of prior units where construction was related to theorem proof. It is recommended that the class be allowed to experiment with some of the impossible construction problems with compass and straight edge; such as, angle trisection, construction of a square having equal area to a given circle, and construction of a cube with volume twice a given cube.
53

VOCABULARY

acute angle

equiangular polygons

point of tangency

alternate exterior angles

equilateral polygons

polygon

alternate interior angles

exterior angle(s)

polygonal path

altitude of geometric figures

exterior of geometric figures polyhedral angle

analytic geometry

face of hali-space

postulate

angle

frustum

prism

apothem

geometric figure

projection of a line

arc

geometric mean

projection of a point

area of geometric figures

great circle

pyramid

arithmetic mean

half-space

proportion(al)

base of geometric figures

heptagon

quadrant

bisector of an angle

hexagon

quadrilateral

bisector of a segment

implication

radian measure

bisects

inconsistent equations

radius

Cartesian geometry

infinite sequences

ray

center of circle

inscribed geometric figures

rectangle

center of sphere

interior of geometric figures

remote interior angles

central angle

intersect

rhombus

centroid

intersection of sets

right angle

chord

isosceles triangle

right dihedral angle

circle

inverse of a statement

right prism

circular cone

lateral edge

right triangle

circular cylinder

lateral face

scalene triangle

circumcenter

lemma

secant

circumscribed figures

length of arc

sector

collinear

length of segment

s~gment

complement

limit

semi-circle

complementary angles

major arc

sides of a geometric figure

concentric geometric figures

mid-point

sine

congruence

minor arc

skew lines

congruent figures

oblique line

space

III

contrapositive of a statement converse of a statement

obtuse angle octagon

sphere square

coordinate geometry

one-to-one correspondence

statement (if-then)

coordinate system

opposite angles

supplement

co-planar

opposite rays

supplementary angles

corollary

opposite sides

tangent (trig.)

correspondence

origin

transversal

cosine

orthocenter

trapezoid

cross-section of geometric figures

parallel lines

triangle

cube

parallel planes

trihedral angles

decagon

parallelopiped

union of sets

degree measure

parallelogram

vertex of geometric figures

diagonal

perimeter

vertical angles

diameter

perpendicular bisector

vertical lines

dihedral angle

perpendicular lines

volume of geometric figures

distance edge of half-plane

perpendicular planes pentagon

x-axis xy-axes

end-points of geometric figures

pi

y-axis

plane angle

y-intercept

54

INTERMEDIATE ALGEBRA

UNDERSTANDINGS TO BE DEVELOPED IN INTERMEDIATE ALGEBRA
Algebra as a logical meaningful structure based on a few axioms trom which all other statements are proved is the fundamental idea to be presented in this course. The first portion of the course is a review of the number system and its properties. The basic algebraic precesses are reviewed. The notion of set and set language need to be firmly implanted in these beginning sections if the pupil is not already thoroughly familiar with them. The foundation of this course is the concept of function as exprsssed in set terminology. This concept brings to many algebraic definitions a new clarity and preciseness. Exponent is extended to include negative and fractional numbers. The exponent is presented as a function. The possibility of the irrational exponent is introduced intuitively with the graph of the exponential function. The number system is extended to include the pure imaginaries and the complex numbers. Again, structure is emphasized. The relation between geometry and algebra is stressed. The plotting of graphs is a key to the understanding of function. Conic sections are thoroughly investigated. Systems, linear and quadratic, are solved graphically and then algebraically to give meaning to the algebraic solution. The three-dimensional coordinate system is reviewed and equations in three variables are interpreted graphically. The year of geometry intervening between Algebra I and Algebra II should facilitate this approach.
57

CONTENT

TEACHING SUGGESTIONS

I. Number Systems
A. Systems of natural numbers and zero
B. Systems of integers
C. Rational number system 1. Closure 2. Density 3. Decimal representation a. Terminating b. Repeating
D. Irrationals 1. Need for irrationals 2. Irrationals as nonterminating nonrepeating decimal fractions.
E. h~al number system F. x = a: solution set
1. Need for extension of real number system

This first chapter should be a careful study of the number systems of elementary algebra. The previous preparation of the class, their interest, and their ability to handle detailed proof will determine the level at which this material should be presented. It is expected that the student will have had some experience with the real number system. Structure is to be emphasized. An excellent reference, one which can be adjusted to different levels of presentation, is the SMSG Vol. I, Intermediate Mathematics, Chapter I.

The material here should be review for the student. The associative, commutative, and distributive laws should be discussed. This is the place to be sure the idea of "closure" is grasped. Examination of the natural numbers under the four basic operations develops a need for other numbers to provide closure under division (the fractions) and under subtraction (the negatives). The natural numbers also provide a good basis for the development of the idea of order. The order relation can be tied to the addition operation using the following property:
For a, b in N a < b if and only if there exists c in N such that a + c = b.

The symbol N represents the set of natural numbers. Solution of inequalities is included here.

Discuss the result of adding zero to the system of natural numbers to form the system of whole numbers. Recall that division by zero is not allowed as it would cause inconsistencies in the system.

Extend the set of whole numbers to include negatives. Show that (-a) (-b)
= ab follows from application of the distributive property.

Thus:

l-a) l-b) =

+a 0 + (-a) (-b) +a [+b + (-b)] + (-a) (-b) +a (+b) + (+a) (-b) + (-a) (-b) +a (+b) + (-b) [(+a) + (-a)] +a. (+b) + (-b) 0 +a (+b) ab

The integers on the number line can be used to show order relations. Lead class to recall numbers not on number line of integers. This leads directly to the rationals.

Define rational number. Discuss closure under the four operations and properties of the rational number system. Discuss and define density (property of denseness). Include decimal representation in preparation for irrational numbers.

Use the discovery method to show need for irrational numbers. Students can
find "holes" in the number line. Point out that rationals fail to provide solution
for all equations of the type x2 - a = O.

Have students form non-terminating non-repeating decimals. " is a good example of one with which they are familiar.

Summarize basic properties of the real number system. Discuss completeness of the number line with irrationals added.
Discuss solution set of the equation x' = a when a is negative and n is even.
This extension will be made later.

58

2. Real solution for x = a a. Rational b. Irrational c. Radicals: nth root of a
3. Irrllitional equations

yab = V'a ybis extended to -.etab = -.eta {Yb
Cover basic operations with radicals including rationalizing denominator.
Stress necessity of checking solution set as derived equation may have roots that are extraneous for the original equations.

II. Polynomials And Factors (Review)

A. Polynomial

1. Definitions 2. Operations

Re-examine meaning of polynomial, polynomial in x and y, polynomial over a given number system. Addition and multiplication operations are to be reviewed here also.

B. Factorization

Factorization is the opposite of multiplication. A sound foundation in multiplication is essential to the understanding of factorization. Emphasis on the distributive, associative, and commutative properties will aid in factorization.

Students should be aware that certain expressions can be factored over some
number systems whch cannot be factored over others. For example a2 + b2 is
not factorable over the real number system but will be factorable when the number system is extended to include complex numbers.
+ + (a2 b2) = (a bi) (a - bi)

III. Rational Expressions (Review)

A rational expression is the quotient of two polynomials. These polynomials may be integral, rational, or real. Hence a rational expression should be distinguished from a rational number.

A. Simplification

"Simplify" is used to indicate removing common factors from numerator and denominator. Stress "the property of one" in simplification.of fractions. "Simplify" is a term of many meanings and has led to some confusion. "Simplify" as used in this guide means to find the simplest name for an algebraic ex pression. See Algebra I, II A.

B. The four basic operations

The "property of one" is also fundamental in complex fractions. The complex fraction can be changed to a rational expression by use of the formula / b. ='/ b

IV. Functions: Linear And Quadratic

A. The function concept

The function concept is one of the fundamental concepts of all of mathematics. The discussion of function should be based on sets. To define a function three things are necessary: a set called its domain, a set called its range, and a rule for pairing the members of the range and the domain.

1. Formal definition of function in terms of sets.

Definition of a function: A function consists of a set A called the 110main, a set B called the range, and a rule (or a table) which assigns to each member of the set A a member of B such that no two different members of B are assigned
to the same member of A.

Have students give examples of functions from the classroom. Be sure to bring out the following ideas:

that the pairing need not be one-to-one. that every member of the domain must have one object assigned to it. that each member of the range must be assigned to at least one member of the set.

Point out that the "rule" as such is not the primary thing but that pairing can occur without knowing just what the "rule" is. All that is required is that for

59

every element of the set A there is assigned an element of the set B. Hence pairing is the important idea, not how the pairing is obtained. Give some examples of function in which the rule is merely an arbitrary pairing.

In the ninth grade students deal with those functions whose range and domain were sets of numbers. Here the student should deal with at least a few functions with more abstract range and domain. For example, the rule of addition assigns a whole number to each pair of the whole numbers.

2. Functional notation

A single letter is used to denote a function. The symbol f(x), read "f of x" or "f at x", is used to mean the element assigned to x by the function f when x is any member of the domain. Note that (x, f(x represents the pairing of the members of the domain and members of the range.

The hardest part of the notation idea is the substitution. If the ideas of domain and range are maintained, this should be clearer for the student. The idea involved here is not new even though the notation may be. For those who have had Algebra I as presented in this guide this would not be the case for the notation for them will be a matter of review.

II

Composite functions should be mentioned briefly here. 3. Functions defined by equa- We have stated that the rule is not the most important thing about a function

tions (formulas)

and that a function can occur without our knowing just what the rule is.However,

many functions do have rules that can be determined and stated as algebraic

equations. The solution set of an equation has been defined and discussed and

this idea of solution set is used to define function. The pair (a, b) represents the

solution set of an equation in x and y. If the first member a occurs only one

time, then assigning b to a defines a function with the set of all first members

as the domain and the set of all second members as the range. Point out that

there are some equations that do not define functions. For example, the above condition is not fulfilled in the equation y2 = x2 as for every x there would

be two values of y. Remind students that there are also functions which cannot

be d'efined by equations.

4. Graph of a function

The set of all pairs produced by the rule of the function is sometimes called the graph of the function. It should be pointed out that this need not result in a geometric figure as in the case of the function which assigns to each person his first name. The set of pairs would be called the graph of the function but could not be represented geometrically. If a function happens to have a domain and a range consisting of real numbers, then the pairs of its graph can be plotted. Note that the graph of a function defined by an equation and the graph of the equation are essentially the same.

5. Functions defined geometri- To define a function geometrically is the reverse of the procedure in the

cally

preceding section. The fact that every point of the plane pairs two numbers,

an x-coordinate and a y-coordinate, is used in defining a function. Some of these

pairings will be functions. Some will not. If we start with a set of points and

are to determine the function of which it is a graph, we must first determine

if the set of points will be a function. If no two points in the set have the same

x-coordinate, then we have a function, the domain of which is the set of all

the x-coordinates of the set and range of which is all the y-coordinates of the

II

set.

This would fulfill the condition that each member of the set which is the

range be assigned to a different member of the set which is the domain. (Note

that if no two points of the set have the same y-coordinate then assigning the

x-coordinate would define a function also. The first .idea of assigning a y-coordi-

nate to a distinct x-coordinate is the method most frequently used.) Then a

set of points defines a function if and only if no two points of the set have the

same x-coordinate. This can be stated geometrically in this way: a set of points

defines a function if and only if no vertical line contains more than one point

of the set. This leads to two conclusions:

Not all sets of points will define functions.

There will be some sets of points which will fulfill the conditions for a function for which we will not be able to write an algebraic equation.

60

6. Functions defined by processes in physics
7. Inverse functions a. Composite functions b. Inverse functions
B. Linear function 1. Defin~tiong
2. Basic property of linear functions
3. ~ear functions having a glven value

A good understanding of the function concept will allow students to find examples of functions in many places. An effort to state some of the processes in physics as functions may serve to reinforce a studenes concept of function. Expressing physical processes as functions is not to be dwelled on at length, but provides an excellent "tie-in" for physics and algebra.

Composite functions are functions that can be defined in terms of other functions. If the range of the function f is the domain of the function g then g can be defined in terms of f. The notation in this case is g[f(x)].

Definition:
Let A and B be sets, let f be a function whose domain is A and whose range is B and let g be a function whose domain is B and whose range is A.

Then we say that f and g are inverse functions if for each x of A, f[g(x)] = g[f(x)] = x.

An inverse function is the function formed by interchanging the range and the

domain of a given function. The range of the given function is the domain of

the inverse and vice versa. Note: Every function does not have an inverse. A

= = function will have an inverse only if there is a one-to-one correspondence
between elements of the domain and of the range. Given y f(x) x + 2 we

have x as a member of the domain and x + 2 as a member of the range. To

form the inverse function we interchange range with domain, obtaining x + 2

as a member of the domain and x as a member of the range.

"

The given function is the ~t of pairs {x, f(x)} or {x, y}. The inverse function will be {f(x), x} or {y, b(y)}. To preserve the previous symbolism using x as
the member of the range, now interchange x and y in the inverse: function ob-
taining the set {x, f-1(x)}. The symbol f-1(x) = f(y) and is used to denote the
interchanging of the variable symbols.

Definition:
= A function is a linear function if and only if it is defined by the equation y
ax + b, where a is a non-zero real number and where b is any real number.

Alternate Definition:

A linear function is a set of ordered pairs (x, y) defined by y = ax + b; where
the domain is the set of real numbers, a and b are real constants, and a # O.

The linear function is a special kind of pairing of numbers with numbers. With a linear function, real numbers are paired with real numbers, and any linear function sets up a one-to-one correspondence between the set of real numbers. (See SMSG Intermediate Mathematics, Chapter on function). From this it follows that every linear function has an inverse: The algebra involved
in finding the inverse is quite simple. If f(x) = ax +b then its inverse can be found by solving y = ax + b for x and then interchanging x and y in the result.

Another property of the linear function that should be stressed is that the ratio of the change in y to the change in x for any two ordered pairs of the set is a constant. Linear functions are 'the only functions which have this property. The property can be stated as a theorem in the following manner: Let the linear function f be defined by y = ax + b, and let p and q be any distinct real numbers.

f(p) - f(q)

If

p-q = a

then f is a linear function.

See SMSG, Intermediate Mathematics, Chapter on linear functions for proof of this theorem and its converse.

Any two pairings of a linear function determine all other pairings. (This fact is closely akin to the geometric fact that two points determine "a line.) In other words given two ordered pairs (Xl' YI)' (x2, Y2) where Xl # x2 there is one and

61

~I

4. Direct variation
C. The Quadratic Function 1. Definition
2. Function defined by y = x2
3. Effect of coefficient of x. y = ax2, a >0, a<o.
4. Effect on constant. Function defined by y = ax2 + c
5. Function defined by y = a (x _k)2
6. Function defined by
+ y = a (x - k)2 p
7. Function defined y = ax2 + bx +c

only one linear function such that y, = f(x,) and y" = f(xJ. To determine
the linear function when given two pairings we substitute x and y in the
standard form of the function y = ax + b for each pairing then solve to find
values for a and b. These values are then substituted in the standard form to give the required linear function. This function should be checked by direct substitution to determineth at it makes the given pairings.
= = In the definition of linear function we required a =F 0 but put no such restric-
tion on b. If b 0 we get a speciai linear function y ax. If we consider the pairings which would follow from this function (x, y) or (x, ax) we see that a will determine the relationship between x and y. If a is 2 (or x is doubled) y will be doubled; if a is lf2 (or x is halved) then y is halved. Any change in x brings the same change in y. This relationship is called direct variation and is defined as:
When x and yare so related that y = ax where a =F () then y is said to
vary directly with x, or y is proportional to x.
Definition:
Let a, b, c be any real numbers. Then if a =F 0 we call the function defined by the equation y = ax2 + bx + c a quadratic function.
Alternate definition:
A quadratic function is the set of ordered pairs (x, y) defined by an equation y = ax2 + bx + c where the domain of x is the set of real numbers; a, b, c are real constants; and a =F O.
The traditional study of qUlldratic function has begun with the study of their graphs. Most of the important facts about the quadratic can be illustrated by their graphs. However, the properties of the quadratic function are not based on the graph; they are based on the real numbers. A discussion of the quadratic function first and then an illustration by the use of the graph would seem more logical. A quadratic function like the linear function is a special kind of pairing of the real numbers. The understanding of the quadratic function is based on two properties of the real numbers: (1) no square of a real number is negative;
va (2) every positive real number a is the square of two real numbers and
-v'll. These properties should be discussed first. Then have students identify
equations which would define quadratic functions. Discuss the domain and range of each.
Now a discussion of the graph of the quadratic function will be in order. The study is outlined here as a successioh of special cases. Begin with y = x2,
= = = progress to y ax2, y a(x - k)2, y a(x - k)2 + p, and then arrive at the
general quadratic function y = ax2 + bx + c. Discuss the nature of the graph in each case. Bring out changes as a, b, c vary. Definitions to be mastered are parabola, vertex, maximum -point, minimum point, axis of symmetry.

V. Quadratic Equations

A. Equivalent equations
B. The equation: y = ax2 + bx + c where y = 0

Define equivalent equations. Show ways in which they are obtained. Illustrate also methods which do not produce equivalent equations.

= Show that no roots are lost by considering the special case 3](2 + bx + c 0

when determining th~ solution set.

-

1. Graphical solution

Let students consider graph of the quadratic function to discover where the solution of the equation will be illustrated on the graph.

62

2. Quadratic inequalities: y =
ax2 + bx + c where y>O; y=ax2 + bx+c where y<O
C. Solution ofax2 + bx + c = 0
by completing the square 1. The procedure and the
formula
2. The discriminant 3. Need for extension of num-
ber system to provide roots when b2 - 4ac< 0 D. Solution by factorhg E. Some properties of the roots of a quadratic equation
1. Sum of roots 2. Product of roots
F. Applications

In connection with ax2 + bx + c = 0 we consider the cases where ax2 + bx +
c =F O. The solution of the quadratic inequalities is found by examining y =
ax2 + bx + c where y =F O. Portions of the graph above the x-axis give values
for x which are solutions for ax2 + bx + c > O. Portions below the x-axis
give solutions for ax2 + bx + c > O. Discuss the different solution sets that
may occur. A graphical illustration plus plotting the solution set on the number line will help understanding here.
The student should master both the procedure of completing the square and the formula which it produces. He will need the process of completing the square again later. He should be able to solve quadratic equations without referring to the formula but he should also know the formula and how to use it. Drill will probably be needed in identifying a, b, c.
The discriminant test for the existence of real roots and its use are important. Point out that the discriminant fails to provide roots for b2 - 4ac < O. This presents a need for extension of our number system to include such cases. It should be made clear that this section applies only to quadratic equations with real coefficients and real roots.
This material was covered in Algebra I. This section should be reviewed in preparation for the section immediately following.
Discuss the different possibilities for roots using the discriminant test. Again emphasize that this refers only to roots which are real numbers. There should be some discussion of the number of roots a quadratic equation may possibly have. (See SMSG, Intermediate Mathematics Chapter on Quadratic Functions for proof of the fact that a quadratic equation has at most two roots.)
Use roots from ax2 + bx + c = 0 to show that the sume of the root is -./.
and the product of the roots is '/. From this it can be shown that if rand s are any real numbers there is a quadratic equation whose solution set is {r, s}.
= Emphasize the following: there is only one quadratic equation x2 + px + q 0
whose solution set is {r, s}; p can be expressed at -(r + s) and q as r; there
would be infinitely many quadratic equations whose solution set is {r, s}; a, b, c are not determined by rand s but the quotients b/. and '/. are determined by them.
"Word Problems!"

VI. The Complex Number System

A. The imaginary number
1. Extension of the real numbers

Review need for extension of the real number system to provide roots for some
quadratic equations. Take x2 + 1 = 0 as an example. We want to preserve the
rules of operation that apply to addition, subtraction, multiplication, and di-
vision of real numbers. If a meaning is given to x2 + 1 = 0 then x2 = -1
must have the same meaning. A new element is introduced to satisfy this equation. It is called i, the imaginary unit, and it has the property i i = i2 = -1. The name imaginary is used in referring to this new number. It should be
emphasized that imaginary numbers are not imaginary in the common use of
the word. This is merely an arbitrary name chosen for them.

2. Operations
B. The complex number a + bi
1. The pure imaginary 2. The real number

The four operations on the imaginary terms such as 3i are performed exactly as if they were monomials. Examine the imaginary number system for the properties of the real number system. The imaginary number system is not closed under multiplication.
= Examine the equation x2 - 2x + 2 O. The solution is neither a real number
nor an imaginary number but a combination of the two. We then define a new
number system, the complex numbers, with numbers of the form a + bi where
a and b are any real numbers. This system will include all numbers of the form
a + bi where a =" O. This is called a pure imaginary number. A number of the form a + bi where b = 0 would be a real number. Hence the complex number

63

system includes as a subset the real number system and also the pure imaginary number system.

3. The basic operations with complex numbers

Most standard texts have a good section on operations with complex numbers of the form a + bi. Include a discussion of conjugates.

C. Graphical representation of the complex number.
1. Th~ complex number as ordered pairs

The following definition of complex number is given as preparation for the plotting of the complex number on the graph. The one-to-one correspondence between the complex numbers and points of the plane is easier to establish with this type of definition.

a. Definitions

A complex number can be defined as an ordered pair (a, b) where a and bare
= real numbers. The pure imaginary is represented by the pair (a, b) where a
o. The real number is (a, b) where b = o. The rule of equality for complex
numbers:
= = (a, b) = (c, d) if and only if a c and b d.

b. Basic operations with complex numbers as ordered pairs. (Optional)

This is included only for the more advanced groups. (See Report of the Commission on Mathematics, Appendices Appendix 4, p. 60-63 for a detailed approach to complex numbers as ordered pairs with definitions and operations included therein.)

2. The graph

a. One-to-one correspondence

Recall from coordinate geometry the one-to-one correspondence between ordered pairs of real numbers and the points of the coordinate plane. Similarly there is a one-to-one correspondence between the complex number plane and the ordered pairs which represent the complex numbers.

b. Argand diagram

The expression "Argand diagram" is used to describe the picture obtained when the point (a, b) of the xy-plane is used to represent the number (a + bi). Points on the x-axis correspond to the real numbers and points on the y-axis represent the pure imaginary numbers.

c. Geometric interpretation of addition

This is covered quite 'well in the standard texts. Continue to use ordered pairs. Show that geometric addition parallels algebraic addition. Subtraction of complex numbers follows by defining subtraction in tenns of adding a negative.

