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A STATEMENT "OF; THE OBJECT-WES
FOR TEACHINQPLANE GEOMETRY
. AND MEANS 0.5 mum .903
' THESE OBJECTIVES

Thesis {arthn Dar.” 9! M. A.
MICHIQAN STATE COLLEGE

'Flmencg Grace Oberlin,
‘ " 1944

 

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A STATEMENT OF THE OBJECTIVES
FOR TEACHIHG PLANE GEOMETRY
AND HEARS OF TESTING
FOR THESE OBJECTIVES

by

Florence Grace Oberlin

A THESIS
Submitted to the Graduate School of Michigan
State College of Agriculture and Applied

Science in partial fulfillment of the
requirements for the degree of

MASTER OF ARTS

Department of’Mathematics

1944

A cm mmcmmr

The author wishes to express her gratitude for the
time, assistance, and advice given by Dr. Paul L. Dressel
under whose guidance this work was accomplished. She is

also grateful for the suggestions of Dr. J. S utherland

Frame and the encouragement of Dr. Vernon G. Grove.

COITTETTS

An Introduction .................................
History of Objectives ...........................
PrOposed Objectives .............................
Analysis ........................................
Four Objectives Pr0posed ........................
Summary of Proposed Objectives ..................
Means of Achieving These Proposed Objectives ....
Behavior Pattern for Objective One ..............
Tests for Objective One
Unit One .....................................
Unit Two .....................................
Unit Three ...................................
Unit Four ....................................
Unit Five ....................................
Unit Six ............. .......... ..............
Unit Seven ...................................
Unit Eight ...................................
Unit Nine ....................................
Unit Ten .....................................
Unit Eleven ..................................
Behavior Pattern for Objective Two ..............
Tests for Objective Two

Unit Twelve 0....0.0.0....OOOOOOOOOOOOOOOQOOQQ

iii

12
15
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19
25

24
28
32
37
42
48
51
54
61
63
69
70

71

Unit
Unit
Unit
Unit
Unit
Unit
Unit
Unit

Behavior Pattern for Objective Three ........

Unit
Unit
Unit
Unit
Unit
Unit
Unit

Thirteen ............

Fourteen 000000000000

Fifteen 00000000000000000000.00000000

Sixteen ...........

Seventeen ................

Eighteen 00000000000..00.000000000000.000
Nineteen 000000000000.0.0...00ooogooooooo
Twenty 00000000000000

Tests for Objective Three

Twenty-One 0000000.0000.000000000000.0000

Twenty-two 000.00.00.00.

Twenty-three 0000000000000...0.00.000000.

Twenty-four 00000000000.00.000.000.000...

Twenty-five 00000000000o.00

Twenty-81X 000000000

Twenty-seven 00.00000000000000000000

75
78
81
84
91
92
95
95
100

101
104
108
112
116
118
119

iv

concluSion OOOOOOOOOOOCOOOOOOOOO.COOOOOOOOOOOOOOO 122

Bibliography 00000000000000.0000 125

Key to Tests 0.00.00.00.0000000000 130

A STATEMENT OF THE OBJECTIYES

 

FOR TEACHING PLAU§_GEOMETRY

 

AND MEANS or TESTING FOR Tessa osJEcrivrs
Ah INTRODUCTION

The effective teaching of geometry in our secondary
schools is seriously handicapped by the haziness of the
objectives for teaching the course. Too few administrators
have a clear concept of the place of geometry in the high
school curriculum; too few teachers know exactly why they
are teaching geometry and for what changes in pupil behavior
they are working; and too few students have any idea of the
purposes and aims of the course. This often leads to a
lack of appreciation on the part of the administration of
the place of geometry in the curriculum, a confusion of
purpose on the part of the classroom teacher, and an
indifference toward the subject on the part of the student.

It is true that at times objectives are agreed upon
and set down in writing, but often these are more of a
defense for teaching geometry than vital goals of the
classroom. Too often they are stated in broad general
phrases, and not interpreted in terms of specific pupil
behavior. We may agree upon objectives and still continue
to work in the same old way, not teaching by them and not
testing for them. we may say in defense of geometry that
it cultivates the habit of independent logical thinking,
and still continue to teach model proofs, and test for

memory. But surely an objective has no value unless it

transforms the teaching of the classroom.