D. Use of complex numbers in solving quadratic equations.

1. The general quadratic
= equation: ax2+bx+c o.

The discriminant can now be used to characterize the roots of any quadratic equation. We no longer have the quadratic equation which has no solution in our number system. Illustrate the use of,the discriminant for all types of roots of the general quadratic.

2. Equations in quadratic form.

Any equation which can be put in the form of the general quadratic equation ax2 + bx + c = 0 can be solved by the methods used for the quadratic equation. The methods involved in solving these equations do not always produce equivalent equations. This is a good place to review and re-emphasize the processes which do and which do not produce equivalent equations. In dealing with equations in quadratic form the methods involved will produce three cases:

(1) Equivalent equations: the solution set of the derived equation is the same as the solution set of the 'original.

() Extraneous roots: the solution set of the original equation is a proper subset of the derived equation.

(3) Loss of roots: the solution set of the derived equation is a proper sub-set of the original equation.

In each case we must verify how the solution set obtained compares to the solution set of the original equation.

3. Equations of degree higher Some of the equations transformable to quadratic equations are of degree

I

than two

higher than two. Emphasize that the complex number has given us roots for. any equation. Some equations of degree higher than two which cannot be put

I

64

11
L

a. Those transformable to quadratic form
b. Those not transformable: factoring as a method of solution

into quadratic form can be handled by special cases of factorization. The quartic and cubic equations we can solve by grouping to remove the common factor. There are certain trinomial equations which can be solved by reducing to the difference of squares. In this entire section the student should be encouraged to follow his own initiative in developing methods of solution. These equations can be solved by a straightforward method but often a little ingenuity provides a quicker solution. Experimentation should be encouraged throughout the study of mathematics, and this is one good place in which to do it.
Note: Many standard textbooks will include equations of this form in a chapter dealing with equations of higher degree. The teacher may find it necessary to select equations from those chapters for use here.
Some students will ask about equations of degree higher than two which cannot be put in quadratic form and which cannot be handled by special cases of factorization. The general equation of the third degree and the general equation of the fourth degree are solvable by formulas similar to the quadratic formula. It has been proved that there are no formulas of this type for equations of degree high than four. Refer interested students to the Mathematics Dictionary, Second Edition, or Men of Mathematics (see Bibliography) for a look at the formulas for cubic and quartic equations.

VII. Equations Of 1st And 2nd Degree In Two Variables

This section assumes previous experience with coordinate geometry. See Ge-

ometry, section xn of this Guide or SMSG Intermediate Mathematics for a

basic introduction if students should not be familiar with the principles of

coordinate geometry.

'

A. Review

1. The linear equation and its graph

Review various forms of the equation of the line. Point-slope form, slope-inter-
cept form, the general linear equation Ax + By + C = 0 where A, B =1= 0
should be discussed. The idea that a straight line is the set of all points equidistant from two fixed points should be presented if this has not been previous.. ly discussed. Check on students' experience with inequalities. If graphs of linear inequalities have not been included see Algebra I of this Guide or SMSG Intermediate Mathematics for a unit on linear inequalities.

2. Formal proof

This section presents an excellent opportunity for work with formal proof, especially for those students already having a firm foundation in the basic properties of linear functions and .equations. The geometric fact that the real number
m = Y2 - Yl
x2 - Xl where Pl(Xl Yl) and P2(X2, Y2) are two distinct points does not depend on the particular pair of points on a line which were used to compute it can be used to prove the following theorem:

If P(xl , Yl) is any point in the plane and m is any real number then the equation of the straight line passing through the point P with slope m is
-= y - YI m(x - Xl)'
The above theorem can then be used in the proof of the following important theorem:
The graph of every linear equation is a straight line, and every straight line is the graph of a linear equation.
See SMSG Intermediate Mathematics for proof of the above in detail. Care should be observed in the manner of presentation. It is essential that students be aware of the nature of proof and what is required for proof. See Algebra I of this Guide for a discussion of proof.

65

\I

l. 'The parabola a. The locus of the para bola. Terms: directrix, focus, axis of symmetry, vertex, latus rectum b. The equation of the parabola
Conic sections 1. Definition of a conic sec-
tion a. The ellipse b. The hyperbola
The circle and the ellipse 1. Derivation of the equation
of the circle a. Center at origin b. Center not at origin 2. Equation for the ellipse
3. Relation between the circle and the ellipse
The hyperbola 1. Equation of the hyper-
bola 2. The equilateral hyperbola
The general equation of 2nd degree
1. Identification of different curves
2. Degenerate curves

The idea that a straight line is the set of all points equidistant from two given lines has been presented. This should lead students to consider next the question, what is the set of all points equidistant from a point and a line? From this we get the formal definition of the parabola. The student is familiar with the parabola and its graph. Some of the parts such as the focus and directrix will be named here. Review
The definition is extended from "the set of points equidistant from a point and
a line" which is the definition of a parabola to "the set of points such that the distance from the point is some constant times the distance from the line". Allow the students to make this extension on their own. They will soon realize that the definition of the parabola does not cover all the ways in which a set of points may be related to a point and a line. The possibilities that occur when the set of points is not equidistant from the point and the line will lead them to the definition of the ellipse and the hyperbola. As always, the accent in teaching is on discovery by the student.
Include the general equation Axl + Bxy + Cy2 + Dx + Ey + F = 0, the para
bola, the sllipse, and the hyperbola as conic sections. Illustrate with a good model how these are obtained from the right circular cone. Also show that the straight line, the circle, and two intersecting straight lines occur as special cases of plane sections of a cone.
Review. See Geometry.
Be sure to bring out the relation of the constant (the eccentricity of the curve to the shape of the ellipse.)
As the constant approaches zero let the student examine the effect on the foci and the directrices.
Again the relation of the curve and the constant is to be discussed. Asymptotes will be a new term for the student. The use of asymptotes as a quick, accurate method of sketching this curve is recommended. Include the equilateral hyper-
bola as a special case when B = 0 in Ax2 + Bxy + Cy2 + Dx + Ey + F = O.
The student should be able to recognize the different curves from their equations and sketch an accurate graph of any of the curves including degenerate curves.

The preceding material on conic sections has gained in importance as part of the high school curriculum in recent years as many students will be going directly into analytical geomtery or introductory calculus froin the high school. This section should give the student as complete a background as the teacher is able to provide.

66

gzg

VIII. Systems Of Equations
A. Systems of equations in two variables
1. Solution sets of systems of equations and inequalities

a. Definition b. Inconsistent systems:
solution set is the empty set. c. Consistent systems: solution set contains at least one element d. Dependent systems: s0lution set of systems issame as the solution set of the component equa tions 2. Equivalent systems of equations a. Equivalent equations b. Definition of equivalent systems c. Use of equivalent systems to find solution set of systems
3. Linear combination a. Principle of linear com bination
b. Use in finding solution set of a system

Review. See Algebra I, Unit XI for basic terminology. Maintain use of set language in describing various cases of solution. Examples and exercises in this iection are needed as a means of clarifying the definitions. Methods of finding the solution set of the systems will be discussed later.

Review. See Unit V" Section A above. It would be advisable to include a reo view of the methods which produce equivalent equations and those which will not. Two systems' of equations are equivalent if and only if they have the same solution set.
If either of the equations of a system is replaced by an equivalent equation, the resulting system is equivalent to the original system.
The principle stated here allows us to determine the solution set of a svstem of equations by substituting equivalent equations for the original equations until we arrive at a system which has an obvious solutio,n set. This method may appear cumbersome, but the important point is that, by using equivalent equations, we are assured that when we arrive at a solution set it will be the same solution set as that of the original system.

The principle of linear combination may be stated as follows:

The system of equations obtained by setting each of two expressions in volving x and y equal to iero is equivalent to the system obtained by pair ing either of these expressions with an equation obtained by setting a linear combination of the two expressions equal to zero.

An example of the use of linear combination is given here:

Find the solution set of the following systems:

r2x -
1,:1:-

y 2y

+-

4 7

= =

0 0

This system is equivalent to the system:

+ + {

2xa(2x -

Y

y

4 -

=O 4)

b(:I:

-

2y

7) = 0

Choose a and b so as to eliminate either x or y from the second equation. Let
a = 2 and b = -1. Then the system become the following equivalent system:

67

11

c. Linear combination compared to eliminaiion method
B. Systems of one linear and one quadratic equation 1. Graphical solution
2. Algebraic solution
c. Systems of equations contain-
ing two quadratic equations 1. Graphical solution
2. Algebraic solution
D. Systems of first degree equations in three variables 1. A three-dimensional cordinate system a. One-to-one correspondence between points in space and number triples

2x- y -4=O {1 x - 1 5 = O The remainder of the series of equivalent systems would be:

2x-y-4=O

{

x-5=O

J6 - y = 0 l.x - 5 = 0

Tb,e solution set of the original system is the same as the solution set of the system:
= y 6
{x = 5
or the set

{(5, 6)}.

The student will no doubt recognize linear combination as essentially the elimination method which he has used previously. The principle of linear combination is the basis of the elimination method. The idea of linear combination of two expressions is important in mathematics and the student will need it again.

By starting with the graphical method of solution, the student will be better able to understand the algebraic method later. Have the students first discuss the way in which a conic section and a straight line may (or may not) intersect. lllustrate the four ways geometrically.
The algebraic solution of the system of one linear and one quadratic equation involves the idea of substitution. The form of the 'ordered pair (x, y) belonging to the solution set may be determined from either equation and substituted in the other.
In finding a solution set for
(a) 3x + y = 5 + (b) x2 y2 = 5
the form of the ordered pair (x, y) determined by the first equation is (x, 5 -
3x). This ordered pair belongs. to the second equation if and only if x2 +
(5 - 3X)2 = 5. Solving the quadratic equation will give a solution set common to both equations (a) and (b). If the ordered pair is found from the quadratic equation and substituted in the linear, extraneous results can be avoided.

As with the preceding systems the determining of the solution set by sketching the graph of the system first gives more meaning to the solution set. Stress that non-intersection implies imaginary solutions instead of no solution.
Again substitution is generally used. Linear combination provides a quicker method fot: certain types. Coq.tinue to require geometric interpretation of the solution set. Continue to use the language of sets to describe the various types of intersection.
The three-dinmcnsional coordinate system is not presented in the traditional Algebra II course. However, that course usually includes the solution of the system of three linear equations containing thre variables. It is felt that the student profits by emphasis on the geometric interpretation of the equation with which he is working and that work with the three-dimensional coordinate systE.m will aid in understanding of this material.
Set up the one-to-Qne correspondence between ordered triples and points in space. TIlustrate the plotting of points, and. be sure each student understands the scheme involved in the plotting. The coordinate planes should be pointed

68

b. Distance formula (optional)
2. The equation of a plane (Proof optional)
3. Solution set of an equation in three variables
4. Graph of the first degree equation in three variables
5. Solution set of a system of two first degree equations in three variables a. Graphic solution b. Algebraic solution
6. Solution set of a system of three first degree equations in three variables
7. Algebraic solution

out. Much confusion can be avoided by a thorough presentation of the techniques.
The development of the formula for the distance between two P<lints is essentially the same as finding the length of the diagonal of a parallelopiped. A model will be helpful in working with the distance formula. The corner of the classroom provides a good illustration.
The student does not have the background in solid analstical geometry necessary for a rigorous proof of the theorem that every plane in three dimensions can be represented by an equation of the form
Ax + By + Cz + D = 0
where A, B, e" D are real constants and not all zero. This theorem can be
presented ~ithout proof or we can avoid the use of ana1)-tic:al tools in the proof. By defining a plane as the locus of all points equidistant from two gi,en points we can arrive at the equation of the plane using only the anaLrtic:al geometry involved in the distance formula. Note: The section on the distance formula has been labeled optional. If the teacher has omitted that se<:tion because of the general level of ability and interest of his students then the proof of this theorem will be omitted also.
Definition: The solution set of an equation in three variables is the set of number triples (x, y, z,) that satisfy the equation. Most student will be able to make the extension from the definition of the solution set for an equation in two variables to that of an equation in three variables quite readily. Students should work with first degree equations in three variables ooth algebraically and geometrically until familiarity with this representation of the plane is assured.
The ability to draw the graph of an equation will give the student insight into the correspondence of the plane and the first degree equation. It will also help in the understanding of the special cases that may occur with systems of two or three equations. The student should be given practice in drawing the graphs of the special cases of the plane.
Have students discuss ways in which two planes may (or may not) intersect. This section should include a study of a system in which the planes are parallel and one in which they coincide.
Let students explore ways in which three planes may intersect. The graphic solution of a system of three variables is quite complicated. Only a few of the more interested students will want to attempt it. The average student will be content with the algebraic solution, but the geometric interpretation of the solution set should be required.
This is the traditional method of solving by successive elimination using a series of equivalent systems.

IX. Exponents And Logarithms

A. FTponents 1. Theory of exponents .

Review basic laws of exponents. Extend idea of exponent to include zero, negative, fractional exponent. Emphasize that zero, negative, fractional exponents are defined to obey previous laws of exponents.

2. The exponential function. Graph of the function.
B. Logarithms

Explore idea of exponent as a function. Consider range, domain of the function. Show that definition of exponents will be extended to include irrational numbers. This can be done intuitively with the graph by filling in missing wints on the curve.
Logarithms have lost their essential value as a computational device because of electronic computers. However, the logarithmic and exponential functions have become more important in mathematics and in other fields. This leads us to minimize computation with logarithms and to lend emphasis to the theory of logarithmic and exponential functions.
69

1. 'Logarithm as an exponent (Graph of log x)
2. The logarithmetic function 3. Common logarithms
a. Use of tables b. Interpolation 4. Basic operations 5. Exponential equations

Emphasize the fact that logarithms behave as exponents. Use logs for bases
= other than 10 to show how they are helpful in multiplication and division. The
graph will help students see log 1 0, log x is undefined for x < 0, positive for x > 1 and negative for 0 < x < 1.
Emphasize that the logarithmic function is the inverse of the exponential function.
Give a moderate amount of computation with emphasis on the laws of exponents and scientific notation.

X. Permutations And Combinations

A brief time should be devoted to the counting process and various problems of enumeration arising in different areas of knowledge. Note that problems may be solved by simple counting of cases but that this method is often ridiculous to attempt because of the extremely large numbers involved. Also our object is to obtain a theory which will handle problems of this nature.

A. Basic' principle

The idea of ordered pairs is extended to include ordered triples, quadruples, quintuples, etc., and the general ordered m-tuple. See SMSG materials for defibitions and for the fundamental principle in terms of set language.

B. Permutations; P (n,r)
1. Permutations of n things all different

Give students only enough exercises to develop confidence in their ability to handle them.

2. Permutations of n things not all different

C. Combinations (selections): C (n,r)

Set language should be maintained throughout. Careful use of the word combination is required. The word selection may cause less misunderstanding.

D. Relationship between P (n,r) and C (n,r)

XI. The Binomial Expansion

A. The binomial theorem

Develop intuitively. Show Pascal's triangle.

B. Binomial expansion

Include fractional and.negative exponents also.

C. General term of the binomial expansion
1. Intuitive approach

Expand (x + y)D for first several terms. Have students analyze relationships
in each term and list rules for writing the general term in terms of the number
of the. ternf in the expansion.'

2. The coefficient of the general term as a combination

After student has gone through the explanation of the general term have him

consider the coefficient of the general term. It should occur to him that this

can be expressed as a combination.

.

3. The coefficient as a summation

Some teachers may prefer to omit this section but it affords an excellent opportunity to demonstrate the conciseness of mathematical language. The student should be able to move with little effort from the notation for combinations into the notation for the summation.

70

.-

iE!C &

XII. Sequences And Series

The ballie definitions for this section are included because they may be s~htly different from the traditional definitions.

A. Definitions

1. Sequence (progression) a. A finite sequence

A sequence is an ordered arrangement of numbers. A finite sequence of n terms is defined as a function a whose domain is the set of numbers {l, 2: 3,
... , n}. The rule of the function associates with each of these numbers another
number {at, a2, , an}. These are the elements of the range and are the terms of the sequence.

b. An infinite sequence

The infinite sequence is a function whose domain is the set {l, 2,3,
of all positive integers. The range of a is the set {at, a2, a3, .. , an, element an of the range is called the nth term of the sequence.

,n ...} }. The

2. Series

A series is the indicated sum of the terms of a sequence.

a. A finite series b. An infinite series

Let {at, then the

a2 , . , an} be indicated sum

a at

given
+ a2

finite
+ ...

sequence of real or complex
+ an is called a finite series.

numbers; The. num-

bers at, a2 , . , an are called terms of the series.

Let a1' a2 Then the

.. , a~, ... be indicated sum

a given infinite sequence
at + a2 . . + an + . .

of real or
. is called

complex numbers. an infinite series.

The number an is called the nth term of the series.

3. The sum of a series

The ~ummation symbol, which was introduced in the binomial expansion, is useful here for indicating the sum of a series. It is best to define only the sum of the finite series here and limit work to examples of that type. The sum of the infinite series will be considered after the concept of limit has been presented.

B. Arithmetic sequences and series
1. Formulas

The teacher will be familiar with the definition of the arithmetic sequence (progression) of the arithmetic series. The formulas can be quite easily devol oped by the student with a few leads from the teacher.

2. Applications C. Geometric sequences and se
ries 1. Formulas
2. Applications D. The limit of a sequence

Here, too, the student can develop the formulas. Note: The idea of the sum of a series is still restricted to the finite series.
The best approach to this idea is the study of several examples of sequences. The student is familiar with the notion of limit from geometry. From a study of several different sequences it can be observed that there are two kinds of sequences distinguishable by the behavior of an as n becomes large. As n becomes large an either approaches some fixed number or does not approach a fixed number. If as n becomes large an approaches a fixed number A, then we say that A is the limit of the sequence.
Definition: The sequence aI' a2, a3, has a limit A if an becomes and reo mains arbitrarily close to A as n gets larger and larger. A sequence that has a limit is said to be a convergent. A sequence which does not approach a fixed number as a limit is said to be divergent.
Note that it is not possible for a sequence to have two different limits for it is impossible to have an arbitrarily close to each of two different numbers for all n sufficiently large. If a sequence is not convergent then it is divergent. Introduce the symbolism
lim an = A.
n~oo
This is read "the limit of an as n becomes very large is A."

71

E. The infinite geometric series
1. The sum of the infinite series

Determining whether a given sequence has a limit and finding its value calls for some knowledge of the properties of limits. The basic theorems concerning limits may be stated for use in finding the limit of a sequence without giving any proof for these theorems. H the teacher prefers to give a more thorough treatment of the concept of limit see Report of Commission on Mathematics Appendices, Chapter 6.
Definition: The sum of an infinite series is the limit of the sequence of its' partial sums if this limit exists. A series which has a sum is called convergent. H no limit exists the sum of the infinite series is not defined, and the series is said to be divergent.

Some geometric series will not approach a limit. For example,

2. Repeating decimals llnd rational numbers

If we use the partial sums
Sl a1 =2
+ + S2 a1 a2 2 4 = 6 S3 a1 + a2 + a3 = 2 + 4 + 8 = 14
then we see that Sl' S2' S3' ... or 2. 6, 14, ... will not have a limit. This series has no sum and the series is divergent.

Other geometric series will approach a limit. For example,

+ + + + ... + + 1/2

1/."

1/8

1/16

1/ 2D

The sequence of partial sums here is

{i/2' 3/4 , 7/ 8, 15/16, ...} and approaches 1 as a limit. This series is said to be convergent.

It should be pointed out ~o students that the word sum here has been given a different meaning. It can be shown by more advanced methods that only in-
finite geometric series having lrl < 1 will have sums. A demonstration of this
will be sufficient here.

= If the formula SD am - 1 is used in an attempt to find the sum of the r- 1

terms of an infinite series it is more profitable to consider it in its equivalent

form,

Sn =

a - am 1-r

. As n

becomes

lar.ge

am 1 - r can be made ar-

bitrarily small. When the number of terms, n, is very large1-a~ r-"-r-- is

very small and Sn becomes closer to-1~a- -- r -. Therefore the sum of the in-

finite geometric series is defined as SD

a 1- r

It should be shown that any repeating decimal equals a rational number which is the sum of an infinite geometric series. Note that a number which cannot be expressed as a terminating or repeating decimal is an irrational number.

The formula s..

a is used to find the rational number which rep1- r

resents the repeating decimals. Students should also learn an alternate method

for changing repeating decimals to their rational representation. The method

is illustrated below.

Given .123123 ... To find the rational number which represents this decimal we use the following procedure:

72

~t n :;:: .123123 ...

Then 1000n = 123.123 .. :

1000n - n = 123

999n = 123

= = n

123/!lltll

41/33:1

73

Argand diagr:am arithmetic progression arithmetic sequence associativity asymptotes
binomial square binomial theorem
characteristic combinations common difference common factor completing the square complex fraction complex numbers conic sections conjugate consistent systems convergent sequence
denseness density dependent systems difference of squares difference of cubes discriminant distance formula divergent sequence
ellipse equivalent equations equivalent systems exponential function extraneous rootl

VOCABULARY
geonietric progression geomet.ric sequence geometric series
harmonic mean hyperbola
iderrti ties identity identity function imaginary number inconsistent system infinite decimal infinite geometric series intercept form interpolation inverse inverse element inverse function
limit of a sequence linear combination log:lrithm logarithm function
major axis mantissa maximum point mid-point formula minor axis
n factorial nth term non-terminating decimal

ordered m-tuples ordered triple ordinate origin
parallel partial sum Pascal's triangle permutations plane point-slope form proper subset
quadrant quadratic equation quadratic function quadratic formula quadratic inequalities
radian rational number system real number system repeating decimal
sequence series sigma notation, ,,~" slope-intercept form solution of an equation sum of a finite geometric series sum of an infinite geometric series sum of arithmetic series sum of cubes summation notation

74

- - - - - - -

-

-- - - ~ - ~ ~

ADVANCED ALGEBRA AND
TRIGON'OMETRY

UNDERSTANDINGS FORADVANCED ALGEBRA AND TRIGONOMETRY
It is desirable that the following objectives will be achieved by the fourth year mathematics course:
Students will have someknowledge of the historical development of algebra including its modern aspects. Students will gain a thorough knowledge of the algebraic number systems and will be able to prove some theorems in each. Students will be able to use the language and notation for sets. Students will gain complete familiarity with concepts of relation, function, and inverse function. Polynomial functions will be explored thoroughly so that methods for finding zeros can be learned, and also methods for writing equations from sets of points can be discovered. Exponential and logarithmic functions will be studied for their importance in mathematical analysis. Trigonometric functions will be examined first as functions of real numbers, then as functions of the general angle, and finally as functions of acute angles in triangles. Students will gain competence in use of logarithmic tables in solution of right triangles and oblique triangles and will be able to solve problems of practical nature by means of trigonometry. Students will be able to solve exponential, logarithmic, and trigonometric equations and to prove trigonometric identities. Complex numbers will be studied graphically both with rectangular and polar coordinates and all the operations including raising to powers and taking of roots will be accomplished in one system or tQe other. Pupils will learn the properties of a group through a study of vectors and the roots of unity.
77

CONTENT

TEACHING SUGGESTIONS

I. Sets Of Numbers

A. Names of sets

Any capital letter may be chosen as the name of a set, such as:

N for the set of natural numbers W for the set of whole numbers J for the set of integers Q for the set of rational numbers R for the set of real numbers M for the set pure imaginaries C for the set of complex numbers.

This section is essentially review of what has been learned in previous courses. The amount of time spent here will be determined by the student's knowledge of set notation. If they have never been introduced to the language of sets, enough time should be used for them to learn the meaning of set, subset, null set, intersection, union, universe. First use very small finite sets for illustrations. Use some examples from geometry.

B. Definition of sets of numbers used in algebra

N = {I, 2, 3, ...} W = {O, 1, 2, 3, ...} J = {... -3, -2, -1, 0, I, 2, 3, ...}
= Q {ajb: a, b. J and b # O} or
{all infinite decimals which either terminate or repeat}
R = {all infinite decimals}
C = {a + bi: a, b Rand i2 = -I}
= = M {a + bi: a + bi. C, a O}

The historical development of the different kinds of numbers can be used effectively in presenting this material. Attention shouldl be called to the fact that it is not possible to list the members of each set and therefore- alternate methods must be used.

C. Relationship of sets

Integers

Positive Zero

(Natural Numbers) Whole Numbers

Rational

Negative

Real

Fractions

- Complex

Irrational Pure Imaginaries

D. Graphical representation of sets.
1. N, W, J, Q and R on number line .
a. Concept of one-to-one correspondence
b. Concept of order

Practice in using the idea of subset and universal set will emphasize the relationships among the sets of numbers. Show that a natural number such as 7 is also the positive integer 7, the rational number 7j l' the real number 7.0000
... , and the complex number 7 + Oi. In demonstrating that the two ways of
designating the rational numbers are equivalent, review methods of changing a repeating decimal to a rational number as outlined in Section X. F. 2 of Algebra II Guide.
This section is also review, and probably little time needs to be spent here except to introduce the idea that although all of the sets are infinitely large they are not all equally numerous. Define "countable" and suggest to the students that they experiment a little on their own with finding ways to set up one-to-one correspondence between the member of other sets and the natural numbers.

78

c. Concept of absolute value
"'d. Intuitive approach to density of sets' and countability of sets
'-'e. Definition of finite and infinite sets
2. Cartesian plane with points representing ordered pairs of real numbers
3. Complex plane.

The students s~ould be familiar with the Cartesian and complex planes, but perhaps they need to be reminded of the one-to-one correspondence between R X R and the points of the Cartesian plane, and between R X M and the complex plane. Explain that these are arbitrary devices that man has adopted to enable him to understand mathematical ideas better and to extend his explorations.

II. Number Systems

A. Definition

A number 'System consists of a set 'of numbers, two operations (generally multiplication and addition) defined on the set, the properties belonging to the set, and a' definition for equivalence hetween members of the set.

B. Operations defined for sets listed in I. B.

In view of the emphasis that is being placed on structure by mathematicians, section II is quite important. Recall the definitions for addition and multiplicatiorr for each set learned in previous courses.

1. Addition

2. Multiplication

C. Equivalence defined for each set

Also determine if the symbol ":;::::" means the same in each case.

D. Properties of each set

Properties to be investigated for each set in I. B. using a, b, c, as any three non-zero members of the set under consideration.
1. a + b = c, a unique member of set (closure) 2. a + b :;:::: b + a (Commutative) 3. Is there a member 0 such that 0 + a = a + 0 :;:::: a? (Identity) 4. For every a is there an (-a) such that a + (-a) = (-a) + a =O? (Inverse) 5. a + (b + c) = (a + b) + c (Associative)
6. a X b = c, a uniq\le member of set (Closure)
7. a X b =b X jI, (Commutative)
8. Is there a member 1 such that a X 1 = 1 X a :;:::: a? (Identity)
9. For every a is there a member '/. such that a X '/. = '/. X a = I? (Inverse)
10. a X (b X c) = (a X b) X c (Associative)
11. a X (b + c) = (a X b) + (a X c) (pistributive)

E. Field' (has all the properties listed) 1. Rational number system 2. Real number system 3. Complex number system

Review the Illeanings of the properties and then allow the students to determine intuitively which Of the properties flach number system has. A good way to record results would be to make a cq'!rt with the properties listed on the porizontal lines and the names of the number system heading the vertical colurtms. When the chart is complete ask the students to name the number systems which are fields and to observe what advantages afield has over systems which are not fields.

F. Integral domain (has all properties except number 9)

Call attention to the fact that the integers have all the properties except number 9 and as.a consequence have certain disadvantages. Identify the set of integers with the two operations, addition anq multiplication, as an integral domain.

G. Subtraction B. Division

Define subtraction for each set and test for c~osure. Define division for each set and test for closure.

79

I. Properties common to all the systems

Discuss the properties which are common to all the systems and the uses of these common properties in algebraic manipulation.

III. Special Properties Of The Integers

A. Definitions

Make formal definitions of equivalence, order, ~nd absolute value. Motivation is needed for effective presentation of Unit III. ''Explain to the students the efforts of the mathematicians in the past one hundred years to organize ariti;lmetic and algebra into logical systems patterned after Euclidean Geometry, which was arranged in this manner in 300 B. C. The present effort has grown out of the need for mathematical models in other areas of research. Aside from the coherence and insight that a knowledge of number systems lends to the study of the processes of algebra, the wider application of such knowledge is more important.

B. Theorems to be proved

Assuming for the integers a multiplication cancellation law (If ac = be and
= c oF 0, then a b) as well as the ten other properties of the integers the fol-
lowing theorems can be proved for any a, b, c f J.
= = 1. a + c b + c if and only if a b. = = 2. a 0 O a 0 for any a f J. = = = 3. If ab 0, then a 0 or b O.
4. (-a)(-b) = ab, also (-a)(-a) = a2
5. If a > band b > c, then a > c.
6. a > b if and only if a + c > b + c.
7. If'a> b, then ac > be for c > O.
8. If a > b, then ac < be for c < O. 9. All equations of the form x + b = c have a unique solution in J.
10. -Ial::; a $ lal
11. labl = lal Ibl
12. la + bl -:; lal + Ibl

There is a delicate balance here between how much to prove and how much to accept by intuition. Not nearly as many proofs can be included in a high school as in a college class. Until there is a wider knowledge and acceptance of modern mathematics by the general public it is well to concentrate on just such proofs as will be needed to justify the algebraic computations. Time is another consideration. To require. high school students to be responsible for a great number of proofs would consume much time that can better be spent in other ways. Care must be exercised in selecting the theorems to be proved and also the sequence in which they are proved. If the series listed here is not acceptable other choices can be made from A Survey of Modern Algebra by Birkhoff and Maclane; Studies in Mathematics, Vol. III, SMSG, Intermediate Mathematics for High School, (Part 1), Sl\iSG. The proofs for the theorems listed can also be found in the above references. The following is a sample proof as it might be presented:

Theorem. If a, b, c f J and a + c = b + c, then a = b.

Proof. Since a + c = b + c, we know that (a + c) and (b + c) represent the
same integer. If we add (-c), which is an element of J by property 4, to each
of these representations then again we would get two different names for the same riumber. (The sum of two integers is a unique integer.)

= Hence (a + c) + (-c) (b + c) + (-c).

(1)

Apply the associative property and the result is

a + [c + (-c)] = b + [c + (-c)].

(2)

But any number plus its inverse is zero (definition of inverse) and therefore (2) becomes

a + 0 = b + O.

(3)

80

~S&

zq

C. Some equations of the form
ax + b = 0 have solutions.
D. Some equations of the form
ax2 + bx + c= 0 have
solutions. E. Division algorithm
F. Prime factorization for integers

Finally, since is the identity for addition, then (3) becomes
= a b.
This proof is the justification for what has been commonly called cancellation, and the statement of the theorem is still called the Addition Cancellation Law.
Since algebra evolved from a need to find methods of solving problems which could not be solved easily by means of arithmetic, equations and the solving of equations are the main ideas throughout algebra. Therefore emphasis should be on what types of equations have solutions in each of the number systems. The students are already familiar with linear and quadratic equations and can easily determine under what conditions these equations have solutions in each system.

For any two integers a and b, with b r 0, if a is divided by b the process may
be expressed in this manner, "I" = q + 'I" where q stands for quotient and l' for remainder. An alternate way of expressing this same fact is a = qb + r.
This is familiar to students as the way they checked division in the elementary school. The formal theorem is stated thus:

Theorem. For given integers a and b, with b r
such that
= a + bq 1', < l' < b.

0, there exist integers q and l'

The proof of the above theorem is too difficult at this stage of training, but
it can be checked out in an empirical way using a > 0, a = 0, a < and remem bering that either q or l' can equal or both can equal zero.

The number a is said to be divisible by b if there exists an integer q such that
a = bq + l' and l' = o. Another way of making the same statement is: b divides
a, or bla. The numbers band q are factors, of a, and a is called the product of band q; a is known as a composite number. If a = bq and band q are composite numbers which have composite factors, then it is conceivable that two people, in finding the prime factors of a, might proceed in a different way. The question naturally arises whether the two sets would be identical. Allow the students to s.atisfy themselves as to the answer and then to formulate the statement of the theorem. Compare their statements with the following:

Any integer not zero can be expressed as (:t 1) times a product of positive
primes. The expression is unique except for the order in which the prime factors occur.

Ask students what would be involved in the proof of the above theorem and whether they have sufficient background to prove it. A good theorem for the students to prove in this sections is:

Theorem. If a is a divisor of band b is a divisor of c, then a is a divisor of c for any integers a, b, c with a, brO.

IV. Special Properties Of Natural Numbers

A. Unique factorization theorem

Since the set of natural numbers is a subset of the integers, the students should be able to see which of the theorems proved would also apply to natural numbers. Ask them to restate the prime factorization theorem so that it will apply to natural numbers, and call attention to the fact that this theorem is called the Fundamental Theorem of Arithmetic.

B. Mathematical Induction Axiom

Introduce this study with several examples, such as:

1
1+3 1+3+5 1+3+5+7

12 22 32 42 and so on.

1. Theorems

Ask students to state this in the form of a theorem and in light of the fact that the set of natural numbers is infinite whether or not they can prove it. Some of the students will see that they do not have adequate tools with which to make

81

'''2. Binomial Theorem C. Some equations of form
ax + b = 0 have solutions.

the proof and at this point the ~Iathematical Jnduction Axiom can be introduced. Induction Axiom: If S is a set of naturai' numbers \\'ith the properties:
(1) S contains 1 (2) If S contains a natural number k. then it follo\\'s that S
contains k + 1 then S is the set of natural numbers. See Allendoerfer and
Oakley for list of theorems to be proved by it.
Sample proof: (Note P. stands for proposition O\'er the natural numbers)
= Theorem or P n 2 + 4 + 6 + ... ~ 2n n(n + 1).
Proof. Let S = {x:x is a nautral number for which P. is true.}
1 S for 2(1) = 1(1 + 1). Assume P. is true for k. Then 2 + 4 + ... + 2k = k(k + 1). If k. N. (k + Il. N and 2(k + 1). N (Properties of natural numbers).
= Hence 2 + 4 + ... + 2k + 2(k + 1) (By property a + b a unique member
of N).
But k(k + 1) + 2(k + 1) = k2 + k + 2k + 2
= k2 + 3k + 2 (Distributive and Associative Laws) = + + (k 1) (k 2) = (k + 1) [(k + 1) + 1]
Therefore since equivalence is transitive 2 + 4 + ... + 2k + 2(k + 1) = (k + 1)[(k + 1) + 1], and this is the same as saying that if k.S, then (k + 1).S.
Finally, then S = {x:x is a natural number} by the Induction Axiom.
After students become accustomed to using the Mathematical Induction Axiom for making proofs, there are some who will be able to understand the proof of the Binomial Theorem.
Determine when a linear or a quadratic equation over the natural numbers has a solution.

V. Fields
A. Properties of ordered field
1. Cancellation law for multiplication
2. Every ekuation of form
ax + b = 0 has a solution
B. Some equations of~he form
ax2 + bx + c = 0 have solu-
tions in rational and real number fields.
C. Complex number field
1. Has all properties of other fields except order.
2. All equations of the form
ax2 + bx + c = 0 have
solution

Since a field has all the properties of an integral domain then all the theorems proved for the integers are also true over the rational, real and complex num-
ber systems. Prove the Multiplication Cancellation Law: If ac = bc, then a = b
for c ,,1= 0, and the theorem: Every equation of the form ax + b = 0 has a
unique solution over a field.
Ask students to recall the use of the discriminant froni intermediate algebra and to make generalizattons concerning where a quadratic equation has solutions over real, rational, and complex fields. Do any review work that is necessary in solving quadratic equations; in particular, justify: If (x - a) (x
- b) = 0, then (x - a) = 0 or (x - b) = O.
Discuss properties of complex number system and complex numbers as roots of equations, but delay the' greater part of the work with the complex number system until the unit on trigonometry has been completed.

Vi. Polynomial In One Variable As Integral Domain

A. Definition of set of polynomials
B. Complex, real and rational polynomials

Let P represent the set of polynomials in one variable. Then P = {P(x):P(x)
is an algebraic expression of the forma"x + a,xn -, + ... + an-,x + an with' n > 0 and a", a" a" ... , an as numerals}. Explain what is meant by the general
polynomial form. Define polynomials as being complex, real, and rational ac-
cording to whether the a's are complex, real, and rational. Determine whether

82

C. Operations D. Properties E. Division algorithm
F. Prime factorization

addition and multiplication are closed on the polynomials. Allow students to make some effort to generalize the definitions for addition and multiplication. Define equi':alence for polynomials. Test polynomials for the 11 properties listed in Section II D.
State the division algorithm for polynomials over a field: If A(x) and B(x) are polynomials and B(x) #: 0, then there exist polynomials Q(x) and R(x) such that
A(x) = B(x)Q(x) + R(x) where R(x) is zero or of degree less than B(x).
If R(x) is zero then A(x) is said to be factorable (reducible) with B(x) and Q(x) as the factors.

VII. Relations And Functions

A. Brief review of definitions and notation.

Begin the study of relations with some simple non-mathematical relations such as "is the brother of" or "sits in front of" until it is evident that a relation is concerned with sets of pairs. Then allow the students to apply the two relations "is taller than" and "is the same height as" to the members of the class.

See how many different pairs can be obtained by "is taller than" and what effect applying the relation successively has on class membership. Then see what effect the relation "is the same height as" has. Take one group that has several members equally tall and test the relation "is the same height as" to see if it is reflexive, symmetric, and transitive. (Let R represent relation).

Reflexive: Symmetric:
Thansitive:

Obviously, John is as tall as John. John R John. If John is the same height as David, is David the same height as John? If John R David, is David R John? If John is the same height as David, and David is the same height as Mary, is John the same height as Mary? If John R David and David R Mary, is John R Mary?

B. Equivalence relation
1. Relexive symmetric and transitive properties
.2 Graphs of relations

Do the same for the relation "is taller than." Now see if the student can draw any parallel between the concrete situation and the relations "is equal to" and "is greater than" for numbers. Make formal definitions for relation, domain, range, properties of a relation, mutually disjoint sets.
"-
Explain that most of the relations in algebra are over the set of real numbers (R) so that an algebraic relation is a subset of R X R. It should be understood in the following section that all the numbers under consideration are real numbers. Review some relations from second year algebra such as:
= = K (a certain relation) {(x, y):x2 + y2 25}
Q (another relation) = {(x, y):x - y = 1}

The graph of relation K is the set of all p9ints on the Cartesian plane whose coordinates are ordered pairs of numbers belonging to set K. A similar statement can be made for the graph of any relation over the real numbers. Suppose a given problem says consider K and Q simultaneously or find the solution set for K and Q. This simply means to find the intersection of the sets K and Q or, in set notation:
n + K Q = {(x, y):x2 y2 = 25 and x - y = 1}

When algebraic methods are used for solving the two equations together, it is
found that K n Q = {(4, 3), (-3, -4)}.1t is obvious that these are indeed com-
mon solutions for the equations or, in the new language, ordered pairs that
belong to each relation. The graph of K n Q consists of the points which are
83

C. Inequality relations D. Definition of function as a
special kind of relation
1. Graphs

the geometric intersection of the graphs of the relations K and Q and their coordinates are respectively (4, 3) and (-3, -4). To sum up: An algebraic equation in two variables is an equivalence relation over the real numbers and as such has a visual representation on the Cartesian plane. When two relations are plotted on the same set of axes, points of geometric intersection of the graphs have coordinates which are the ordered pairs that belong to the algebraic intersection of the relations. The idea may be extended to relations in three variables and three-space.
The solving and plotting of inequalities have already been developed in Elementary and Intermediate Algebra, but now the statement of an inequality should be recognized as a relation and put into set notation:
x>y + 2 means R] (relation) = {x, y):x>y + 2} + + x2 - 5x 6 > 0 means R2 = {x, y):x2 - 5x 6 > O}.
To solve systems of algebraic statements in which one or more of them are inequalities means to find the intersection of the relations defined by the statements. The visual representation of these intersections is very effective. The amount of time spent on this material will be determined by the student's ahility to handle inequalities algebraically and graphically.
There are many definitions for function, but following a study of relations the logical one is: A function is a relation in which no two ordered pairs have the same first element. An alternate definition, and one which may mean more to the students is: A function consists of a set A called the domain, a set B called the range, a formula, a rule, a table or a graph which associates with each member of set A exactly one member of set B.
Some non-mathematical functions should be given illustrations, such as: (1) Let f be the name of the function "is the atomic number oL"
If A = {the symbols for names of all elements} and B = {1, 2, 3 .... , 103} then f = {(H, l),(He, 2), ... _., (Lw, 103)}. Explore different notations for mapping as f(H) = 1 or f:H~ 1
f(He) = 2 He-->2
(2) Let g be the name of the function "is the state capital oL" If A = {names of
= cities which are state capitals} and B = {names of the states of the United
States} then g {(Atlanta, Georgia), (Hartford, Conneticut), and so on}_
Most algebraic functions have as their domain the real numbers, as their range a subset of the real numbers, and an equation which defines the rule for as-
sociating the members of the ordered pairs. Given the equation 3x + y = 1,
a function can be derived which can be stated thus:
= = f {(x, y):y 1 - 3x} or alternately,
f = {(x, f(x:f(x) :z: 1 - 3x}

The graph of f is the set of points on the Cartesian plane such that each point
of the set has as coordinates an ordered pair belonging to the function f. The
= + graph of the function f is also the graph of the equation 3x y 1. The fol-
lowing are two of the methods used for listing the ordered pairs belonging to
the function f.

f: x --> 1 - 3x 1 --> -2 2 --> -5

f: f(x) = 1 - 3x
= f(1) -2
(2) -5

The first is read "under the function f, x maps into 1 - 3x, 1 maps into -2" and so on. The second one is read "given the function f such that f of x equals 1 - 3x, f of 1 equals -2, f of 2 equals -5" and so on. A third method is simply making a table of values for the ordered pairs:
84

2. Constant and identity functions
3. Absolute value function 4. Greatest integer function *E. Composition of functions
F. Inverse relations and functions

For f (x) = 1 - 3x

l-x~!-l--1-2-1 I f (x) I -2 I -5 !

Choose some first and second degree equations and determine from their graphs which are functions and which are relations. Give students a chance to discover for themselves the rule that if a vertical line cuts the graph in two
= points, the graph is that of a relation and not a function. Derive some functions
from such equations as y (x - 3) I (x - 2) and call attention to the fact that 2 cannot be in the domain of the function since division by zero is not permitted. Also show the difference between the functions derived from
y = (x - 1) (x + 2) I (x - 1) and y = x +2.

Determine whether a reasonable function can be derived from any algebraic equation.

The amount of time spent on this unit on functions should be sufficient to give students a thorough understanding of function. Enough practice with use of function notation and plotting of graphs should be given that students will feel secure in their knowledge of functions. The success or failure of the Fourth Year Mathematics Course will depend on how well the concept of function is implanted in the minds of the students. Additional materials and lists of equa tions can be found in Allendoerfer and Oakley, Principles of Mathematics and SMSG Elementary Functions, Part I. Some time should be spent on special functions such as the identity, constant, absolute value, and greatest integer function. Information about these can also be found in the above references.

An important idea concerning function is that of composition of functions. Suppose f is a function such that f: x ~ 2x + 1 and g is a function such that g: x ~ 3x2, then these two functions can be combined in two different ways, f (g (x) ) or g (f (x) ). In either case the function inside the parentheses is ap-
plied first. It might seem that this would produce the same result, but con-
= sider the results for x 2,

then f (g (x = f (12) = 25
= = and g (f (x g ( 5) 75

Note:

f (g(x) ) is some times shortened to fg (x) and g (f(x) to gf (x). For the function fg, which is called the composite of f and g, the domain is only those members in the domain of g such that all g (x) are in the domain of f. It is possible then that the domain of fg is a restricted set of real numbers. A good analogy for composite functions and also some exercises are found in SMSG Elementary Functions, Part I.