As Me Call points out, "Measurement is indispensable
to the growth of scientific education."1 And measurement
in itself has no value, unless we know what we are measur-
ing and are measuring for attainment of the same goals we
hope to achieve. It is my belief that we will actually
never teach for the objectives we adopt until accurate
means are developed for measuring pupil achievement along
the lines of these same objectives. Unless we can test
for these, we can never be certain of what we are accom-
plishing. Unless we can test for these, the students
will never actually make the stated objectives their
objectives. Until then they will be hollow phrases
incapable of realization.

Therefore, in order to have effective teaching of
geometry in the classroom we must

1. carefully select the objectives for which we

are teaching,

2. analyze these objectives in terms of desired

changes in student behavior, and

5. devise means of testing each of these objectives.

1W. I. McCaII. How to Measure in Educatigp.. The
.Mac.Millan Company, New York, 1923. Pp. 91.

HISTORY OF OBJECTIVES

From the very beginning two great aims have stood
out in the teaching of geometry--the teaching of geometry
for knowledge and the teaching of geometry as a way of
thinking.

The conditions of our world led man to an early
discovery of the common geometric relationships. To
survey his fields he needed a method of laying off a
perpendicular. To buy and sell land, he must know of
area. In the exchange of agricultural products, a knowl-
edge of volume became necessary. To navigate the sea
successfully, he learned properties of plane figures and
spherical surfaces. The early Egyptians found value in
geometry in the field of mensuration. The Nile overflowed
its banks each year and surveying was necessary to reestab-
lish boundary lines. In the construction of the pyramids
and temples geometry proved its usefulness in architecture.
In Babylonia also, practical geometry was developed at an
early date, and was studied for its usefulness in the
practical affairs of the ancient world.

Demonstrative geometry, as a logical system of
reasoning, developed in Greece. Here geometry was studied
for its own sake, and valued for the beauty and perfection
of deductive reasoning. It was considered as having ‘

value for "the training of the mind and the cultivation

of the soul."2 Hence, among the Greeks we may say that
after its early development, geometry was studied as a
form of logic or method of reasoning rather than for its
practical use.

The Romans were chiefly interested in geometry for
the practical applications. But some leaders believed it

beneficial in the training of an orator.
The Arabs studied geometry for its practical applica-

tions to astronomy and surveying. They also considered
it an aid to the study of algebra and physics.

The medieval church schools and universities did not
consider geometry very important, and so taught it merely
for the purpose of mensuration and surveying. But with
the coming of scholasticism, geometry was again studied
to sharpen the logical faculties of the mind. With the
rise of modern European countries, geometry was taught
chiefly for its disciplinary values with some emphasis
upon practical applications.

By the eighteenth century geometry was taught to
prepare students for the technical colleges and univer-

sities where geometry was needed for the study of advanced

mathematics and science.
In Great Britain and the United States the disciplinary -

function of mathematics was dominant until the beginning of

 

2
J. Shibbi. Recent Develo ments in the Teachi of
Geometry. IMaple Press 50mpany, York, I952. P5. §6§-252,

the twentieth century. In 1877 G. A. Wentworth declared,
"Mathematical training is not so much the attainment of

information as the discipline of the mental faculties."3

In 1893 the following statement appeared in the report of
the committee of ten: "Geometry possesses remarkable quali-
fications for quickening and developing creative talent.
Whatever the training (in independent geometrical work)

may accomplish for him geometrically, there is no student
whom it will not brighten and strengthen intellectually as
few exercises can."3 In the latter half of the nineteenth

century the secondary schools of both Great Britain and

the United States taught geometry to prepare pupils for
college and for the study of science and advanced mathe-
matics.

Since the beginning of the present century the aims
and values of the teaching of geometry have been under
almost constant discussion, and repeated revision. They
have been tuned to the spirit of the times and harmonized
with the new points of view of education. The statement
of these objectives has changed from the passive to the
active voice. And the student is no longer thought of as
a piece of clay which we must mold into the desired form,
but as an active participant thinking and.planning for

himself. Mathematicians have been awakened from their

 

.....15*
J. Shibbi. Recent Developments in the Teaching of

Geometry. ‘Maple Press Oompany,vYork, 1952. Pp. 203-204.

thoughtless acceptance of the geometric objectives of
past ages, and have been searching anew for the real
objectives of teaching geometry in the world of today.

In 1912 came the report of the Committee of Fifteen
emphasizing the need for the reconsideration of our objec-
tives and stressing such points as the function concept,
space intuition, accuracy and precision of thought and
expression, logic in the everyday affairs of life, per-
sonal pleasure, and an appreciation of and control over
forms existing in the material world.

In 1923 came a report from the National Committee
on.Mathematical Requirements -- a report revolutionary
in the field of the teaching of mathematics, entitled,
"The Reorganization of’Mathematics in Secondary Education."
True--the report followed the traditional division of the
aims of mathematics into three general classes -- practical,
disciplinary, and cultural. But there was a vitalizing
and broadening in the defining of these aims. There was
a shifting of the emphasis, a fresh appraisal of the
relative importance that was far from traditional. And
above all there was the daring spirit of the committeemen
who challenged many of the tenets of the past, and blazed
the trail to the future. These men threw out a daring
statement for 1923 when they said that the practical aims
may without danger be given secondary position in seeking
to formulate the general point of view which should govern

the teacher. They declared that the primary purposes of

the teaching of mathematics should be to develop those -
powers of understanding and of analyzing relations of
quantity and of space which are necessary to an insight

in and control over our environment and an appreciation
of the prOgress of civilization in its various aspects,
and to develop those habits of thought and of action which

will make these powers effective in the life of the indi-
vidual. Furthermore, they stated that all tOpics, processes,

and drills which do not directly contribute to the develop-
ment of these powers should be eliminated. There is dynamite
in that statement even today after years have elapsed.

The fifteenth yearbook of the National Council of
Teachers of Mathematics entitled, "The Place of Mathematics
in Secondary Education," restated the general philosophy

and outlook of the 1925 report. The objectives were devel-

oped with special care. First there was consideration of

the objectives of education as a whole; then a statement
of the particular objectives of secondary education.
These the committee classified as follows:

1. Ability to think clearly.
a. gathering and organizing data
b. representing data
0. drawing conclusions
d. establishing and judging claims of proof
2. Ability to use information, concepts, and general
principles.
5. Ability to use fundamental skills.
4. Desirable attitudes.
a. respect for knowledge
b. respect for good workmanship
o. respect for understanding
d. social-mindedness
e. open-mindedness
5. Interests and appreciations.
6. All other objectives.

Finally the committee attempted to show how the teaching
of mathematics fitted into this general pattern and con-

tributed to each of these aims.

In 1940 the committee on the Function of’Mathematics
in General Education (a committee of the Progressive Edu-
cation Association) upheld a different point of view in
their report, "Mathematics in General Education." Instead
of trying to fit mathematics into each of the objectives
of general education, this committee spoke of the special
role of mathematics, and believed that mathematics was most
valuable because of the unique contribution it had to offer.
Their emphasis was upon mathematics as a means of developing
techniques of reflective thinking. They called attention
to the fact that with specialization had cOme an increasing
mathematization of all fields, and believed emphasis should
be placed upon those concepts basic to problem solving
which are crucial in democracy, and pervasive in mathematics.

In "Nature of Proof" Harold P. Fawcett declared that
demonstrative geometry was no longer justified on the
grounds that it was necessary in order to give the student
knowledge of geometrical facts, as all of these can be
gained through an informal approach to geometry based
upon observation, measurement, and construction. He
believed the most important values to be derived from
the study of geometry were an understanding of the nature
of proof, a familiarity with postulational thinking, and
an ability to use it not only in geometry but in every
field of thought.

PROPOSED OBJECTIVES

Today the objectives advanced for the teaching of
geometry are numerous. Between different groups there
Sis much disagreement as to these objectives, and as to
their relative importance. Thus I have found my attempt
to list these objectives an almost impossible task. I
have limited this more intensive study to the last twenty-
five years. I have secured these objectives from many
sources of which the most fruitful have been professional
magazines, yearbooks of educational groups, geometry text-
books, and professional books on the teaching of mathematics.
I have attempted to avoid repetition of the same objective
expressed in different forms, and have taken the liberty
of rewording them into brief compact statements. The list
of these objectives follows.

1. To familiarize students with the great basic

propositions.
2. To familiarize students with properties of common
geometric forms.

3. To teach model proofs and develop memory.

4. To discipline the mental faculties.

5. To learn the facts of mensuration.

6. To secure a thorough understanding of measurement.

7. To develop a sense of form.

8. To exercise powers of Spatial imagination.

9.'To sense relationships and dependencies.

10. To develop functional thinking.

ll.
12.

14.
15.
16.
17.

18.

19.

20.
21.
22.