For any given set of ordered pairs the order within the pairs can be reversed so that a new set of ordered pairs is produced, which is called the inverse of the .original. This means that the set of numbers that was originally the domain is now the range and vice versa. Consider what would be the result of interchanging domain and range of a relation. Is' the result necessairly a relation? Do the same for function. Take an example of a finite function such as f:

ID 1 2 3 4 5 6 IR a a b b c a

and reverse order of pairs

TD

a

a

b

1;>

Ca

IR 1 2 3 4 5 6

I --_I

From the definition of function, it can readily be seen that the result is not a
function since several of the pairs have the same first element. Allow the students to determine for themselves when a function has an inverse, and then see if they can state" a formal definition using the notation f~l as the inverse of the function f. Note: A function of f has an inverse f -, if f (f ~l(X) =
= f" -- 'l(f (x) = x. Take a given Iii-lear equation such as x - y 2; derive the
function and plot the graph from a set of at least three points: Reverse co-

85

ordinates of the points ;md plot the new points. From the coordinates of the

new points and using the line formula y - y1

YI - Y2

write the equation of the line which passes through the points. Compare with the original equation. Let the students do a number of these, and then ask them to make some generalizations.
"'Some theorems that can be proved here are:
(1) Every linear function gives a one-to-one correspondence between the set -of all real numbers and the set of points on a line.
(2) Every linear function has an inverse.
(3) If f is the linear function defined by y = ax+ b, its inverse f -I is defined by y = (1!a)x - b/a.
The proofs of the above theorems can be found in SMSG l'ltermediate Mathematics, Pait I.

VIII. Circular Functions

Circular functions are introduced at this point to reinforce the understanding of function and inverse function.

A. Unit circle

Consider an object whirling about a fixed point in a plane so that its path is a circle. How could you describe its position at any point along its course? In thinking about this problem the students will probably arrive at two solutions:

(1) If a fixed point is taken as the starting point, then the object will have moved a definite number of units of. distance at any point in its flight.

(2) If reference line are drawn so that they intersect perpendicularly at the center of the circle, then the object can be located by its distance from each of the lines. The following model can then be set up for the situation just described with points of the circle representing the object at various positions along its course.

y
P,

x'

x

B. Sine and cosine functions of real numbers defined

'I,

If the starting point is taken arbitrarily at 0, 0), then the circle which represents the path of the object is called a unit circle since it has one as its radius. The distance of the object from 0,0) measured along the arc is clearly associated with exactly one point of the circle so that a function is determined.

f: d,
d~
d,

-+ P,
-+ P~
-+ Po

86

But each point can be represented by its coordinates so that the function might be

f:

-+ (x" y,)

--+ (x2' Y2) -+ X3, y,)

Since the range consists of pairs of numbers it is more convenient to make two functions from f and give each a name as

Sine:

d, -+ y, d" -+ y, d" -+ Y3
0.

Cosine: d, -+ x, d" -+ x. d" -+ X:t

C. Domain and range
p. Radian
t. Graphs of sine and cosine
function 1. Periodicity 2. Amplitude 3. Phase F. Inverse of sine and cosine
G. Sin x and Cos x and valariables in the real number system
H. Use of infinite series to evaluate sine and cosine

Let it be decided that when the point moves in a counterclockwise direction the distances wil be positive and when clockwise they will be negative. Now considering that the point can make any number of revolutions in either direction it is evident that the domain of the two functions is the set of real numbers. Determine the range for each function.

Since the length of the radius is the unit of distance used, then in a circle there would be 211' of these units which are called radians.

The lengths of the arcs in radians can be determined for the familiar angles such as 30, 45, 60, 90 and so on. By choosing positions on the cIrcle for pointsso that the angle between the line connecting the point with the origin and the positive x-axis is one of these angles, enough ordered pairs can be found to plot the graphs of the sine function and the cosine function. Notice what happens to the graph when the point has made one complete revolution.
+ Show that sin (211' d) = sin d and make a generalization from this Dis-
cuss periodicity, amplitude and phase.

Reverse the order of the pairs used to graph the functions and determine if the resulting relation is also a function. Investigate what could be done to the domain of the original function so that it will have an inverse. The symbols used for circular functions are: sine (x), which is abbreviated sin x; cosine (x), which is abbreviated cos x. Inverse of sin x is sin -IX.

It is evident that sin x and cos x are variables in the real number system and as such can be used iIi equations, inequalities, and systems of equations. Do some exercises using sin x and cos x as variables. Students should already be familiar with the use of natural tables from previous courses; Plot the graphs of some functions of the form y = k sin x and also of the form y= sin kx where k = 1/'2' 11 a' 2 3, etc., and have students make some generalizations froni the results.

The following infinite series can be used to approximate the sine and cosine functions to any degree of accuracy:

cosX=

1 - -x2+ - x-4- +x-6: "x-U_ " 2' 4! 6! 8!

IX. Polynomial Functions, Equations, And Graphs

A. Definitions
B. Brief review of quadratic function

A polynomial function is one defined by the polynomial equation
aox" + a1 x"-j + ... + a" _ + 1 X a" = 0
If a :F 0, then n is called the degree of the polynomial. A linear function is a polynomial funcition defined by polynomial equation of degree one, a quadratic function by a polynomial equation of degree two, cubic of degree three, quartic of degree four, and quintic of degree five. Note: The ploynomial function of degree zero is the constant function, and this is altogether different from the zero polynomial. Linear functions have already been discussed in ,the section of functions, and the following material deals with polynomial functions of degree n =. 2.
87

C. Theorems
D. Methods of finding zeros 1. Synthetic division 2. Synthetic substitution 3. Approximation 4. Descarte's rule of signs
~E. Tangents to graphs of polynomial functions 1. Application to graphs 2. Slope function 3. Maxima and minima

The following theor.ems should be proved:
1. Remainder 2. Factor
3. Location (accepted without proof) 4. Upper bound for zeros of polynomial 5. Rational zeros 6. Integral zeros 7. Fundamental Theorem of Algebra
(accepted without proof)
8. Number of roots 9. Complex conjugates 10. Multiplicity of roots
One of the main concerns in a study of polynomial functions is that of finding points (if any) where their graphs intersect the x-axis, since the abscissae of these points are solutions of the equations of the polynomials which define the functions. A definition which should be learned here is: Let f be a polynomial function. If a is a number in the domain of f with the property that f(a) = 0, then a is called a :zero of f; a is also called a zero of the polynomial
which defines f. The set of zeros of f = {x: f(x) = O}.
The students already know methods of finding zeros of quadratic functions. The history of efforts to find formulas for solutions of polynomials of higher degree can oe found in a number of books such as E. T. Bell, Men of Mathematics. A convenient method of finding the ordered pairs of a polynomial function in order to plot its graph is the synthetic substitution method as described in SMSG Elementary Functions, Part I. A full treatment of polynomial functions can also be found in that reference, with proofs for the theorems listed. See Allendoerer and Oakley, t:undamentals of Freshman Mathematics for further treatment.
This section may be of interest to the better students. Reference material can be found in SMSG Elementary Functions, Part I.

4. Newton's method
F. Methods of finding the equation of a polynomial when iome ordered pairs of function are known

Another main concern in a study of polynomial functions and equations is to be able to write a polynomial equation which will pass through certain specific points of the graph. The students will know how to write a linear equation when two points are known and a quadratic when the roots are known. Two methods for writing higher degree equations which will pass through specific points can b~ found in SMSG Elementary Functions, Part I, Appendices.

x. Exponential Function

A. Brief review of exponents

Do any necessary review with exponents, radicals, and irrational equations. The
study of exponential functions should start with a review of laws of integral,
rational, and real exponents, emphasizing proofs of the laws of exponents. A set
of theorems with suggestions for proofs can be found in Allendoerfer and
Oakley, Principles of Mathematics, Appendix to Chapter IV. It is especially
important to prove the theorem a' > a' if x > y for a > 1, and the related
theorems.

B. Graphs of exponential functions to base 2 and base 10
C. Extension of exponents to include irrational numbers
D. Solve exponential equations

The functions over the real numbers defined by the equation y = aX, for a > 0
and a # 1 is called exponential function with base a. Take a = 2 and plot the graph of f: x ~ a', Discuss the effect of changing a' to ka' for any positive
k. Assume the graph continuous and investigate the possibilities of irrational
values for x and corresponding values for' f(x) .. These may be determined by
methods of approximation. Estimate some values for f(x) when x is irrational.

88

E. Applications
'T Tangent to graph of exponential function

Also estimate some values for x when f(x) is known. On the same set- of axes plot the graph of f: x -+ 10' and go through the same sort of process. It shoulJ be clear to some of the students by this time that any positive real number can be expressed as a power of any other positive real number greater than one.
= Ask for suggestions for methods of solving 2- 10' and 2' = 10-. Leave
question unanswered for a while. Solve other exponential equations and discuss some practical problem where the exponential function is used, such as rate of growth of bacteria or compound interest. A discussion of tangent lines to exponential graphs can be found in SMSG Elementary Functions, Part 2.

XI. Logarithmic Function

A. Graph B. Theorems C. Special bases
1. Base 2 2. Base 10 3. Base e

Take the table of values used for plotting f(x) = aX, interchange the range
with the domain, and plot the new graph. Determine if the resulting graph is a
function. Give y = log. x as the equation which defines the new function, and
then state the formal definiton of logarithm. Do some exercises changing from one notation to the other. Prove the theorems on logarithms that follow diretcly from the definition. Discuss the real number e as a base for logarithms; have the students do some research into its history, discuss its importance in mathematics, especially the fact that it is the only known function whose slope
function fl(x) = f(x). Present the method by which values for numbers to
base e are obtained. (See: SMSG Elementary Functions, Part 2.)

D. Conversion from one base to another

Present and prove the formula for changing a number from one base to another Do some exercises using logarithms to review what has been taught in previous courses and to check for degree of student's mastery of processes.

XII. Trigonometric Functions of the General Angle

A. Definitions

Define the general angle as any angle generated by a radius vector as it rotates either clockwise or counter clockwise from its initial position along the positive X-axis of a Cartesian coordinate system. As the radius vector assumes any terminal position the angle which is thereby determined has associated with it three real numbers; r, the length of the radius vector (arbitrarily taken as positive) and x and y, the coordinates of the end-point of the radius vedor. From these three real numbers there are six possible ratios -and these ratios are the six functions of the general angle. Make definitions for sine, cosine, tangent, cotangent, secant, and cosecant functions. Define standard position, triangle of reference and related angles. Allow students to discover for themselves the method of finding the related angle for any positive angle (those for which the radius vector rotates counter clockwise) and for any negative angle (the radius vector rotates clockwise). Also let them find the relationship between the functions of positive and negative angles of the same absolute value.

B. Law of Cofunctions

Use discovery method for developing the law of cofunctions.

C. Quadrantal angles

Define quadrantal angle and then taking the radius vector of unit length determine the values of the six functions of quadrantal angles both positive and negative unit! student can arrive at a generalization.

D. Domain and range of the trigonometric functions

Using the entire set of angles as domain for each of the six functions establish the range for function.
Example:

Sine Function
D = {e : e is an angle}
R = {a e Rand -1 ~ a 'C.1}
NOTE: e can be expressed either in degrees or radians, and students should be
able to translate from one to the other with ease.

Identify the two big problems involved at this point:

89

E. Graphs of functions
F. Inverse of functions
G. Identities I. Reciprocal 2. Quotient 3. Pythagorean

1. Given any 0 as the first member of an ordered pair belonging to a specified function, find the real number which is associated with it.

2. Given any real number as the second member of an ordered pair bl?longing to a specified function, find the angle which is the first member of the pair.
Use natural tables for this work and do enough exercises until students feel secure in their use of the tables and of interpolation.

Using the natural tables to find the second members of the ordered pairs and

using the

first members the angles

00 ,

15 0 ,

300 ,

45 0 ,

60 0 ,

75 0 ,

90 0 ,

1050 , 120 0 , etc., draw graphs of the six functions by expressing the angles in

terms of radians and plotting them along the positive X-axis. Plot the functions

which are reciprocals of each other on same graph in order that some rela-

tionships will be more obvious. Determine the period for each function.:'

Review material of Section VIII. E and extend to include tangent, cotangent, secant, and cosecant.

Emphasize that if the functions are to have inverses which are also functions
then the domain must be restricted so that _90 0 < 0 < 90 0 Give students

ample opportunity to be come familiar with the differeriiforms of inverse no-

tation. They should be able to use them interchangeably.

=i Example: Sin - 1.5

is eauivalent to arcsin .5 = ~

The reciprocal and quotient identities are easily derived from the definitions of the functions. The Pythagorean identities should be proved. Line representation of the functions using a unit circle will help fix the Pythagorean identities in the student's memory.

_-+"ki= - - - - - - - - - : 7Q

OR=cos8 PR=sin8 MN=tan8 ON=tan8 KQ=cote OQ=csc8

H. Conditional equations I. Law of cosines

Probably only the eight fundamental identities should be memorized and then all proofs of identities be done in terms of these eight. The key to proving many identities lies in the ability oJ students to realize that the definition of multiplication of rational numbers is reversible, i.e.,
ab a b a b
Cd ==c-d or d c
Review some examples from algebra to reestablish the difference between an identity and a conditional equation. Do exercises with trigonometric equations until students can readily recognize which one is to be proved and which one is to b solved. If a trigonometric equation is to be solved have students recognize that the solution set in some cases, is infinite and that a least upper bound and a greatest lower one must be established for the set.
The analytic proof for the Law of Cosines is more consistent with this course than the proof from plane geometry.
90

(bcosA,bsillA)
c

(c. 0)
B

J. Sine -'lnd consine of ~um and difference of two angles
1. Tangent and contangent of sum and differnce of two angles

Place any triangle BAC in standard position ona coordinate system and write the coordinates of each of the three vertives. Then use the distance formula to prove.
+ a2 = c2 b2 - 2bc cos A
and similarly,
+ 1>2 =a2 c2 - 2ac cos B + c2 = a2 b2 - 2 ab cos C
Show that the Pythagorean theorem is a special case of the Law of Cosines
The La"Y of Cosines can be used to derive the formula' for the cosine of the difference of two angles and the other formulas follow directly from this one. A great deal of practice in algebraic manipulation can be reviewed in deriving the tangent and cotangent of the sum and difference of two angles the double angle, and half-angle formulas, and others. As much of this should be done as time permits. The minimum number of formulas for the students to memorize should include the sine and cosine of the sum and difference of two angles. With the new formulas added to the eight fundamental identities more work can be done in proving identities and solving equations.

2. Double-angle formulas

3. Half-angle formulas

XIII. Trigonometry Of Triangles

A. Right triangles
. 4w of Sines and half-angle
fonnulas

Define the six functions of an acute angle in a right triangle in terms of the sides of the triangle. Do some computation using logarithmic tables. At this time there is a need for a discussion of significant figures, precision and accuracy. of measurement. A ful discussion of this topic can be found in a great number of high school text books. In general, results of computations are sufficiently reliable if the following rules are followed:

1. The final result of a series of computations cannot be more accurate than the least accurate number used in the computations.

2. For computation with triangles, accuracy in measurement of the angle should correspond in the following way with accuracy of linear measurement of side:

Angular measure to nearest
5 1 10
5" I"

Significant figures in linear measure
1 2 4 5. 6

Prove Law of Sines and develop the half-angle formulas.

91

~ --_.

C. Oblique triangles
D. Applications 1. Physics
2. Geometry "'3. Navigation "'4. Gunnery

This section is standard material in high school trigonometry books. If three independent parts of a triangle can be measured, and at least one of these is a side, then it is possible to solve for the other parts of the triangle. The four cases are:
1. One side and two angles 2. Two sides and angle opposite one of them 3. Two sides and the included angle 4. Three sides
Set up a procedure for attacking each type of problem. Be very specific about the instances where logarithms may be used and others where they may not. If students prefer to use logarithms then develop the formulas which can be used instead of the Law of Cosines.
There are many uses for trigonometry in a study of physics. In particular, there are problems dealing with angular velocity, angular acceleration, angular momentum, moment of inertia, composition and resolution of forces or velocities, and geometric optics. The sine curve is used in a study of alternating currents and the phase angle between the sine curve of the voltage and that of the amperage is used to determine the power factor for a partieular electric circuit. Time will not permit the solution of a great many nor .very difficult problems (this should properly take place in a physics class), but enough work should be done that some transfer of learning will take place. Students should also realize that trigonometry is a valuable tool in the sciences.
The students have already had some practice in using trigonometry for indirect measurement of lines and angles. Now they can develop new formulas
for areas of triangles such as A = 1/ ~ ab sin C, etc. and also for sector and
segment of a circle.
A few problems concerning navigation could show the student the important part that trigonometry has played in the development of the science of navigation.
Define the mil as a unit of angular measure and then work some problems dealing with the actual use of trigonometry in finding range of targets.

XIV. System of Complex Numbers

A. Algebraic and vector addition
B. Rectangular and polar coordinates and conversion from one to the other

This unit may be introduced by a brief study of vectors. Some students will probably have had experience with vector and scalar quantities in physics, and the transition to abstract matl-jematical vectors should be made without creating confusion for them. A suitable unit on vectors can be found in Appendices to the Report of the Commission on Mathematics by the College Entrance Examination Board. Properties of complex numbers and plotting of complex numbers with rectangular coordinates can be reviewed briefly. Addition, substraction, and scalar multiplication can be done graphically by means of geometric vectors and algebraically by means of abstract vectors and thE' results compared. See if the students can build a number system with vectors and study its properties. A reference that can be used here is NCTM, Insight~ into Modern Mathematics. Chapter VI.

C. Multiplication and division cf complex numbers in polar form, De Moivre's Theorem

Introduce the use. of polar coordinates for representing complex numbers, and allow students to discover for themselves methods of conversion from
the form a + bi to P (cos 6 + i sin 6) and vice verse. Do multiplication and
division of complex numbers in latter form. Show that De Moivre's Theorem follows directly from the theorem for multiplication of complex numbers.

D. Powers and roots of complex numbers

Use De Moivre's Theorem for raising complex numbers to powers and for taking roots of complex numbers.
92

xv. Properties Of A Group

A. Roots of unity B. Geometric interpretation

After the students have had sufficient practice in taking roots of complex
numbers, have them find the six sixth roots of 1 + Oi. Have the roots checked by actual multiplication of roots in standard form, a + bi. Using these six
complex numbers as a set with two operations, multiplication and addition, test for the closure property for each operation. When it is determined that the set is closed under multiplication, find the identity and inverse for each element. Determine if the commutative and associative properties hold. List the properties and then identify this system as a group. Plot the roots on the complex plane and show that they are the vertices of a regular hexagon inscribed in a unit circle. Do further exercises of a similar nature.

93

VOCABULARY

The vocabulary for this course shall include the vocabularies of first and second year algebra and also

of geometry with these additional words:

absolute value .of complex number amplitude of periodic function amplitude of complex number angular velocity antilogarithm approximate number approxim.ltion of numbers argument of complex number asymptotes
characteristic of logarithm circular function cofunction components of vectors conjugate conjunction continuous function conversion cosecant cosine cotangent coterminal angles cycle

discontinuous functions disjunction
exponential function
function
greatest integer function
initial side of an angle interpolation inverse function logarithmic base logarithmic function
modulus multiplicity of roots
negative angle
period periodic function polar coordinates .polynomial function precision pythagorean identities

quadrantal angle quotient identities
radian radius vector reciprocal functions reciprocal. identities reference triangle remainder theorem resolution of vectors restricted domain resultant of vectors
scalar scientific notation significant figure sine function standard position of angle
tangent function terminal side of angle trigonometric equations trigonometric identities
unique factorization
vector vertex of parabola vertical component

94

MATHEMATICS, DEPARTMENT EQUIPMENT

Laboratory space ~ a special room or space in each classroom with laboratory sink and tables.

Calculating machine

Storage facilities

Construction. materials acoustical tiles balsa wood binders - plastic and metal cellophane tape - colored and clear chalk -:- colored elastic cord - colored graph paper hamker hand coping saw hand drill paper cutter paper staples peg board plastic sheets (1/16" thick) plastic sheets for overlays pliers plywood scissors screw driver wooden dowels

Typewriter with mathematical symbols'
Bulletin board
Chalkboard drawing instruments compasses - wooden or metal geometry stencil set gravity protractor parallel ruler protractor straightedge suction cups T-square triangle
Chalkboard graphs stencils or panels polar radian scale rectangular

Duplicating machine stencils paper
Projection equipment opaque projector overhead projector 16mm projector slide prujector projection tables screens

Display space for books, pamphlets, and models Map rail for hanging charts and models Pegboard for demo}1stration and displays Slated globe Workbench

INSTRUMENTS AND MODELS

adjustable triangle

geometrical solids - plastic

(with movable side points)

dodecahedron

binomial cube

hexahedron

Burns Boards

icosahedron

octahedron

carpenter's square

tetrahedron

Cavalieri's Theorem model

compasses (for student)

parallel lines device

cone (with circle, ellipse, hyperbola,

Pythagorean Theorem model

and parabola)

cone, sphere, and cylinder volume ratio set radian and circle demonstrator

eube and pyramid volume ratio set

rulers - student

cubic decimeter

English scale

metric scale

Daudelin's cone

unmarked

dissectible pyramid

leometrical regular plane figures

segmented sphere slide rules (for student) space geometry kit space rings kit space spider
tapes - English and metric scales transit triangles trigonometry unit
circle device transectible prism
vernier caliper
yardsticks

95

SUGGESTED REFERENCES FOR BOOKS, MATERIALS, AND AIDS Every secondary mathematics teacher should have available for guidance in selection of materials, instruments,
and aids the following suggested references. The double asterisks e") have been placed in front of the refer-
ences that are considered basic. "':'Berger, Emil J. and Johnson, Donovan A., A Guide to the Use and Procurement of Teaching Aids for Math
ematics. National Council of Teachers of Mathematics, 1201 Sixteenth Street, N. W., Washington, D. C. Rosenbaum Robert A. and Rosenbaum, Louise J. Bibliography of Mathematics for Secondary School Librari,s.
Wesleyan University, Middletown, Connecticut, 1957. ""Schaaf, William, L. The High School Mathematics Library. National Council of Teachers of Mathematics
Standards. for Materials and Equipment for the Improvement of Instruction in Science, Mathematics, and Modern Foreign Languages. Council of Chief of State School Officers, 1201 Sixteenth Street, N. W., Washington 6, D. C.
"Title III, National Defel\Se Education Act in Georgia. State Department of Education, Division of Administration and Finance and Division of Instruction, Atlanta 3, Georgia. (July 1, 1960).
""Urbancek, Joseph J. (ed.), Chicago Schools Journal Supplement Mathematical Teaching Aids, 1954 Chicago Teachers College, Chicago, Illinois.
Books for High School Asimov, Isaac. Realm of Numbers. Houghton-Mifflin. 1959. Bakst, Aaron Mathematics: Its Magic and Mastery. Second Edition. D. Van Nostrand, 1952. Bakst, AarGn. Mathematics: Puzzles and Pastime. D. Van Nostrand, 1959. Bell, E. T. Men of Mathematics. Simon and Schuster, 1937. Brandes, L. G. Mathematics Can Be Fun. J. W. Welch, 1956. Box 1075, Portland, Maine. DeGrazia, J!>seph. Math Is Fun. Emerson Books. 1954. Eves & Newsom. Introduction to the FC)undations and Fundamental Concepts of, Mathematics. Rinehart and Co., Eves, H. An Introduction to the History of Mathematics. Rinehart and Company. 1957. Gillii.ow & Stern, Puzzles in Mathematics. Viking Press. 1958 Gamow, George, One Two Three . . . Infinity. Viking Press, 1957. Hobson, Squaring the CirCle. Chelsea Publishing Co. H153. James &, James. Mathematics Dictionary. Revised Edition. 1958. 'Kline, Mathematics in Western Culture. Oxford. 1953. University Press. Newman, J. R. The World of Mathematics four volumes. Simon & Schuster, Inc., 1956. Ore, Oystein, Number Theory and Its History. McGraw-Hill. Ransom, ~illiam. One Hundred Mathematical Curiosities. 1954. J. Weston Walch. Ravielli, Anthony. An Adventure in Geometry. Viking Press, New York, 1957. Reid, Constance. From Zero to Infinity. Crowell Co. 1955. Sanford, Vera. A Short History of Mathematics. Houghton-Miflin & Company. 1940. Turnbull, H. W. The Great Mathematicians. Methven.
96

PAMPHLETS AND PAPERBOUND BOOKS Adler, I. The New Mathematics. John Day & Company, 1959. Adler, I. The New World of Mathematics. Deal Press, 1958. Aiken and Beseman. Modern Mathematics Topics and Problems. McGraw-Hill Company, 1959 Bakst Aaron. Mathematics Puzzles and Pastimes. D. Van Nostrand, 1959 Bonola, R. et a1. Non-Euclidean Geometry. Dover Publications. Christian R. R. Introduction to Logic and Sets. Gamow, George. One, Two, Three .. Infinity. Mentor Jones, S. 1., Mathematics Clubs and Recreations.
Mathematical Nuts. Mathematical Wrinkles. Distributed by the author at 1122 Bellvidere Drive, Nashville, Tennessee. Moritz, Robert E. On Mathematics and Mathematicians. Dover Publications, Inc. 1959. Mott-Smith, Geofrey. Mathematical Puzzles. Dover Publications, Inc. 1954. Polya. How To Solve It. Doubleday Anchor Books, Doubleday and Company, 1956. Sawyer, W. W. Mathematicians Delight. Penguin Books, Inc., 1956 Sawyer, W. W. Prelude to Mathematics. Penguin Books, Inc., 1956 Moroney, M. J. Facts from Figures. Pelican Book Company.
97

PROBABILITY
AND
STATISTICS

UNDERSTANDINGS TO BE DEVELOPED IN PROBABILITY AND STATISTICS
An introduction to probability and statistics is presented as a suggestion for a one semester study in a fifth year of high school mathematics. It is intended to teach basic statistics, not a particular kind used in specialized field. It is designed, however, to provide students with a general knowledge of probability and statistics and their relation to each other as they are used in many fields. The Commission on Mathematics of the College Entrance Examination Board has prepared a book, Introductory Probability and Statistical Inference, which is suitable for a textbook.
101

CONTENT

TEACHING SUGGESTIONS

I. Introduction

Point out the relation of statistics and probability, the meaning of each term, and the history of each. Emphasize the importance of modern statistical methods, based on probability theory, in the sciences, psychology, government, industry, economics, sociology, and agriculture.

II. Statistics
A. Organizing and presenting data 1. Frequency tables 2. Graphs
B. Analyzing the data 1. Measure of central tendency
2. Measures of scatter

Two visual forms of presenting data are the tabular and the graphic. Stress the making of frequency tables from grouped and ungrouped data. Include the use of class boundaries, mid-point of the class, tallying, frequency, cumulative frequency, relative frequency. Introduce the different kinds of graphs-the dot diagram, the cumulative graph, the histogram, the cumulative polygon.
Define the measures which describe the important features of a distribution, namely the measures of central tendency and those of scatter. Teach how these measures are obtained and compare the relative merits of mean, median, and mode, and also those of range, variance and standard deviation. It will be necessary to review the summation symbol and subscript numbers before teaching some of these concepts.

III. Probability And Its Relation To Statistics

A. Introducing probability 1. Experiments

It is recommended that an intuitive approach to probability be used at first, associating it with the outcomes of experiments. Practical tests may be made using coins, dice, cards, or thumbtacks.

2. Formal treatment 3. Review

After moving toward the theory of probability by experimenting, one may understake a more formal treatment of it. For this, it is necessary to review set language, operations with sets, and permutations and combinations, or selections.

4. Language of probability

Among the' important concepts of probality theory to be taught are: sample space, events, compound events, mutually exclusive events, conditional probabilities, independent events, mathematical expectation, and randomness.

B. The binomial distribution 1. The normal curve

Review the binomial theorem. Introduce criteria for a binomial experiment, extension to n trials, and use of binomial probability tables. Show that in a very large number of items the cumulative relative polygon approaches a smooth curve known as the normal curve.

2. Uses of the normal curve
C. Applications of the binomial distribution

Study uses of the normal curve that are independent of the binomial distribution.
Acceptance sampling, a branch of statistical qaulity control methods, and hypothesis testing are two widely used applications of probability theory or its subdivision, the binomial distribution. Attention should be given to constructing sample plans and tests of significance.

D. Uses of samples
1. Basic ideas about sampling

Samples are used to estimate a population characteristic. Clarify fundamental ideas about sampling, such as sampling with and without replacement, distribution of sample means, distribution of sample standard deviation, and standard error.

2. Drawing samples

The importance of the method of drawing samples, particularly the basic random sampling method, should be emphasized.

3. Chebyshev's theorem

Chebyshev's teorem, which gives a mathematical foundation for the use of samples, should be studied.

102

acceptance number acceptance sampling alternative hypothesis arithemetic mean average
bibnomial coefficient binomial distribution binomial experiment binomial probability
Central limit theorem Chebyshev's theorem combination commutative law of sets complementary events conditional probality critical set of values cumulative freququency cumulative graph cumulative polygon
de Morgan's law descriptive statistics disjoint sets distributive law of sets dot diagram

VOCABULARY
element empty set event exhuastive events
finite set frequency distribution frequency table
historgram independent events infinite sets intersection sets
law of large numbers level of significance lower quartile
m-subset mathematical expectation mathematical induction mean median Monte Carlo methods mutually exclusive events
normal distribution null hypothesis

partition of sample space Pascal's rule Pascal's triangle percentile population Permutation probability function proper set
random drawings random number random sample range
sample sample points sample space set set function standard deviation statistical hypothesis statistical quality control subset
union of sets universal set upper quartile
variance Venn diagram

103

ANALYTIC GEOMETRY

J

UNDERSTANDINGS TO BE DEVELOPED IN ANALYTIC GEOMETRY
In analytic geometry the student should gain an understanding of the correspondence which makes the subject possible, should learn the properties of lines and conic sections traditionally taught in this course, should see the power and the advantage of the analytic method, and should get a foundation which will allow him to begin a course in calculus upon entering college. The student in this course should have an opportunity to discover relationships, to solve a variety of numerical problems, and to prove theorems. The course in analytic geometry will be one of the last high school courses for the student. It should be a rigorous course taught at a mature level.
107

CONTENT

TEACHING SUGGESTIONS

I. Introduction

A. History

Before the 17th century, algebra was thought of as dealing with numbers, and geometry as dealing with sets of points in space. Early in the 17th century several people saw that it was possible to interpret geometric theorems algebraically and algebraic theorems geometrically. Notable among these people were Pierre de Fermat and Rene Descartes. In 1637 Descrates published a famous work on geometry in which he applied with great ease algebraic method to geometry problems which were very difficult using the purely synthetic methods of the ancient Greeks. Descartes' contribution was important not only for the study of geometry but also as a foundation for the calculus developed in the latter part of the century by Newton and Leibniz. Its main importance is that it unifies algebra and geometry.

B. Coordinates

Point out to students that a unifying of algebra and geometry necessarily involves a correspondence of numbers and points, the basic elements of algebra and geometry. Once such a correspondence is assumed, the axioms and theorems of the real numbers and of geometry may be used in proving theirems and in solving problems.

1. On a line
II

From work in eariler courses, students may be familiar with establishing a coordinate system on a line; however, if they are not, this idea should be developed very carefully. It is one of the basic concepts of analytic geometry. Given a line we assume that it is possible to select two points and a positive direction on the line and to assign to the two points the coordinates 0 and 1, the point whose coordinate is one lying in the positive direction from zero.

A

B

o

1

Using the length of AB as a unit of measure we can now assign coordinates to other points of the line. We assume then a one-to-one roorespondence between the points of the line and the set of real numbers such that if X and Yare two
points whose coordinates are x and y and if 0 < x < y, then X is between A
and Y. Students should be led to realize that the choosing of a positive direction on a line is a matter of convention.

2. In the plane a. Rectangular

We may introduce coordinates into the plane in the following way. Choose two perpendicular lines and assume like coordinate systems on both lines. Let their point of intersection have coordinate O. Call the point of intersection the origin and the two lines the x-axis and the y-axis. The projections from a point in the plane to the two axes give the coordinates of the' point, and in this case there is a correspondence betwen an ordered pair of real numbers and each point of the plane.

It is customary to consider the x-axis horizontal and the y-axis vertical, and positive directioas as being to the right and up respectively; however, again this is convention and there is nothing sacred about it. In fact, there are coordinate systemsi nw hich the axes are not perpendicular and others in which systems of curves are used instead of straight lines. Rectangular coordinates are chosen for plane analytic geometry because they are easiest to work with.

b. Polar

Points in the plane may also be specified another way. If a starting point and
a fixed direction from that starting point are chosen( called respectively the pole and the polar axis), then a point may be specified by giving its angl~ from the fixed direction and the distance from the starting point in the dI-
rection given by the angle.

108

o

p

c. Relation between rectangular and polar.

Suppose 0 is the pole and 6P the polar axis. Then the point Q may be specified
by a number pair (A, r) representing the angle A and the distance r from the pole. The correspondence between Q and number pairs (A, r) is called a polar coordinate system. Students should note that this is not a one-to-one correspondence. For each point Q there are many angles A which correspond to
Q. If the restriction is made that 0 ~ A < 2... then there is a one-to-one cor
respondence between points and number pairs.
If rectangular and polar coordinates are selected so that the origin and pole are represented by the same point and the positive x-axis and polar axis are coincident, it is easy to establish the relationship between them.

y

p
The point Q
has rectangular coordinates (x, y). Using trigonometric ratios sin A = /'
= = = and cos A x/" we can obtain the equations y r sin A and x r cos A.
These relationships allow us to change from one coordinate system to the other.

II. The Straight Line

A. Directed line segments, midpoint of line segments, length of line segments

1. Directed line segments

If direction on a line or line segment is important it can be indicated in several ways. We are already familiar with positive and negative directions on a line. An agreement can be made that direction will be indicated by the order in which end points are written.

A

B

2. Lengths of line segments

The segment AB can be considered to be in one direction and BA in the opposite direction. Different textbooks use different notations for this idea. It is important to choose one notation and then be consistent.
The best approach to this idea is probably to develop the distance between two points on the coordinate line through use of examples, followed by a formal definition, and then extend the idea to include lines in the plane.

109

3. Midpoint of a line segment
B. Distance formula
C. Slope of a line 1. Horizontal and vertical 2. Parallel and perpendicular
D. Equations of a line 1. General form 2. Point-slope form 3. Slope-intercept form 4. Two-point form 5. Intercept form 6. Polar form
E. Theorems from plane geometry to be proved

Definition. If A and B are points on a coordinate line, with coordinates a and
b respectively, the length of the segment AB, designated by AB is defined to
be AB = Ib - al. This number is also called the distance between A and B.

The equations for the midpoint of a line segment should be developed first on a coordinate line, then extended to any line segment in the plane.

x

M

y

x

m

y

Suppose XY is a segment on a coordinate line By definition its length is

Iy - xl . We are looking for a point M such that XM = MY It follows from

this that 1m - xl = Iy - ml and

m- x=y- m

2m=x+y

x+y m= -2.-

The coordinate of the midpoint of a coordinalaJ line segment is therefore the arithmetic mean of the coordinates of the endpoints.

[n developing the formulas for the coordinates of the mid-point of any line segment in the plane be sure that students are aware of the theorem from Euclidean geometry which makes it possible.

The distance formula for oblique line segments follows directly from the definition of length of a coordinate line segment and application of the pythaforean theorem. The distance formula for oblique line segments is also applicable to the cases of vertical and horizontal line segments. Drawings to illustrate the various cases are important in development of formulas.
Approach this intuitively through use of "rise" and "run," before giveing definition of inclination and slope. Be sure to treat horizontal and vertical lines and lines with negative slopes. Let students try to discover the relationship between coefficients in the equation of a line and the slope of the line.
Give students equations of parallel and perpendicular lines to find slopes of. Let them discover the relationship of the slopes in each case. This exercise should precede proofs of the theorems of slopes of parallel and perpendicular lines. Include in the exercise lines which are parallel to the coordinate axes.
In developing the equations for a line it is probably advisable to state the general form of a line and then show how the various equations can be put into general form. Students will need practice in finding equations of lines given various information, but they also need to understand the equations and how they are developed. Horizontal and vertical lines as well as oblique lines should be given.

There are many theorems about lines which can be proved at this stage. The following is a list of some of them:
1. Two lines perpendicular to the same line are parallel.
2. If a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
3. The line segment joining the midpoint of two sides of a triangle is parallel to the third side and equal to one-half of it.
4. If a line bisects one side of a triangle and is parallel to a second side, then it bisects the third side.
5. The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
6. Opposite sides of a. parallelogram are equal in length.

110

7. The diagonals of a rectangle have equal lengths.
8. The diagonals of a square are perpendicular.
9. Diagonals of a parallelogram bisect each other.
10. The lines joining the midpoints of successive sides of a quadrilateral form a parallelogram.
It would be well to compare synthetic and analytic proofs. Students should observe that some proofs are simpler using the analytic method but that others lend themselves to the synthetic method. Students should also observe that choice of axes is important in analytic proof.

III. Con ic Sections
A. General equation
B. Circle 1. Standard form of equation of a circle
2. Circle with center at the origin
3. Finding equation of a circle which contains three given points
4. Test for a circle
IS. Length of a tangent to a circle from an exterior point
6. Polar equation of a circle C. Parabola ellipse, hyperbola
1. Definitions

After a study of the various conic sections it might be well for students to try to determine the relationship between the values of the coefficients in the general equation and a particular conic section.

Since the circle is a major topic in Euclidean geometry, it is treated separately. It should be related to the ellipse later.

The standard form of the equation of the circle can be simply derived from the definition of a circle and the dis.tance formula.

Definition. A circle is the set of all points in the plane which are equidistant from a given point.

If (h,k) is the given point and r the distance, we seek an equation for all points (x,y) which are distance of r from (h, k). By the distance formula,

v(x - h)2 + (y - k)2 = r

or

(x - h)2 + (y - k)2 = r2

which is called the standard form of the equation of the circle.

Taking the standard form of the equation of a circle with center at (h , k)
(x - h)2 + (y - k)2 = r2,

we can easily determine that if the center is (0, 0) the equation becomes x2 + y2 = r2.

Students should know construction of this from Euclidean geometry. Giving this problem as an individual exercise will test their knowledge of many ideas previously presented- midpoint of a line segment, distance formula, slope of perpendicular lines, etc.

By changing it to standard form, a quadratic equation may be tested to see if it is the equation of a circle.

Example:

x2 + y2 + 8x + 6y = 11 = 0 (x2 - 8x) + (y2 +6y) = 11 (x2 - 8x + 16) + (y2 + 6y + 9) -
+ + (x - 4)2 (y 3)2 = 36

11 + 16 +9

Therefore this is the equation of a circle with center (4, -3) and radius 6.

If we define the tangent line to a circle at a point P on the circle, as the line containing P which is perpendicular to the radius at P, students should be able to solve this problem without help by using the distance formula and the Pythagorean theorem.

By using the relations between rectangular and polar coordinates, one can find the polar equation of a circle, given the cartesian equation.

These should be defined both as intersections of plane and cone and in terms of foci and directrices.

111

2. Derivations of general equations

With a proper choice of coordinate axes, the student should be able to derive the formulas from the definitions.
Example: Derive the formula for the parabola.
Definition: A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). Let F (p, 0) be the focus and x = - p the directrix of a given parabola.

-p I I I
I I
I
I
I

3. Plotting 4. Translation of axes 5. Polar equations 6. Applications

By definition a point P(x,y) belongs to the graph of the parabola if and only if
PF = PQ y,.,.(x-_----,p)'""2-+-y-"..2 = y"..(x-+-p-,)'"'2-+,.-----;(y-_-y'"")2-
(x _ p)2 + y2 = (x + p)2
x2 _ 2px + p2 + y2 = x2 + 2px + p2
y2 = 4px
Therefore y2 = 4 px is the equation of a parabola.
In plotting graphs of the conic sections ideas of symmetry, of boundedness of an ellipse, of asymptotes of a hyperbola should be emphasized.
Translation of axes should be introduced with the idea that it simplifies the equation and allows one to tell more about the graph from examination of the simplified equation.
Polar equations of the conic sections may be obtained by substitution of the
equivalences x = r cos e and y = r sin e in the cartesian equations.
Students should have opportunities to sketch graphs, to find equations given certain conditions, to determine kind of conic section by examination of equation, to simplify equations by translation of axes, to prove theorems. and to derive equations.

IV. Parametric Equations And Loci

A. Various parametric equations Sometimes it is desirable to express each element of an ordered pair such as

for the same curve

(x,y) in terms of a third variable k. When we do this we find that it is neces-

sary to have ~ pair of equations x= f (k) and y = g (k) to represent a

curve algebraically. We call these parametric equations. If we eliminate the

iI

parameter k ~e obtain the cartesian equation for the curve. Many loci problems are more eaSily treated by use of parametric equations.

Since. the parameter can be chosen in different ways, a variety of parametric equatIOns may represent a single locus. Consider the line which contains the two point~ (1,4) and 3.0). x = 1 + 2 k and y = 4 - 4 k (k a real number) are para~etnc equations of this line; however, x = 3 - 2k and y = 4 k are also equations for the same line.

112

B. Locus problems

Plotting the graph of parametric equations is quite simple. Each value of k

used in the two equations gives the coordinates of a point of the curve.

= = Given the parametric equations x a cos k and y b sin k, what is the
graph? In this case if the equation were in cartesian form it might be easier

to recognize. To eliminate k:

=: cos k

and sin k = ~

x2

y2

i2 and sin2 k = b2

8x22

+

y2 1)2=

I

and

we

recognize

this

as

the

equation

of

an

ellipse.

In some cases one form of an equation is superior to the other.

113

-------

VOCABULARY

The vocabulary for this course shall include vocabularies from the previous high school courses together with these additional words:

asymptotes
boundedness
conic section coordinate axes coordinate line
Descartes directrix
ellipse
focus

hyperbola
Inclination latus rectum locus
major axis Minor axis
parabola parameter

parametric equations polar axis polar coordinates pole
rectangular coordinates
segment slope symmetry
translation

Books for Advanced High School Students
Allendeorfer, C. B. and Oakley, G. O. Fundamentals of Freshman Mathematics. McGraw-Hill Book Company, Inc. Courant & H. Robbins. What is Mathematics? Oxford University Press. Exner & Rosskopf. Logic in Elementary Mathematics. McGraw-Hill Book Company. Fisher, R. C. and Ziebur, A. D. Integrated Algebra and Trigonometry, Prentice Hall, Inc., 1959. Hogben, Lancelot. Mathematics for the Million. W. W. Norton & Company, 1951. Kline. Mathematics and Physical World. Thomas Y. Crowell Company, 1959. Meyer, Jerome S. Fun With Mathematics. The World Publishing Company, 1952. Pedoe, D. The Gentle Art of Mathematics. MacMillan Company. Richardson. Fundamentals of Mathematics. MacMillan Company, revised 1958. Ruchlis & Engelhardt. The Story of Mathematics. Harvey House, 1958.

114

. ' - .,V

,

,.

_.

.'



_

,

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 a 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.
117

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) + (1 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:

Ul

Ul
~ <
Ul ~
0
::I:l
Eo<

Ul
A ril P:: A
Z
$

Ul
Z
ril Eo<

Ul
ril
Z
0

Ul
::I:l
Eo<
Z
ril Eo<

::I:l
Eo< A ril P:: A
Z
~
::I:l

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 decimal fraction is named 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, etc. Make a list of numbers heard in conversation, other classes, TV, etc., during a given 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.

118

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
D D D Cardinal

"

""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
Ixxxxxxxx I xxxx means one eight. four units, and
Ixxxxxxxxxx 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.

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.

119

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.....

0123

"-
0 0..... 1 2 3

1 1 ..... 2 , 3 4

2 2 3 4, 5

33456

~

~

~

~

5 5 6 7 10

- "

~

~

7 7 10 11 12

45~ 7 I'l 5 ~ 7

5 6 7 10

6 7 0 11

7 10 1 12

12 13 .~

~~

1 \2 13 14

'1.2 13 14, 15
"-
13 14 15 16,

Multiplication Table

..... X

0

1

2

~

4

5

~

7

0 i"0 0 0 p 0 0 ) 0

I'
1 0 1, 2 3 4 5 ~ 7

.. ." 2 0 2 4 6 10 12 4 16

u

~

'u"

"~

In

22 25

4 0 4 10 4 1'20 24 30 34

5 0 5 12 7 24 31 36 43

. u

u

u

.,. 2 30 36 44" 52

7 0 7 16 25 34 43 52 61.....

c. Converting from base ten to base eight.