23.
24.
25.
26.

27.
28.

10

To prepare students for college.

To prepare students for work in advanced mathe-
matics and science.

To develop proficiency in the use of the straight-
edge and compass.

To learn useful constructions.

To draw artistic designs.

To develop clear and accurate expression. _

To give knowledge of the historical deve10pment

of mathematics.

To appreciate the power of geometry in the deve10p-
ment of civilization.

To appreciate beauty in the geometric forms of
nature, art, and industry.

To think naturally, logically, and systematically.
To learn the form of rigorous deductive reasoning.»
To withhold judgment until sufficient evidence is
given. -

To check authority and verify results.

To analyze complex situations into simpler parts.
To develop an inquiring or questioning type of mind.
To see the need for assumptions, definitions, and
undefined terms behind every body-of logic.

To cultivate the habit of independent investigation.
To make the student critical of his own and other's

reasoning.

29.

31.
-LA‘32.
53.
54.

55.
36.

57.

as.
59.
4o.
41.
42.

11
To show usefulness of system and order as help in
memory and understanding.
To develop urge to check and prove everything.
To follow directions exactly.
To teach respect for good workmanship.
To develop persistence in the face of difficulty.
To deveIOp moral and spiritual ideals as reverence
and honesty.
To make pupils more Open-minded.
To teach neatness.
To increase unwillingness to let a mistake go
uncorrected.
To teach the student how to study.
To become aware of and insist upon precision.
To learn the practical applications of geometry.
To provide pleasure.

To develop new types of recreational activities.

12

ANALYSIS

In studying and analyzing these objectives there are
several facts which may be noted. First of all, there are
some few of these objectives with which I heartily dis-
agree. For example, objective number three, to teach
model proofs and develop memory, is contrary to the nature
of geometry. Geometry is a creative subject -- not one
built upon memory and blind following of directions.
Others of the objectives, as number thirty-two, to teach
respect for good workmanship, and number thirty-six, to
teach neatness, certainly represent desirable traits.
But these are general objectives of education which
apply to all subjects in the curriculum. Geometry should
make its contribution. But these objectives in themselves
are surely not the important ones for teaching geometry as
all could be acquired by some other means. Other objectives
'listed,as number fifteen, to draw artistic designs, seem
to me to be merely by-products of the course, and not
objectives in themselves. But most confusing of all is
the large number of objectives. No teacher can success-
fully teach for forty-two objectives. For efficiency we
must select a small group of vital objectives and

concentrate on these.

13

FOUR OBJECTIVES PROPOSED

Which objectives are then of the most vital importance
in the teaching of geometry? I would like to present four
objectives and to defend my choice by telling why I con-
sider these of the greatest importance.

First, through the study of geometry the student
should learn to think critically. This objective is
important both to the individual from a personal view-
point, and to our democratic state whose judgments and
decisions are the judgments and decisions of its citizens.

It is valuable for any individual to be able to
analyze his life problems, and arrive at a correct solu-
tion. It is important to know when somebody else is using
incorrect reasoning and arriving at a false conclusion.
Political parties, advertisers, pressure groups, salesmen,
religious groups, and labor unions, all at times advance
propaganda and attempt to do the individual's thinking for
him; or perhaps to indoctrinate the individual without
allowing him to think at all. As civilization becomes
more and more complicated, it is increasingly important
that the individual, for his own protection and happiness,
be able to think critically.

Since our government is a democracy, its nobility,
its.success, and its efficiency depend upon the reasoning
ability of its citizens. The founders of our republic
were aware of the relationship between education and

democracy, for good citizenship is not justa matter of

l4
obeying the law, but thinking critically about political

issues, and contributing to the solution of social problems.

Perhaps in war time, we realize more than ever that
the strength of a democracy depends upon its individuals.
A good citizen must be critical of sources of information
of news. He must be on his guard against snap judgments.
He must think logically and not be a target for prOpaganda
advanced by subversive groups.

Morris Raphail Cohen declares that, "Surely it is
good for a democracy that its citizens shall constantly
ask their governors to give a rational account of their
stewardship, that men and women shall demand that new
proposals as well as the established institutions should
defend themselves before the bar of Reason."4

A second objective for the study of geometry is to
develop the ability and desire to use language clearly
and precisely. Us cannot discuss any tepic intelligently
unless definitions for all important terms are agreed upon.
we must appreciate the necessity of using words exactly
to present our own thoughts clearly. We must be able
readily to understand the written and spoken language of
others. For careful thinking in any field of activity

involves using language clearly and precisely. And

 

4

Edward.m, Glaser. An Experiment in the Development
of Critical Thinki . Bureau.ofrPublications,‘Teachers
Coilege, Columbia gniversity, 1941. Pg. 1.