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 table!' 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.,gbl 26.'Oh' 52e I fil:h I

Look in addition table for 6 + 4. It is 12.
Write 2 in its proper place.
2 + 2 + 1 (group regrouped) gives 5.

Change from base ten to base eight

3141

8 / 314
8/39 8/4
o

remainder 2 7 4

120

d. Converting from base eight to base ten.
3. Numeration and operation! in other bases.
F. Natural numbers
G. Whole numbers

Now read up the column of remainders, and your number is 472",h'.
Change 3125"", to base ten.
+ + + 3125" oh' = (3 X 83) (l X 82) (2 X 8) 5(1)
= (3 X 512) + (l X 64) + (2 + 8) + 5
= 1536 + 64 + 16 + 5
= 1621.,,,
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 123 4 5 6 7 8

5 ~nits

~
3 units

The sum is 8.

121

B. Using the number line to show properties under fundamental operations 1. Commutative property of addition and multiplication
2. Associative property.

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 >

'-.r--' '-.r--''-.r--'
222

Example: Two threes equal three twos or 2 threes equal 3 twos.

This process can be "undone" by division (the inverse of multiplication).

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

~

<o

123

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 num ber line that 8 - 3 "'" 3 - 8.

Multiplication may be taught using the number line.

3

3

~

o 1 2 3 4 5 678

~
222

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 -7- 3 oF 3 -7- 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

Show on number lines:

5
123
,-A-..~~
o123 4 5 67

122

'----y--J'----y'----y-----'

12

3

'--y----'
3

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
,------'------.,,

3

3

~

o1 2 3 4 5 6 7 8

'--y-J '--y-J '--y-J '--y-J

2

2

22

6

6

6

- - - , , . - -_ _--A.-_ _~\ .~--~'-----,

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

3. Distributive property with respect to multiplication over addition
4. Closure

Multiplication is associative. By use of examples lead students to discover that division is not associative. Does (8 -;-. 4) -;-. 2 = 8 -;-. (4 -;-. 2) ? Show 3 7 and 3 5 on the number line. This may be shown first on the number line:
3X7

o7

14

21

'--v----' '--v----' '--v----'
7+ 7+721

3 X5

o 5 10 15

'-y--' '-y--' '-y--'
5 + 5 + 5 = 15

21 + 15 = 36

o7

14

21

36

Now have pupils discover that 3 sevens and 3 fives can be grouped and made
+ into 3 X (7 5) or 3 twelves.

1.......... 1 .......... 1 ........ .. 1

--0-----

12

24

36

On the number line 3 twelves is 36.

This is the distributive property with respect to multiplication over addition.

Practice: 5248 means 5200 + 540 + 5.8

Lead pupils to see that 374 X 132 uses the distributive property repeatedly

(374. 100) + (374 30) + (374 2) or

(300 100) + (70 100) + (4 100) + (300 30) + (70 30) +

(4 30) + (3002) + (702) + (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.

123

C. Classifying whole numbers. 1. Number names 2. Even and odd numbers
3. One, the identity element for multiplication
4. Zero

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 not for division.
Any num~r 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. lor 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 an 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 multiplication 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.

a. Identity element for addition
b. Division by zero is impossible
5. Factors and primes

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 it
repeated addition the sum of three zeros is 0 + 0 + 0 and this is zero.
Division is the inverse of multiplication.
0/Il = 0 means 0 X 5 = 0
Bring out that 5/ 0 = 0 is impossible, as this would mean 0 X 0 = 5. Help p~pil
realize that division by zero is meaningless and impossible. 0/0 has no meamng 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.

124

..U

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 have 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

12.5

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, 35, etc., are names 01' numerals for the same number. Other names may be 30/2, 45/ 3, etc. Another numbeI may be named as 4/ 3, 8/6, 12/9, etc. 30/2, 45/3, 4/ 3, 8/6 and 12/9 are rational numbers. Explain that symbols for natural numbers wer.e 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 = a /
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 = a/., b "'" 0, because b x = a
If a and b are numbers, b "'" 0, the symbol a /. is called a fraction.
Remember that sometimes a/. (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, 1a=a1=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/. 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/. = ke/ kd == e/.

126

_I

Reciprocals
5. Equivalent fractions
6. Operations with rational numbers a. Addition
b. Subtraction

Recall definition of a rational number.
Now consider bx = 1
x = 1/, and
b rib) = 1
If the product of two numbers is 1, the numbers are called reciprocals or mul. tiplicative inverses of each other .
Example: a is called the reciprocal of '/., and 1/. is called the reciprocal of a.
Lead pupils to discover that if /, '/. = 1, neither a nor b is zero.
Definition: The reciprocal of the rational number' /, is the rational number'/.
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, and give full proof. + + = 1 1/ 3 1 1/3 l ' 1/ 3 + + (l 1 1) 1/3 = 3. 1/ 3 = 1
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 denomi nator (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 lh and llh.

Pupil could also locate the points that divide the unit segment into fourths or thirds.

By use of the number line show lh - 4.

12. 7

c. Multiplication

Locate lh. Count from lh to the left lf4. The point located is lf4. lh-lf4=lf4
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

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 11 2 of 3 or 11 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

2Xl

2.1

1

2-- X ""5 = 2X5 =""""2.5 = 5

d. DivisIon of rational num bers

Example:

7

5

15 X 7"

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
75 _ 1
3~-3

This may be done mentally, or by showing the division of 5 and 7 by using a slanting line, as:
r'/)
3'/)'
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 An 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 lh's in 1, and generalize that 3 + lh is 6.

12.8

C. Decimal fractions as an extension of rational numbers 1. Decimal notation
2. Operations with decimals a. Addition
b. Subtraction

Through enough applications pupils generalize that division by '/n is equivalent to multiplication by n.

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 Ih, we havelh = c.

We look for c such that when multiplied by lh gives 1 (lur a). The number multiplied by lh 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(10) + 2(1/101) + 3(1/ 10,) + 4(1/ H") + 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 problem 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 develop speed and accuracy. Have pupils show the details as above, on several problems, and then allow the regrouping to be done mentally.

1209

c. Multiplication d. Division

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 multipli cation of this type.

In work in multiplication students multiplied as this example indicates:

1

1

1

100 X-WOO = 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 one decimal place

Call attention to the fraction 1/100' Show that it means 1/10' or .01.
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 (1/1~') Your answer or solution set is the product of 37 X 26 with five decimM places pointed off.

750,000 ..;- 2500 means
(75 X 104) ..;- (25 X 102) = 3 X 102 = 300

This application is made with decimal fractions also.

Examp.le:

.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 1/10 = 2

130

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 -;- .oo2

(8 X 1/10') -;- (2 X 1/10") =

4 X 1/10""1 =

4 X 1/10.....
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

80 . 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

'Ihese 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 di vided 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:

True

False

3+4=7

3+4=6

3+4 oF 6

3+4 oF 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-
places the variable in the equation. In 3x = 9, 3 is a root.

Approach inequality with a reminder that an inequality expresses the fact that one number is either greater than or 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

131

C. Order of decimals
D. Ratio 1. Meaning 2. Other meanings
E. Proportion 1. Property I.
2. Property IT 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.

o 1 2 3 4 5 6 7 8 9 10
10 10 10 10 10 10 10 10 10 10 10

6/
6/

10 10

occurs
> 5/10

to or

the 5/ 10

right of
< 6/10

5/ 10

so

If fractions are not decimal fractions but common fractions, convert into decimal form and compare.

Example:

2/ 5 and 4/ 11
2th/ 5us=0.400.4>0 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

20/ 55 "", .36

> .40 .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 oF O.

3/5 , 6/ 10, 9/ 15 ....
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 IT for proportion is suggested here.

Property I. if a/b = c/ d b oF 0 and d oF 0, prove ad = bc

If alb = c/ d, then the property of 1, a;. X did = C/d X bib or,

= ad/ bd

Cb/ 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 = be.

Develop Property IT the same way. If ad = bc then alb = c;"

If ad = be

= ad/ d

bc / d

Dividing both members by d

a = b X C/.

Property of 1

Dividing both members by b

= a/ b

C/ 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.

132

v. Measurement
A. Measuring things that are continous
B. Subdivision and measurement
C. Metric system

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 is not 20%. While 5 is 20% of 25, it takes the ratio N/ 1 00 to give the per cent.
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).

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.

133

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 matte graphic presentation of frequency distribution.

No. Yrs. car used
5 4 3 2 1

Frequency Distribution Tally
I
II +++t-
III I1II

Frequency
1 2 5 3 4

134

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

/\

4

a

3

"""W

\
\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, ob tuse).
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 them selves the relation of circumference to diameter.
135

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,
a-b

Div. +, Alb, 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

136

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
zero

137

I

CONSUMER

:

MATHEMATICS

UNDERSTANDINGS TO BE DEVELOPED IN CONSUMER MATHEMATICS
This course is designed for the junior or senior who has had General Mathematics, and is not pursuing the college preparatory program. This presupposes that an understanding of the fundamental processes of basic mathematics have been developed. Stress should be placed on the fundamental laws of arithmetic to stimulate more repid and accurate calculations. The student should come to realize and understand the need for and the techniques of budgeting and awareness and mathematical understanding of the many forms of consumer credit, with appropriate guides of desirable and undesirable use of credit, are of major importance in our society of mass advertisement. Any member of society should be informed about his taxation system. He should understand the various types of revenue and should develop skills for computing taxes. As the sciences advance, the increased use of statistical methods demands of the average reader a basic understanding of the techniques and terms of statistical methods. The increasing number of types of insurance has placed a new importance on man's ability to select the appropriate policy to fit his needs. The trend for the average consumer to invest in stocks and bonds makes it desirable that he read and interpret intelligently current data and statistics.
141

CONTENT
I. Statistics
A. Selection of data
1. Size of sample 2. Representative sample
3. Comparable data 4. Relevant data B. Presentation of data 1. Data table
2. Frequency table
3. Histogram

TEACHING SUGGESTIONS
This unit may be introduced by having students bring to class clippings from newspapers and magazines of various uses of data or averages. Selection of certain lines from commercials and advertisements will provide an excellent opening for discussions concerning selection and presentation of data. For example: "two out of three ..." raises the questions of (1) how many were questioned, and (2) from what areas of the country? "Selling for as low as ... " raises the questions of (1) as high as what, (2) any hidden price, (3) includes what at this price and (4) what is the most common or popular selling price?
The effect of the size of a sample for statistical purposes may be illustrated with discrete data by having the student take three coins and toss them together at the same time recording the results each time. Would the per centages based on eight or sixtyfour tries give a better indication of expecta tions? Would a greater number of tries be more reliable?
The need for data to be representative of the population being studied should be stressed. Would the average height of the boys in the class room be repre sentative of the height of all boys in school? Would the like or dislike of a certain type of music in a certain block of town be representative of the musical taste of the comunity?
The technique of simple random sampling and stratified random sampling should be introduced.
It should be stressed that for data to be comparable two things must be true. (l)The data must be obtained under the same conditions or under very similar conditions. A good example of this is the daily point of discussion of how hot or cold it was, all depending on the locations of the thermometers. (2) Data must be given in the same units of measure and with the same degrees of precision.
Would data on the smoking habits of students be relevant to their grades in school? Would data on the number of miles driven in a week by students be relevant to their grades?
Have students bring in data tables from such sources as the World Almanac and the U. S. Census Tables. Select examples of tables presenting a comparison of three or more items such as age, height, and weight and point out the dif ficulty with this arrangement of recognizing any pattern of behavior of the data. It should be made clear that the table is the best means of presenting data in detail but does not allow any averages or patterns in the behavior of the data to be made selfevident.
The constructions of frequency tables for ungroup and grouped data should be presented with ample problem work to allow the student to become fa miliar with their arrangement. (It would be advisable to select three or four sets of data to be developed through sections B and C.) Data of local interest to the student may be used, such as subject grades by numerical or letter value, achievement scores, and statistics on football team members.
Carefull discussion and arrangement of grouped data should include the use of class intervals, class boundaries, tally column, frequency colunm (f), cum ulative frequency column (CF), mid-point column (z), relative frequency column (RF), cumulative relative frequency column (CRF). Data should be limited to small groups to save time in computational work (See The World of Statistics by Johnson and Glenn.)
Neat and precise construction of frequency histograms and cumulative frequency histograms should be stressed. The use of the left vertical scale for be stressed in the frequency histogram. In the cumulative frequency should be stressed in frequency histograms. In the cumulative frequency histogram,
142

4. Frequency polygon and cumulative frequency polygon
C. Statistical measurements 1. Central tendency a. Mean
b. Median
c. Mode
2. Measurement of dispersion (spread) a. Range
b. Deviations from the mean
c. Standard deviation d. Quartiles

the use of the left vertical scale to indicate the cumulative frequency and the right vertical scale to indicate the right vertical scale to indicate the cumulative relative frequency should be stressed. Cumulative graphs for "measurements equal to or less than . . . and "measurements equal to or greater than ...." should be given as problems.
An intuitive approach of the transition from the area of the bars of the histogram to the area under the smooth curve line of a polygon should be made at this time. With proper emphasis on the above section, this should not take too much time. The frequency polygon indicates the midpoints, allows for a simpler graph for indication of averages and dispersion, and gives a graphic approach to the "normal curve."
A concise review of construction of tables, histograms, and polygons should be held at this time. Strong emphasis should be placed on the correct reading of data from completed graphs.
The proper treatment of measurements of central tendency or "average" calls for precise definitions and their careful application.
The computation of the mean for ungrouped and grouped data should be presented. Careful consideration of what this average tells one about the data should be discussed, remembering the later introduction of median and mode. Groups of data where the mean is not typical or is a misleading measurement should be mixed in with other sets of data.
The median as another average should be compared with the mean as a measurement, to give a picture of the total group. With careful selection of a small group of scores beforehand, the distincition and appropriateness of each average is representative of the group may be clearly emphasized. For ungrouped data the median must be located by taking the center score when the scores or items are arranged in order. (Remember this item or score mayor may not actually exist.) The quickest method of locating the median for grouped data is by referring to the cumulative frequency polygon and the coordinater for the .50 on the CRF scale.
The mode as an average (the most frequent or common item) may best be located by reference to the histogram or frequency table.
Selected situations should be given with the mean, median, and mode recorded. Class discussion should be encouraged concerning which average best presents a true picture of the data as a whole. Emphasis should be placed on the correct use of and distinction between the two terms average and normal. Have students locate average presented in publications and see how many times it is clearly stated which average is being given.
Anyone of the three measures of central tendency does not present sufficient information by itself. The various measures of dispersion need to be added.
The simplest measure, which may be used with any of the three statistical averages, is the range. For ungrouped data this would be the difference between the low and high item. For grouped data it would be the difference between the mid-points of the lower and upper class intervals. Spend sufficient time discussing the new picture presented when the three statistical averages are presented with the range.
A small group of data should be used to explore the behavior of the sum of the mean. This would lead to a clearer understanding of the mean.
The introduction of the standard deviation should be optional depending on the ability of the class. A general discussion may be presented with strictly an intuitive aproach. No attempt should be made to compute the standard deviation of a set of data.
The lower (Ql) and upper (Q3) quartile and the median (Q2) measurements should be introduced and related to the median as measurements of dispersion.

143

D. Interpretation of data 1. Interpolation 2. Extrapolation 3. Inference

This concept might best be presented through the use of th cumulative frequency polygon and the CRF scale.
Careful consideration should be given the three sub-headings of this section as their misinterpretation or misuse accounts for the vast majority of false conclusions concerning statistical measurements. This section will serve as an excellent review and summary of the preceding sections. Ample illustrations of (1) reading between the data, (2) reading beyond the data, and (3) inferring causes or drawing conclusions which are not justified by the data being presented, should be used. The understanding of this section should be considered the goal of this unit, not the ability to compute or quote definitions.

II. Budgeting
A. Reasons for using a budget B. Purpose of a budget C. Setting up a budget
D. Basic concepts of a budget
E. Problems related to budgeting F. Sources of material

A statement of planned spending based on expected income is called a budget.
Budgets are helpful in two ways. They tend to curb foolish and unnecessary spending, and they point out the best places to save or a better way of spending.
The main purpose is to control the money earned before it is spent.
Have pupils study prepared budgets for the family of various incomes, for government, for schools, and for churches Require them to keep a personal record of money received and spent over a certain period of time, then to analyze the record to discover a pattern of spending. From this, each pupil may prepare a personal budget. By a similar procedure he may set up a family budget.
No one budget plan will meet the requirements of all families, but basic items common to all families are food, clothing, and housing. Four general headings to be considered in making a budget are necessities, improvement, giving, and saving.
In the budget there should be a list of items for which expenditures are made, the yearly cost of each, and the cost of each expressed as a percentage of the total expenditure. The yearly expense added to the savings should of course total the yearly income, and the percentages should total 100%.
In connection with budgeting, investigate such related ideas as cash records, thrift in buying food and clothing, ownership of homes versus renting.
Newspapers and magazines are good sources of material to be used in the problems.

III. Credit Systems
A. Introduction
B. Installment buying
1. installment plans a. Contract between retailer and customer b. Use of financing companies c. Installment loan

The portion of consumer mathematics which affects probably mote people than any other phase is consumer credit. It is concerned actually with borrowing money although we think of it more as borrowing time to pay.
To introduce :. study of consumer credit, discuss opening charge accounts, kinds of charge accounts, credit cards, credit rating.
Look into the trend of installment buying, cost of paying by installments, dangers and advantages.
Explain that there are mainly three plans for installment buying.
The retailer holds the contract and collects the payments directly from the customer.
The retailer sells the contract to a financing institution to which payments are made.
The company makes an installment loan directly to the customer.

144
I II

2. Financing charges
3. Financing homes and automobiles
C. Paying cash by mean of borrowing money 1. Loan companies 2. Credit unions 3. Banks

Teach pupils how to compute financing charges in terms of a yearly percentage by using a formulas such as
2mI
R = + P (n 1)
Where R is the annual rate, m the number of installments paid in a year, I the difference in the cash price and the installment price, and n the total number of installment payments.
Since almost everybody buys houses and automobiles on an installment plan, a special study of the details concerning those two items is advisable. Information may be obtained from federal saving and loan companies and automobile financing companies.
Contrast installment buying with borrowing money with which to buy. Have pupils secure information concerning loans from loan companies, credit unions, banks. Study collateral, promissory notes, and bank discount.

IV. Taxes
A. Uses and sources of tax money
B. Kinds of tax 1. Income tax: state and federal
2. Property tax a. Types b. Assessed value and rate

We have come to expect local, state, and federal governments to provide us with many comforts and conveniences. Streets and roads; fire and police ptotection; public parks, libraries, and hospitals; mail collection and delivery; inspection of products and business establishments; armed forces for protection against foreign enemies; welfare agencies to care for the unfortunate are all costly and must be financed by taxing the people.
All three forms of government are financed by taxes. Most of the money to support local and state government comes from the tax on personal property and real estate. Many states also have tax and personal income tax. Money to finance the federal government comes from many different sources. There are about 225 different forms of federal tax.
Information needed on state and federal income tax forms is of four types:
1. Income 2. Deductions 3. Exemptions 4. Computation of tax
Witholding forms, tax forms, and directions for filing both state and federal taxes should be studied. Students should be provided forms to fill out and taxes to compute. Hypothetical cases should be given so that students will have to make judgments about the best form to use and the most advantageous way to compute tax.
A property tax is charged on real and personal property. Land or anything attashed to it makes up real property. This includes buldings, mines, oil wells, etc. Personal property includes all property that is easily moved from place to place, such as automobiles, jewelry, furniture, livestock, stocks, bonds etc.
Property taxes are levied in much the same way in all districts. Property is assigned an assessed value usually a percentage of the market value, by an assessor. A tax rate is computed by the local government, and the amount of tax is found by multiplying the assessed valuation by the tax rate. The tax rate is figured by comparing the cost of government with the total assessed value of all property in the tax district. To compare taxes in two different tax districts, one must consider both rate and assessed valuation. A high valuation and a low tax rate may yield the same amount of tax as a low valuation and high rate. This informtion will probably be more meaningful to students if data for use in computation come from the local government.
145

3. Others a. Sales b. Import duties c. Exercise tax d. Licenses e. Inheritance f. Social security g. Tolls on roads and bridges
v. Insurance
A. Property 1. House (against theft, fire, flood and hail) a. Computing premuims
2. Car a. Bodily injury. b. Property damage c. Comprehensive

There are other miscellaneous sources of revenue for the three levels of government. They are more important to the tax payer than to the government since they furnish relatively small amounts of tax money. Students should gain information. about state sales tax, licenses, and tolls on roads or bridges in their vicinity. They should learn about the special tax on alcoholic beverages, cigarettes, and gasoline; the luxury tax, duties on imported goods. They should have information not only about the amount of such tax also about its use when it is designed for a specail purpose. The teacher should emphasize that it is the duty of every responsible citizen to pay his taxes, but for his own protection he should learn what he must pay and how to figure the amount for the greatest advantage to him.
Emphasize the importance and purpose of insurance.
Prepare bulletin boards which show different kinds of insurance.
Answer the questions: What is insurance? Who are actuaries? How did insurance begin? What does age of driver have to do with premiums? What does the accident rate of the community have to do with premiums? Will the rates be the same paid monthly and weekly?
Discuss. insurance as an investment.
Discuss the insurance which the school has for the members of the athletic teams and for students who regularly ride in school buses.
List the kinds of insurance the students have in their homes or know about.
List the different kinds of property insurance available in your community. Available for every member of the class:
The Mathematics of Life Insurance, A unit for High School Mathematics Classes from
Institute of Life Insurance 488 Madison Avenue New York 22, New York
Activities:
Have class discuss kinds of insurance about which they desire information.
Have volunteers make appointments for interviews with insurance agents of the various types selected and then make class reports on their findings.
The report on a given type may be given at the time the work on that topic is being developed by the class.
Project: Investigate insurance rates in your community. Choose some pieces of property and find out the fire insurance rate on each one. Select houses of different kinds and in different sections of the community to compare the rates.
Answer these questions: What goes in to computing the premiums? These ideas will go into the answer: The cost is always based on the risk that the insurance company is taking. The material used in the construction of the house is important. Fire protection available and record of losses from fire are considered. The location of fire hydrants plays a part. The water supply is an important factor. Do some investigation on premiums not only in your community but in other cities and states.
Since the rates differ from location to location, study not only the material found in textbooks but the rates of he insurance companies of the local community. Find out the rates for the different car insurance in your community.
Check on the rates for drivers under 25.

146

d. Collision e. Fire f. Theft 3. Other personal possessions
B. Personal insurance 1. Life
2. Health, accident and hospitalization
3. Annuities a. Straight life b. Installment refund annuity c. Cash refund annuity d. Deferred life annuity

Check on the rates for cars used for business.
Discuss the meaning of the words: I;>odily injury, property damage, comprehensive, damage, and collision insurance, cash surrender value, face, face value, premium adjuster.. Collect data on insurance and property losses. Have students make graphs or bring them in from other sources.
Help students decide that it is most economical to send heavy packages by freight and light packages by express or parcel post.
Have some students make investigations about the costs of insuring and of sending packages by express parcel post, freight.
Investigate the use of air express, air freight and air parcel post.
Inquire at the post office concerning these matters.
Investigate insurance on all kinds of valuables such as furniture, jewelry, furs, clothes, silver, cameras, musical instruments collections of all kinds.
Check on what information is necessary when insuring items.
Familiarize students with terms used such as: policy endownment, policy holder, insurer, beneficiary, premium, insured, dividends, cash value and mortality table.
Discuss kinds of policies such as: term insurance, whole or ordinary life, canceling policies, limited payment, life and endowment insurance, group and industrial.
Invite an insurance man to visit the class as a resource person.
Write to different insurance companies for information.
Discuss these questions: Could a life insurance company operate without using a mortality table? Is it important for a life insurance company to keep records of the number of its policy holder who die at each age? Will medical research have any effect on the mortality tables in the future? How? Are there other factors which will affect the tables?
Introduce this unit by asking the following questions: Estimate the number of persons injured during the year by causes other than cars. Which are most common? Estimate amount lost in income during the year caused by illness. Ask students to bring to school graphs showing these items just discussed.
Call to the attention of the student that the intended purpose of an annuity is to make up for loss of wages.
Point out that in this case the annuitant receives payment regularly for his entire life. However, if at death money is left over it goes to the insurance company and not to his beneficiary.
Point out that this type differs from straight life in that the money left over at death goes to his beneficiary.
Point out that in this type the beneficiary receives a lump sum.

VI. Investments
Characteristics of a good investment

If a person makes a living wage and manages his money well, he should have money to invest. Points to consider in choosing an investment include the following
1. Safety 2. Salability 3. Income
Investment is an individual matter. There are no set rules to cover all situations and all people. One person may want a completely safe investment

147

which can easily be turned into cash with a definite income. Another may be willing to gamble on something which is not as safe but which might pay higher returns. Anyone interested in investments should consult an investment counselor.

B. Kinds of investments 1. Life insurance 2. Savings bank account 3. Social security

The members of the class should investigate the points previously mentioned for the different types of investments. This is a good opportunity to have resource people in to speak to the class, to home members of the class make individual or group investigations, to have panel discussions, individual projects, or related activities.

4. Real estate

5. Stocks and bonds I

148 I
II

ad valorem duty annuity approximate number assessed value average (mean)
bank discount bank reconciliation statement bar graph barometer beneficiary binary operation brokerage budget
capital stock circle graph class boundaries class intervals collateral commission common stock consumer credit continuous data coordinate axes corporation cumulative frequency polygon cumulative frequency table cumulative relative frequency
date of maturity decimal system depreciation deviation dimension discrete data discount Dow-Jones Stock Average

VOCABULARY
employee endorsement equity estimate excise tax exemption exemption expenditure extrapolation
face value face of note frequency frequency distribution frequency table
Gallup poll gross profit gross sales grouped data
histogram
indentity element indirect measurement inheritance tax internal revenue interpolation liability insurance limited payment
mean median meter metric system mid-point mode

mortgage mutual insurance companj!
negotiable normal curve normal distribution
per cent of decrease per cent of increase percentile perimeter pictograph power precision premium principal
quardrant quartile - upper, lower
range ratio reciprocal reconciliaton relative frequency round off
savings account score social security standard deviation statistics system of numeration
tax rate
zero

149

Volume m.
REFERENCES FOR TEACHERS
A. Periodicals
Mathematics Student Journal. National Council of Teachers of Mathematics, 1201 Sixteenth Street, N. W., Washton 6, D. C.
School Science and Mathematics. Central Association of Science Mathematics Teachers. Oak Park, TIlinois. The Arithmetic Teacher. National Council of Teachers of Mathematics, 1201 Sixteenth Street, N. W., Washington
6, D. C. The Mathematics Teacher. National Council of Teachers of Mathematics, 1201 Sixteenth Street, N. W., Washing-
ton 6, D. C.
B. Books
Adler, Irving S. Magic House of Numbers. Doubleday Anchor Co., 1957. Adler, Irving S. The New Mathematics. The John Day Co., 1958. Allendoerfer, C. C. and Oakley, C. O. Principles of Mathematics McGraw-Hill, 1955. Archibald, Raymond C. Outline of History of Mathematics. Mathematics Association of America, Inc., 1949. Bell, E. T. Men of Mathematics. Simon and Schuster, 1937. Bell, E. T. The Queen and Servant of Sciences. McGraw-Hill, 1951 Birkhoff, G. and MacLane, S. A Survey of Modern Algebra. The Macmillan Co., 1959. Breuer, Joseph. Introduction to the Theory of Sets. Prentice-Hall, Inc., 1958. Courant, Richard and Robbins, Herbert What is Mathematics? Oxford University Press, 1943. Cundy, H. M. and Rollett, A. P. Mathematical Models. Oxford University Press, 1952. DeVault, M. Vere. Improving Mathematics Programs. Charles E. Merrill Books, Inc., 1961 Fliegler, Louis A. Curriculum Planning for the Gifted. PrenticeHall, Inc., 1961. Haag, Vincent H. Studies in Mathematics, Volume III, Structure of Elementary Algebra. School of Mathematics
Study Group, 1961. James, G. and James, R. C. Mathematics Dictionary. D. Van Nostrand Company, Inc., 1959. Kelley, John L. Introduction to Modern Algebra. D. Van Nostrand Compnay, Inc., 1959. Kemeney, John G., Snell, J. L. and Thompson, Gereald L. Introduction to Finite Mathematics. Prentice-Hall, 1957. Labarre, A. E. Jr. Elementary Mathematical Analysis. Addison-Wesley' Publishing Co., 1961. Larsen, Harold D. Arithemtic for Colleges. Revised Edition Macmillan Co., 1958. Meserve, Bruce and Sobel. Mathematics for Secondary School Teacher. Prentice-Hall, Inc. 1962. National Council of Teachers of Mathematics. Insights into Mod ern Mathematics. Twenty-Third Yearbook. The
Council, 1957. National Council of teachers of Mathematics. The Growth of Mathematical Ideas, K-12. Twenty-fourth Yearbook.
The Council, 1959. National Council of Teachers of Mathematics. Multisensory Aids in the Teaching of Mathematics. Eighteenth
Yearbook. The Council, 1945. Newman, James R. The World of Mathematics, 4 volumes. Simon and Schuster, 1956.
Studies in Mathematics, Volume n. Euciledean Geometry_Based on Ruler and Protractor Axiom. School Mathematics
Study Group, 1961.
Studies in Mathematics. Volume m. Structure of Elementary Algebra. School Mathematics Study Group, 1961.
Tarski, Alfred. Introduction to Logic. Oxford University press, 1946.
C. Articles, Pamphlets, and Bulletins
A Guide to the Use and Procurement of Teaching Aids for Mathematics. Prepared by Berger and Johnson. Publishers: National Council of Teachers of Mathematics, 1201 Sixteenth Street, N. W., Washington, D. C., 1959.
Chicago School-Journal. Mathematical Teaching Aids. Compiled by Joseph J. Urbancek, 1954. Chicago Teachers College, 6800 Stewart Avenue, Chicago 21.
Designing the Mathematics Classroom. National Council of Teachers of Mathematics, 1201 Sixteenth Street, N. W., Washington 6, D. C., 1959.
Martley, M. C. Patterns of Polyhedrons. Mich: Edwards Bros., 1948. Johnson, D. A. and Glenn, Wm. H. Exploring Mathematics on Your Own. The Pythagorean. Theorem. Webster Pub-
lishing Compnay, 1960. Johnson, D. A. and Glenn, Wm. H. Exploring Mathematics on Your Own. Topology - the Rubber Sheet Geome-
try. Webster Publisihng Compnay, 1960. Johnson, D. A. and Glenn, Wm. H. Exploring Mathematics on Your Own. The World of Statistics. Webster Publish-
ing Compnay 1961.
150

Planning Schools for New Media. A Guide for Boards of Boards of Education, S~hool Administrators, and Architects. The project reported on in this publication was conducted persuant to a contract with the United States Office of Education. Available from Division of Education, Portland State College, Portland, Oregon, 1961.
Ransom, Wm. R. Geometry Growing. Washington, D. C.: National Council of Teachers of Mathematics, 1954. The Secondary Mathematics Curriculum. Reprinted from the Mathematics Teacher, May, 1959. Publishers: Nation-
al Council of Teachers of Mathematics, 1201 Sixteen-th Street, N. W. Washington 6, D. C. D. Course of Study
College Entrance Examination Board. Report of the Commission on Mathematics. and Appendices, Commission on Mathematics. New York. 1959.
High School Mathematics, Units I IV. University of illinois Committee on School Mathematics. Urbana: University of illinois Press, 1960.
Mathematics for High School. Elementary Functions, Part I and II. Yale University: School Mathematics Stu d y Group, 1961. Mathematics for High School. School Mathematics Study Group, 1961.
Mathematics for High School. Geometry, Parts I and II. School Mathematics Study Group, 1961. Mathematics for High School. Intermediate Mathematics, Parts I, II, and III. School Mathematics Stut\y Group 1961. New York State Education Department, Bureau of Secondary Curriculum Development, Handbook for Eleventh
Year Mathematics. Albany, N. Y. 1959.
Universtiy of illinois Committee on School Mathematics. High School Mathematics Units I, II, m, IV" V VI. Ur-
bana: University of illinois Press 1960. E. Mathematics Tests
Myers Sheldon S. Mathematics Tests Available in the United States. National Council of Teachers of Mathematics. 1201 Sixteenth Street, N. W. Washington 6, D. C. 1959.
151

..... ~_.~-~-~

GLOSSARY FOR TEACHERS

ABSCISSA. If the ordered pair of numbers (a, b) are the coordinates of a point of a graph, the number a is the abscissa.

ABSOLUTE ERROR. One -half the smallest marked interval on the scale being used.

= ABSOLUTE VALUE. The absolute value
Ia I = -s. On the number line absolute

of the value

real num ber a is the distance

is denoted by Ial.
of a point from

If a >
zero.

0

then

I al

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 symbol 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. If A and B are two points of a circle with P as center, the arc AB is the set of points in the interior of
LAPB on the circle and on the angle.

AREA OF A SURFACE. Area measures the amount of surface.

ARGAND DIAGRAM. Two perpendicular axes, one which represents the real numbers, the other the imaginary numbers thus giving a frame of reference for graphing the complex numbers. These axes are called the real axis and the imaginary axis.

ARITHMETIC 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 and 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 additiontables, subtraction tables, multiplication tables, division tables, power tables, logarithmic tables, etc.

152

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".
CIRCLE. The set of points in a given plane each of which is at a given distance from a given point of the plane. The given point is called the center, and the given distance is called the radius.
CIRCULAR FUNCTION. A function which associates with each arc of a unit circle (as measured from a fixed point of the circle) a unique point. The sine function associates with the measure of an arc the ordinate of its companion point, and the cosine the abscissa of the point.
CIRCUMFERENCE. The length of the closed curved line which is the circle.
CLOSED CURVE (SIMPLE). A path which starts at one point and comes back to this point without intersecting itself represents a simple closed curve.
CLOSURE. A set is said to have the property of closure for any given operation if the result of performing the operation on any two members of the set is a number which is also a member of the set.
COLLECTION. Elements or objects united from the viewpoint of a certain common property; as collection of 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 commuta tive; 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 1.
CONCRETE. Belonging to things; that which is directly experienced by our senses.
CONJUGATE COMPLEX NUMBERS. The conjugate of the complex number a + bi is the complex number a - bi
= CONJUNCTION. A statement consisting of two statements connected by the word and. An example is x + y 7
and x - y = 3. The solution set for a conjunction is the intersection of the solution sets of the separate state-
ments.
CONDITIONAL EQUATION. An equation that is true for only certain values of the variable. Example: x + 3 = 7.
CONIC, CONIC SECTION. The curves which can be obtained as plane sections of a right circular cone.
CONSISTENT SYSTEM. A system whose solution set contains at least one member.
CONSTANT. A particular member of a specified set.
COTERMINAL ANGLES. Two angles which have the same initial and terminal sides but whose measures in degrees 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. {l, 2, 3, 4, J CONVERGENT SEQUENCE. A sequence that has a limi L 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.
153

DECIMAL EXPANSION. A digit for every decimal place.

DEDUCTIVE REASONING. The process of using previously assumed or known statements to make an argument {or 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 SoX- + a,x- - 1 + ... + an_txt + an is said to be of degree n if ao~'

DENOMINATOR. The lower term in a fraction. It names the number of equal parts into which a number is to be divided.

DEPE.1WENT 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".)

DllIEDRAL ANGLE. The set of all points of a line and two non-coplanar half-planes having the given line as a common edge. The line is called the edge of the dihedral angle. The side or face consists of the edge and either half-plane.
= DIRECT VARIATION. The number y varies directly as the number x if y lex 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. = = = = 4x3lf.! 4(3+lf.!) (4X3) + (4xlf.!) 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 "'" 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 "=".

II

EQUATION. A sentence (usually expressed in symbols) in which the verb is "is equal to".

EQUIVALENT EQUATIONS. Equations that have the same solution set.

EQUIVALENCE RELATION. Any relation which is reflexive, symmetric, and transitive.
Reflexive: a = a
= = = Symmetric: If a = b then b = a.
Transitive: If a b and b c, than a c.

EQUIVALENT FRACTIONS. Two fractions which repre sent 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.

154

EXPONENT. In the expression all the number n is called an exponent. If n is a positive integer it indicates how many times a is used as a factor.
all = ax ax xa
'----v-J
n factors Under other conditions exponents can include zero, negative integers, rational and irrational numbers.
EXPONENTIAL EQUATION. An equation in which the independent variable appears as an exponent.
EXPONENTIAL FUNCTION. A function defined by the exponential equation y = as where a>O.
EXTRANEOUS ROOTS. Those roots in the solution set of a derived equation which are not members of the solution set of the original equation.
EXTRAPOLATING. Estimating the value of a function greater than or less than the known values; making inferences from data beyond the point which is strictly justified by the data.
= FACTOR. The integer m is a factor of the integer n if mq n where q is an integer. The polynomial R (x) is a
= factor of the polynomial P(x) if R(x) Q(x) P(x) where Q(x) is a polynomial. Factorization is the process of find-
ing 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 "'1,," 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 11
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.
155

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.
INTERSECTION OF SETS. If A and B are sets, the in tersection of A and B, denoted by A n B, is the set of all
elements which are members of both A and B.

INVERSE. The opposite in order or operation. Thus counting backward is the inverse of counting forward; subtraction is the inverse of addition; division is the inverse of multiplication.

INVERSE FUNCTION. If f is a given function then its inverse is the function (provided f is oneta-one) formed by interchanging the range with domain. The symbol for inverse of f is f_l.

INVERSE VARIATION. The number y is said to vary inversely as the number x if xy = k where k in a constant.

IRRATIONAL EQUATION. An equation containing the variable or variables under radical signs or with fractional exponents.

= JOINT VARIATION. A quantity varies jointly as two other quantities 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 of two or more numbers is thE common multiple which is a factor of all the other common multiples.

LEAST UPPER BOUND. An upper bound b of a set S of real numbers is the least upper bound of S if no upper bound of S is less than b.

LESS THAN. An arithmetical relation which indicates that one value is smaller than another. On the number line, the point standing for the smaller number always lies to the left of the point standing for the larger number.
Example: seven is less than ten, 7<10.

LINEAR EQUATION. An equation in standard form in which the sum of the exponents of the variable in any term equals one.

LINEAR MEASURE. A measure used to determine length.
= LOGARITHM. The exponent that satisfies the equation bE 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:

aal2 bbi2 cci2 ddl2 ]

[

as a4

bs b4

Cs c4

ds d.

156

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 introduced. MEDIAN. In a frequency distribution, the measure that is in the middle of the range is called the median. In geometry, a median of a triangle is a line joining a vertex to the midpoint of the opposite side. MODE. In a frequency distribution, the interval in which the largest number of measures fall is called the mode. Alternately, in a frequency distribution, the measure which appears most frequently in the group is called the mode.
MULTIPLE. If a and b are integers such that a = bc where c is an integer, then a is said to be a multiple of b.
MULTIPLICATION, A short method of adding like groups or addends of equal size. It may be illustrated on a number line by counting forward by equal groups. MULTIPLICATIVE INVERSE. The multiplicative inverse of a non-zero number a is the number b such that
ab = 1. It is usually designed by 1/. or a -'.
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.
N-FACTORIAL. The expression "nl" is read "n factorial". nl = n(n-1) ... 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. Th 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 (102), 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, vm, 7+1, 10--2, 1./.
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 give n to the variables.
Example: x+5 = 7
ORDERED M-TUPLE. A linear array of numbers (a" a., as, '." 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.
157

PARTIAL PRODUCT. Is used in elementary arithmetic with regard to the written algorithm of multiplication. Each digit in the multiplier produces one partial product. The final product is then the sum of the partial products.

PARTIAL QUOTIENT. In long division, any of the trial quotients that must be added to obtain the complete quotient.

PERIMETER. The sum of the measures of the sides of a polygon. The measure of the outer boundary of a polygon.

PERIOD. The number of digits set off by a comma in an integer or the integral part of a mixed decimal. In a repeating decimal the period is the sequence of digits that repeats.

PERIODIC DECIMAL. The decimal representation of a rational number in which a sequence of digits repeats.
= Example: 1/7 = .142857142857 ...
Sometimes given as 1/7 .142857
PERIODIC FUNCTION. A function from R to R, where R is the set of real numbers, is called periodic if, and
only if, f (x) is not the same for all x and there is a real number p such that f(x+p) = f(x) for all x in the do-
main 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.

j.

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,x - I + ... + a._,x + a. 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 =F O.

PYRAMID. A pyramid is a polyhedron, one of whose faces is a polygoB of any number of sides and whose other faces are triangles having a common vertex

QUADRANTAL ANGLE. If the terminal side of an angle with center at the origin coincides with a coordinate axis, the angle is called a quadrantal angle.

QUADRILATERAL. A quadrilateral is a polygon formed by the union of 4 line segments.

QUINARY SYSTEM. A system of notation with the base 5. It requires only five symbols or digits: 0, 1, 2, 3, 4.

RADIAN MEASURE. Angular measure where the unit is an angle whose arc on a circle with center at vertex of angle is equal in length to the radius of the circle.

RADIUS. Any line segment with endpoint at the center of a circle and the other endpoint on the circle is called a radius of the circle.
RADIUS VECTOR. A line segment with one end fixed at the origin on the cartesian plane and rotating from aD initial position along the positive x-axis so that its free end point generates a circle.

158

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
"/b is called a rational number.
RAY. Let A and B be points of a line.-Then ray A-+B is the set which is the union of the segment A- B and the set
AB. of all points C for which it is true that B is between A and C. The point A is called the end-point of
RECIPROCAL. Multiplicative inverse.
RECIPROCAL FUNCTION. Pairs of functions in the se t of real numbers whose product is 1
Example: (Sin </ (Csc </ = 1.
REFERENCE TRIANGLE. For any angle on the cartesian plane with vertex at the origin, the triangle formed by the radius vector, its projection on the x-axis and a line drawn from the end of the radius vector perpendicular to the x-axis is called the reference triangle.
REFLECTION IN A LINE. A point P has a mirror image P' in the line AB if P, P', and AB all lie in the same plane with P and pi 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 multiplication, as
= 12 ones 1 ten, 2 ones
= 16 inches 1 foot, 4 inches
(b) Changing larger units to smaller units in subtraction and division as
= 1 ten = 10 ones
1 year 12 months
RELATED ANGLE. For any angle on the cartesian plane, the related angle is the angle in the reference triangle formed by the radius vector and x-axis.
RELATION. A relation from set A to set B (where A and B may represent the same set) is any set of ordered pairs (a, b) such that a is a member of A and b is a member of B.
RELATIVE ERROR. Ratio of the absolute error to the measured value.
REPEATING DECIMAL. A decimal numeral which never ends and which repeats a sequence of digits. It is indicated in this manner: 0.333 ... or 0.142857.
RESOLUTION OF VECTORS. The process of finding the vertical and horizontal components.
RESTRICTED DOMAIN. Domain of a function or relation from which certain numbers are excluded for reasons. such as (1) division by zero is not permitted, (2) need for the inverse of a function to be a function.
RIGHT ANGLE. Any of the four angles obtained at the point of intersection of two perpendicular lines. The angle made by two p,erpendicular 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 10k where a is a real number containing at most three significant digits such
that 1 < a < 10 and k is any integer.
159

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 seg ment is a geometrical figure while the distance is anum ber which tells how far A is from B.
SEQUENCE. An ordered arrangement of numbers.
SERIES. The indicated sum of a sequence.
SIGNIFICANT FIGURE. Any digit or any zero in a numeral not used for placement of the decimal point. Example:

703,000

.0056

5.00

SKEW LINES. Two lines which are not coplanar are said to be skew.

SLOPE. The slope of a given segment (PI P2) is the number m such that m pair (Xl' YI) and P2 is the ordered pair (X2, Y2)' SOLUTION SET. The truth set of an equation or a system of equations.

Y'l - Y1 where PI is the ordered x2 - X,

SPHERE. The set of all points in space each of which is at a given distance from a given point. The given point is. called the center of the sphere and the given distance is called the radius.