15

nowhere is correct language more necessary than in
geometry. Every word must have a definite and specific
meaning. One word misused may destroy the entire meaning
of a prOposition or corollary. It is for this reason
that geometry has a unique function to fulfill in teach-
ing the use of language. ancett presents the hypothesis
that the language of mathematics as a medium of communi-
cation may be used as a means of improving all communica-
tion. Such fundamental concepts in geometry should be
studied as will help the student develop the ability and
the desire to express his own thoughts more clearly, and
to understand written and spoken language more readily.
In the third place, the study of geometry will enable
the student to appreciate better the world in which we
live. Geometry should be studied for its cultural value.
Centuries ago Plato said, "God eternally geometrizes."
Nature is filled with geometry. The bees may never have
proved that regular hexagcns fill a plane space without
overlapping or leaving space between them, but for can-
turies they have been making hexagonal honey cells.
Geometric forms are in evidence all about us. The sun
and moon and planets are spherical; and before life
existed on the earth, the bodies of the solar system
were whirling about the sun in ellipses. The trunks of
trees and stems of plants are circular, snowflakes are
hexagonal or triangular pattern, and crystals of various
substances take the form of hexagonal prisms, octahedrons,

and cubes.

16

The student should have an appreciation of what
geometry has done for our pleasure, comfort, convenience,
and progress. Geometry is used in design and decoration.
Many commercial articles such as wall paper, cloth, rugs,
vases, watches, curtains, bedspreads, compacts, furniture,
and clothing make use of geometric designs because of
their beauty and simplicity. dechanical drawing, art,
and surveying are all based upon geometry. Architecture
is a field most fertile, for geometry is involved in
every part of a building. The floor, the walls, the
windows, and the doors are all geometric forms. Triangles,
circles, hexagons, squares, and other geometric figures
are used freely in decoration of public buildings, churches,
and cathedrals. Modern transportation, bridges, tunnels,
and irrigation reservoirs would be impossible without
geometric calculations. The applications of geometry
are increasing day by day.

In order to understand and appreciate this great
world in which we live, a knowledge of geometry is essen-
tial. It should be taught so as to show the student how
basic it is in comprehending certain elements in the civi-
lization they are to share.

And finally, geometry should be taught so as to put
new meaning into arithmetic and algebra, and to lay a
foundation for advanced mathematics, the sciences, and
specialized vocational training. I feel this objective
has probably been overemphasized in the past in that it

17

was considered the central objective and crowded the
others out. However, I feel that it has a vital place
among the objectives, and am somewhat concerned that in
many modern discussions this objective has been entirely
omitted. Surely geometry should put additional meaning
into the other branches of mathematics already studied.
Surely we should familiarize students with facts and
principles about form and space, and the great basic
propositions needed as a foundation for a study of
advanced mathematics, the sciences, and Specialized
vocational training. The geometry teacher has an

obligation here which should not be overlooked.

In
for the
1.
2.

3.
4.

18
SUHHARY'OF PROPOSED OBJECTIVES
summarizing, the chief objectives I am proposing
teaching of geometry are to help the student
to think critically,
to develop the ability and desire to use language

clearly and precisely,

to appreciate better the world in which we live, and
to put new meaning into arithmetic and algebra,

and to lay a foundation for advanced mathematics,

the sciences, and specialized vocational training.

 

It‘llilltlf .

19'

MEANS OF ACHIEVING T ESE PROPOSED OBJECTIVES

Deciding upon the four major objectives for the
teaching of geometry should not be the end, but the

beginning. Often in the past we have stOpped here,
believing that somehow the objectives would work them-
selves out if we but agreed upon them. Objectives to be
effective cannot be stated and then forgotten, but must
be kept constantly in mind by the administrators, teachers,
and students. A

The students should be active participants in the
plan. This implies a great deal more than being told a
few times through the year of the valuable training they
are getting from the course. Is it too much to assume
that the same students whom we believe are capable of
understanding plane geometry, are also capable of under-'
standing the goals of the course and helping plan means
for attaining these and evaluating their progress toward
achieving them? I believe the students should be let in
on the whole plan. Each individual student should know
the values he may hope to achieve from the course. As
different parts of the work are undertaken, he should
realize to which objectives of the course they are con.
tributing. He should be urged to help plan.ways of making
the course more efficient toward achieving these ends.
Furthermore, he is entitled to know from time to time the

progress he is making along each of these objectives. For

20

only when the student becomes an enlightened participant,
in place of a blind worker, can we hope to attain maximum
achievement.