SQUARE. Formed by four line segments of equal length which meet at right angles.

STANDARD DEVIATION. The square root of the arithmetic mean of the squares of the deviations from the mean.

STATISTIC. An estimate of a parameter obtained from a sample.

SUBTRACTION. To subtract the real number b from the real number a, add the opposite (additive inverse) of
b to a. a -b = a+(-b). Also, a -b = c if and only if a = c+b.

SUCCESSOR. The successor of the integer a is the integer a+1.

n

L: L: SUMMATION NOTATION. The symbol at. The symbol

,the Greek letter "sigma," corresponds to the

k=::m
first letter of the word "sum" and is used to indicate th e 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: at = a2+a3+a,+8o.

k II 2

00

2: When the summation includes infinitely many terms it is written

at. In this case there is no last term aoo

k
because 00 is not a number. The symbol 00 is used simply to indicate that the summation is infinite.

SYMMETRIC PROPERTY. 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, B, and C are three non-collinear poi nts in a given plane, the set of all points in the segments having A, B, C as their end-points is called a tr iangle.

UNEQUAL. Not equal, symbolized by oF.

160

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 ItS.
VARIABLE. A letter used to denote anyone of a given set of numbers. Another name for variable is placeholder in an equation.
Example: x+5 = 7.
VECTOR. In physical science, a quantity having magnitude and direction. In mathematics a vector is a matrix
r of one row or one column as (a1 b1 c1) or a1 1
l::1a2 1 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 0 r enclosed within it is called the volume.
161

FLOW CHAitT

====================~K====~===========================2===============:===========3=============\= ========~~4~===========

5 I

6

7

8

I I I 111 I ELEME9NTARY

GENE9 RAL

10

ALGEBRA

MATHEMATICS GEOMETRY

MEDIINATTEER ALGEBRA

ADVA12NCED ALGEBRA

11 or 12 Consumer Mathematics

NUMERATiqN
============~======~========~====================~==================~====================
HISTORY

SYMBOLS FOR NUMBERS AND COUNTING 1. Hindu-Arabic
2. Roman 3. Others

Cardinals through 12
Ordinals through fifth
Reading and writing Numbers 15

0 Sets (GroupsW

RECOGNITION OF

to 3

''

CARDINALITY OF Vocabulary

SETS (Groups)

introduced

by using ob-

jects

More than

and less than

,...,

Reading and writing to 99 Counting by 2's, S'.s, lO's to 99 Reading and writing number names through 10 Ordinals through tenth
Sets (Groups) to 10 One to one matching Members of a set Vocabulary extended to understand ing of more than, greater than, less than, and others

BASES FOR NUMER-

ATION

4

1. Decimal

Notation (Base 10)

and Place value

2. Notation and

Place Value for

Other Bases

'....Place value of ones-

and tens

..



Ci

VARIABLES





Counting experiences extended through 200 by l's, lO's, 5's, 2's; through 20 by 4's; through 30 by 3's; through 1000 by lOO's
Ordinals through fifteenth Reading and writing Roman numerals
through XII
Recognizing, combining, separating, and rearranging through 18
Symbols for sets used informally Sets with objects demonstrated Vocabulary of sets extended

Reading and writing 3 digit num'-h . Understanding of place value concept o~
ones and tens strengthened Understanding of place value through
hundreds Decim point with money expressions


Counting through 300 by l's, 2's, 3's, 4's, 5's; through 1000 by lO's and lOO's; backwards from 100 by l's, 2's, 5's, lO's
Ordinals through thirtieth Reading number names through 100 Writing number names through fifty Reading and writing through XXX

Counting by 6's, 7's, 8's, 9's, and back ward from 100 to 1
Ordinals through fiftieth Reading and writing through LXXX and
recognizing D, C, M Base 5 introduced

Abstract symbols to indicate members of a set
Symbol for empty or null set Use of braces ~ ~

Union of sets (U) Intersection (n) Subsets

Jte-ading and wl'iting 4 digit numerals Reading and writing 9 digit numerals

(perhaps 5 and 6 digit)

Understanding of place value through

Understanding use of 10 as the base of 1000's

our system of numeration

.t..> Zero in l's, lO's and lOO's place
Regrouping numbers through 100

c

Counting forward and bac~rd Ordinals through hundredt~ Reading and writing beyond LXXX Understanding L, C, D, M, and their
combinations Counting in base 8 Understanding of sets str ngthened

Numerals to 9 places

I

Reading and writing dec~rn~ls

Place value through millio s

Place value in base 5 and 8

Use in equations; in problefll solving
Use of 0 and N in equations

History of Roman numerals
Concept and use of ordinal and cardinal numbers extended
Reading and writing with facility Comparisons with Arabic system Meaning of bar over letter Base 4 and 7 Undestanding of sets continued
Numerals to 12 places Base 5 and base 8 reviewed Place value in base 7 and base 4 Continued

Appreciation of decimal system through historical development of other number systems Knowledge of of Roman numerals maintained Computation with base 5, 6, 7 Undestanding of sets continued
Use in mathematical sentences and formulas

Historical development of numeration, Hindu-Arabic, retaught and extended through research Symbolism extended to include Babylonian and Egyptian Computation with bases 8 and 12 Mastery of set concept
Term variable introduced Application made in number sentences

Irrational numbers
Set notation Set notation

Research to sho'A how historical background helps pupils discover what constitutes the modern num ber system
Knowledge of Roman numerals and other symbols maintained Bases 8, 12 and others reviewed
Interpretation of sentences expressed in set notation
Usage reviewed and extended

HiSO:ory of efforts to solve equations of degree greater than two

Historical developments of taxatiOn, inv<.>stment and credit system

Pure imaginary number system Complex numbers; the symbols i

Polar form of complex numbers

Intuitive consideration of idea that all infinite sets do not have same number of elements

Logarithms to different bases and method of conversion

Maintain skills developed in our decimal notation

Quadratic variables Equations with three linear vari ables

Sine x and cos x as variables in real number system

--------------------+-----------~~----------------r---------------------------------r---------------------------------~-------------------------------- ~ ----------------------~-------I-U~se:;i:n-a:r:e:a-f~o:r:m:u~l~a:s;~in~p~la:c:e~v=a~lu~e~~E~x=p=o~n~e=n~t=s----t~7Us-e~i~n~b~a-s_e_s+-~P~o-si~t~iv_e_,--n-e=g-~-~R~e-v~ie-w---a-n~d~e-x---I---------~--N-e_g_a-ti~v-e-,-----I-F--u-rt_h_e_r__in----~-------------

EXPONENTS

in other number bases

used to emphasize role of base and position in the decimal system

other than ten

ative integral exponents Zero exponents Multiplying and divid4J.g

t e n sion

fractional, irrational exponents

vestigation with irrational number as exponent

SERIES AND SEQUENCES
PROBABILITY AND STATISTICS

Simple averages

Mean, median and mode

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 computing simple and compound interest
Computation and interpretation of data. Percentile in terms of normal curve

=============b=====================================================================db==================~='=============~===================~=======================~======~======k======~==~==~======

ADDITION AND SUB TRACTION
MULTIPLICATION AND DIVISION
FRACTIONS (Common)
PROPERTIES FACTORING
PROPERTIES OF 1 and 0

K Readings for
combining and separating groups Readiness
Dividing and sharing equal parts of a whole (one half)

Combining and separating groups through ten with symbols
Regrouping of 2's and 3's to form other groups
Understanding and writing !., of a whole 2
Understanding and writing !., of a whole 4

I

2 II

3

FLOW CHART

4

5

'===============6=======================7=======~~~'==8====~=E=L:EA:LM:GE:E9NB:=RT:AA:R=V==lM==A:GT:HE:NE:E~:RA:A=T:LIC=S=I===:';~~10;~!~======1N=T==EA:RL:G~:E~B:DR:IA:A=T=E=======A=AD:LVG:A1E:~B:=RCA:E:D======MC=a~t=1hn=e:m=~:a~t:i:rc::s;

I
Mastery of 45 easy addition and subtract'on facts Development of the other 36 basic facts Use of higher decaf e without grouping
Adding twos and ddubles
Separating twos and doubles

Mastery of 81 basic addition and subtraction facts Adding 1, 2, and 3 digit numbers with 4 addends with internam zeros and regrouping Subtracting 2 and 3 digit numbers with internal zeros and regrouping
Discovering and mastering basic multiplication and division facts with 2's, 3's, 4's, 5's, and 10's
Multiplying by 1 digit multipliers Dividing by 1 digit divisors

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
Developing multiplication and division facts with 6's, 7's, 8's, 9's
Multiplying by 2 and 3 digit multiplier Dividing by 1 digit numbers with and
without remainders Understanding what is meant by aver-
age

OPERATIONS
Addition of 7 digit numbers Subtraction with regrouping
Multiplication with 4 digit multipliers using zero in different places
2 digit divisors Tests for divisability begun Rounding divisors to get first quotient
figures

Commutation with high degree of mastery
Mastery of multiplication facts Use of 4 digit multipliers 3 digit dividers Tests for divisibility extended Casting out 9's Distributive law

Mastery Mastery

1\~astery
I

Algebraic expressions

Mastery Algebraic expressions

Mastery Mastery

Meaning of 1 and of an object ;- ;-

11 1
2 3i 4

15 81- of a whole and of a group

Familiarity with: ..!..
of a group, :0

Use of symbols for fractions and use of terms numerator and denominator
Use of decimal point in reading and

of an object'-;'

writing money expressions

Comparing fractions Finding parts of whole and of groups Simplifying fractions Use of fundamental operations in deal-
ing with money Relationship between the cent sign and
decimal point

Equivalent fractions Addition, subtraction, multiplication
and division of fractions
Adding and subtracting tenths and hundredths with regrouping
Decimal mixed numbers Estimation of results

Multiplying and dividing fractions Short cuts of multiplying and dividing
by 10's, 100's, and 1000's Estimation and rounding off Use of reciprocals Distributive law Multiplying and dividing decimals

Fractions in simplest form Division of fractions by use of one and by the common denominator method

Reteaching

Complex frac tions Rational expressions Properties of a field

1

Reteaching and practice in computation

I

Understanding of cummative associative Use of proper terminology and distributive properties

IReview

~eteaching

I

Use of factors in dealing with fractions Use of factors in more complex frac-

Natural numbers

Use of terms factor, prime, multiple

tions and in getting common denominators

(primes)

I

Common monomial factor Differences of

l

squares

Perfect squares

Quadratic tri-

nomials

Discovering generalizations: n+ o= n; Understanding of the generalizations in Reviewing generalizations

Continued

n-o= n; n-n=o; nXO=Oj oxn=o reference to 0 and 1

Use of property of 1 in simplifying

lxn=n; n+l=n

fractions

Continued

Continue If ab= O then Reteaching a=O or b=O

Addition and sub- Vector addition of traction with com- complex numbers plex numbers

Multiplication and division of complex numbers, i2 = 1 Rationalization of denominators
Rational expressions Complex algebraic fractions

Multiplication and division of complex numbers in polar form DeMoivre's theorem Powers and roots of complex numbers

I Properties of field, integral domain, group

Basic laws reviewed Special cases of factorization ineluding trinomials
reducible to difference of squares

Factor theorem, synthetic division

'

Use of property of Idea that 1 and 0

one in simplifying need not resemble

fractions

1 and 0

PROBLEM SOLVING

TYPES OF PROBLEMS

Daily activity

One step problem Gradual introduction of abstract terms

Pictorial represent~'tions
Development of pr blems with word s:-. mbols and number S) mbois Relating all four p ocesses in simple o al and written proble,ns

Analyzing and solving one step lems
Estimation of answers Checking of computation

prob-

Analyzing and solving problems involving four fundamental processes
Various types of problems Estimation of answers Checking of computation

Using four fundamental operations Using ratio to symbolize rate and com-
parison Finding averages Using equations related to processes
taught

Finding quantities in various situations Three basic per-

Expressing numbers as fractional num- centage prob-

erals Variables

with

solution set containing

lems Life situation

one or more digits

problem

Processes with variable extended to in-

elude:

- a-

X
-

b c

Extension
I

Linear equations Quadratic equations Systems of linear equations

Extension

Proof of statements concerning of geometric figures

Application of function to physical processes VVorded problems involing one variable, two variables, quadratic variables

Solutions of simul: taneous equations using set language for representing solutions Practical problems using trigonometric functions

LOGIC

Readiness through step by step reason- Extended ing

Analyzing problem solving procedure Estimation of answers

Extension
I

Statement, con-

verse, inverse,

and contrapositive

-

Deductive and

inductive rea-

soning

Conjunction Disjunctwn

CONCEPT OF PROOF

Readiness with concrete materials

Matching and othe simple procedures used to show correctness of answers Comprehending in verse process

Using Laws of Arithmetic as reason Checking

E x te nded

Checking reason- Extension Direct and indi-

ableness of

rect proof

solution set

Extension

Direct and indirect proof used

Direct and indirect proof of selected theorems

Algebra as a logical system with facts capable of being

proved from a basic

set of postulates

I

Proofs emphasized throu~tho11t course

--

FLOW CHART

======================K==========================================3=================-===4==========~===s====~~======='========' ' 1

[ RELATIONS

EQUALITY

Readiness

Mathematical sentence6

Principles of base ten numeration used in understanding a collection of coins Experiences with measurement

Understanding names for the same number
Commutative law of addition and multiplication

Associative law of addition and multiplication

Use of concrete objects to show

-I = -2

2

4

Understanding of commutative and associative Jaws extended

Use in equation and problem
solvinf Use o many names for the same number

Extension to include deci- Simple number sentences mals

INE.QUALITY

Readiness

Mathematical sentences

Opportunity to Jearn 5 is less than 6, and 4 is greater than 3 Use of number line to show addition

Demonstration with counters and number line that sume numbers are greater than or less than other numbers

Skill in making comparisons:

. unequal, greater than, less

than Symbols:

<

> . .p

Use of terms with proper
.symbols:
< >

Use of symbols in prob!em solving

Nature of inequalities Number sentences with inequali-
ties

8 PropertiE>s of equations
Nature of inequalities Number sentences with in
equalities

9 ELEMENTARY
ALGEBRA
Linear equations Quadratic equations Systems of 2 equations with 2 unknowns
Linear inequalit ies Properties of inequalities

GENE9 RAL

I 10 GEOMETRY

MATHEMATICS

Solving number Equality as iden

sentences

tity in defini-

tio ns

Review and extension of in equalities

Exterior, remote interior angles of triangles
Longer side and sum of other two sides

11 INTERMED IA T E
ALGEBRA Linear equations, quadratic
equations Syst ems of linear equations Systems of quadratic equations Systems of linear equations with
three variables Quadratic inequalities Graphs of quadratic inequali-
ties

12 ADVA NCED ALG EBRA

I 11 or 12 CONSUME R M 4THEMATI CS

Reflective, symmetr ic and transitive properties
Proofs of theorems

Axioms of inequalities Proofs of theorems

FUNCTIONS VARIATION

I

I

Graphs of

Graphs of simple compari-

simple com- sons

pari.sowl

I

Introduction of concept: Linear Quadratic

Linear, quadratic funct ions Exponental function Logarithmic function

Idea of inverse function Detailed study of circular
polynomial, r ational, exponental, logarithmic, trigonometric func tions

Ratio and proportion in number sentences

Direct Inverse Joint

Application in formulas and in mathematical s e nt ences

Direct variation: Y= mx+ b where b=O

RATIO AND PROPORTION
PERCENT
UNITS OF MEASURE ESTIMATION

Identifying equal groups Understanding. of l. l. l.
2 t It 4

Demonstration of 1 to 3 or 2 to 5 correspondence with objects
Use of problems of rate tn>e in clarifying multiplication and division

Ratio used to express rate and comparison

Meaning of terms used in r ates and comparisons

Use of colon between terms Findin& missin& te.rma
Solving problems of per cent

~tio as the comparison of one integer with another (except 0) te as a I;atio cale drawr g

Used in solving per cenb Scale drawings

Readiness experi-

l

ences through classroom activitiM related to 100%
.

Developmental experiences; clock, calendar, mon-a, inch, foot, yar weight

Telling time: to hour, and one half hour Recognition of coins

Identifying and US ing standard measures as needed

Skill in esti- Estimating using ob-

mating

jects in classroom

Classroom experiences: 100%, Experience's extended 50%, 25%

Identification and use of standard units of measurement to include relationships between units
Sums and differences by rounding 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

Use ratio to introduce per cent
Conversion from one unit to another Meaning of square measure Rounding large numbers

I

Per cent compared with Per cent bYj use of ratio and procommon and decimal portion

fractions

'Pse of per c,ents greater than 100

Meaning of three caaes of and less than 1

per cent

ftoundinJ: off per cent

Use in problem solving 1 pse of per fents in business

Meaning of cubic measure Principle of regrouping
in multiplication and division
Estimating in linear, square, and cubic measure

MI EASUREMENT
d rmffeamsnur~inr'wwei"ghnt aonodunmtinasgs dvantages of metric measure over English system ccuracy and precision
I
>ttport:ance ~of precision

I I

Extend use of per cents
. Nautical measure, longitude
and latitude Square r oot Measures of science Continued

Review and reteaching of r at io Properties of proportion

Sine, cosine, tangent Similarity of geometric figures Area of sectors

Extend use of per cents

..
y2 on number
line Scientific notation

Linear, area, volume, degree, r adian

Estimation as compared with accuracy and precision in measurement
I

Trigonometric functions as ratios in r ight triangles

Paying cash through :Loans and using carryin g charge.

Radians as units for measuring arc, angles Methods of conversion to degrees

-Extension o-f-
use of metric measure

--

-

IK I 1 I

2

POINTS, LINES AND P L AN ES

Recognition of rectangle, line, circle, square, triangle

Explore sq_uare, triangle and rectangle

Straight line, square corner

FLOW CHART

I

3

I

4

I

5

6

7

I

Iden~i1fy lines and surfa es

Parallel lines and right angles

GEOMETRIC CONCEPTS

Using number lines to show relations
Lines, points, plane, ray, line segment, and end point defined
Perpendicular lines

Study of llliles: right, acute, obtuse

Points, lines, planes, and their properties

Recopizlng 1, 2, 3, dimensional space

I I I ELEM:NTARY

GENhl,L

ALGEBRA

MATHE . TICS

10 GEOMETRY

I

11 INTERMEDIATE
ALGEBRA

I

12 ADVANC~D ALGEBRA

I 11 or 12 Consumer . Methe~n~~tlcs

I
Points, lint, and planes aj sets

Ruler, postulate, rays, segments, con~ent segments; para el lines; lines and planes; p~ndicular lines an planes

One-to-one correspondence between ordered pairs and points on the xy plane. One-to-one correspondence between ordered triples and points in three-space

Graphs of functions and relations on Cartesian plane
Addition and subtraction of geometric vectors

GEOMETRIC FIGURES
CONSTRUCTION
ANALYTICAL GEOMETRY
GRAPHS
I: HARTS

Cube
Readiness Readiness

Cylinder, sphere

Recognizing tri angle, circle, square, cone, cube, cylinder, and sphere

Identifying and reproducing circle, square, triangle, rectangle, cube, sphere, cone, cylinder

Making, collecting and using geometric figures Finding area and perimeter of square and rectangle

Defining closed curves, parallelogram, triangle (equilateral and isosceles), rectangle, sqnare, quadrilateral, . circle, polygon
Studying area and perimeter Recognizing prism, pyramid and
rectangular solid

Usin! term: degree . in lonii tu ~ latitude problems
Exten ing study of area Finding volume Using term "congruent" Law of Pythagoras

Extension of concepts of 6th grade
Perimeters of the triangle, quadrilateral, pentagon
Areu of the triangle, squarei rectangle, parallelogram, eire e
Symmetry, similar triangles Measurement of angles

Area. and volume Measurement of angles

Construction of simple figures by means of straight edge and protractor
Constructing figures to scale

Extension of simple construetion of scale drawing
Three demensional figures from drawings
Using Compass

Bisection of angles and lines
Pe~endiculars
Ang es equal to given angles

Construction of inscribed polygons Perpendiculars to a line at a point on the line and E:.erpendiculars to a line om a point not on the line

Application of measurem nt of geometric figures

Rays, angles, triangles, polygons, quadrilaterals, circles, spheres, prisms, cones, cylinders

Conic sections

' Review of funda
mental construetions Skill in use of the straight edge, co~pass, and the p otractor

Inscribed and circumscribed circles of triangles Inscribed and circumscribed polygons of circles Segments of line in proportion

Derivation of slope-in- 3-dimensional geometry

tercept form

Equation of plane

Introduction of . 3-di-

mensional graph

GRAPHS AND CHARTS

Observing and manipulating

Reading thermometers
Making sin;fale graphs of aily expenences

Makin simple bar and icture graphs of cl ssroom experie ces Drawing simple
mneaipg1; 1boofrhsochoodol,and class

Making and inter preting simple graphs Extending ability to draw maps and make and read scale drawings

Picture graphs Bar graphs Line and circle graphs Reading maps

Extension of construction of graphs and maps to include large numbers and scaling

Construction of graphS: Histograms, bar, broken line, circle, pictograms
Scale drawings

Readiness Observing and
constructing simple charts

Making simple charts Using information on charts Developing awareness of the function of a chart

Making weather chart' , number chart of basic facts

Making and interpreting charts extended

Reading simple charts and tables

Reading of charts and tables extended to include interEretation and construction rom statistical data

Reading charts

Extend construction of graphs

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~umber line Coordinate L~aenaer and quadratic equations

Interpretation and constiuction of graphs

Introduction of 3-dimensional graph Linear equation Quadratic equations in form; x2+y2=r2

Graphs of linear equations, Quadratic equations Solution of systems

Frequency distribution chart and table

Elementa1 statistics Extension of grade 1 and 8 skills Inequalities in a plane

Table of trigonametric functions

Table of Logarithms

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Tangents to curves; areas under curves
Intensive study of straight line and circle with proofs of theorems from plane geometry
Ex~tensfioornm oof fsetuqduyatioofnsc,o~nri,casr&mpeto-r1c equations, rational orms
Plottin~ graph of function defined by a ebraic eluation
Study of graph o exponential and logarithmic functi.on
Writ:t equation from points on grap Advantage of quantity and quality buying. Home owner ship ve1'11111 rent.