How are we to teach so as to attain these objectives?
A whole book might be written simply outlining plans and
methods to be followed in the classroom. But if a teacher
followed the book carefully, would he then be sure that
his teaching was actually contributing to these objec-
tives? Could he then feel satisfied with the progress
he was making?

I strongly believe that real progress cannot come
until we find means of testing each one of the objectives.
The teacher and the students must be able to measure from
time to time the growth of achievement along the definite
goals for which they are working. ‘Only then can they be
sure of what they are really accomplishing. And so I
believe that finding accurate means of measuring pupil
achievement along the lines of the four proposed objec-
tives is absolutely essential if they are to be of any
value. I believe that the forming of such tests is the
most important single thing.which can be done toward
bringing about their achievement. Material to be covered,
class procedure, methods, and plans, important as they
are, will more or less take care of themselves once we

clearly see our goal, and can measure our progress toward
it.

21

Having decided then that we must devise tests for
our objectives, let us think once more of our four
proposed objectives, which were:--to help the student

1. to think critically,

2. to develop the ability and desire to use language

clearly and precisely,

3. to appreciate better the world in which we live, and

4. to put new meaning into arithmetic and algebra,

and to lay a foundation for advanced mathematics,

the sciences, and specialized vocational training.
Looking at them from this viewpoint, we immediately see
that they are stated in rather general terms. But we
cannot test for these until we have broken them down into
concrete changes in pupil behavior. For each objective
let us ask ourselves, "What changes should this objective
bring about in the student? In what ways will he react
differently at the end of the course than at the beginning?"
Only when thus broken down in terms of pupil behavior can
we test for an objective.

So the remainder of this paper will consist of two
parts-~a careful analysis of each objective in terms of
student behavior, and suggested tests for measuring these
changes. Because of the necessary limit of the scope of
this paper, and because the traditional tests have measured
the fourth objective better than the other objectives, this
detailed study will be limited to the first three of the

proposed objectives. Therefore the tests which follow

22

must not be considered as a complete appraisal of
geometry, but merely as tests of the first three
objectives. The fourth objective is important and
tests for this objective must be included in any well
balanced program of testing.

23

BEHAVIOR PATTERN FOR OBJECTIVE ONE
TO THIEK CRITICALLY

What does this general objective mean, interpreted

concretely in terms of pupil behavior? What changes will

there be in the student because he has taken plane geometry?

At the end of the course he should be able:

1.
2.
3.

49
5.

6o
7.

80

9o
10.

11.

to recognize problems in life situations,

to find the hidden assumptions in arguments,

to distinguish between important and irrelevant
factors in any situation,

to recognize dependencies and relationships,

to associate general statements with specific
instances,

to understand the nature of proof,

to distinguish between correct and incorrect
reasoning,

to draw conclusions and identify the types of
thinking used,

to be consistent in his thinking,

to examine statements critically to see if argu-
ments are sound and based on accepted facts, and
to have a scientific system for solving his own

problems.

1.

2.

24

UNIT ONE
TO RECOGNIZE PROBLEMS IN LIFE SITUATIOHS

plgECTIOUS. Below are given facts from certain
real life situations. What problems do these facts

present to you? Make a very careful exact statement
of each of the problems involved.
A certain high school now has a student body of about

800 members, and still continues the traditional stu-

dent council of 12 members. These representatives
consist of the presidents of each class, and two
representatives elected by each class. There are ‘
approximately 125 Seniors, 150 Juniors, 225 sophomores,
and 300 freshmen. There are 24 homerooms. The student
council is taking over more and more responsibility.

What problems are involved?

In an industrial area a certain high school building
was built for a maximum of 500 students. There are a
gymnasium, a library, 16 classrooms, and 250 lockers.
There are now over 800 students enrolled, and 26

teachers employed. What problems are involved?

25

5. The Junior class of a certain high school consists of
150 members. They will present their class play
April 13th and 14th. They are planning their J-Hop
for April 28th. What mathematical problems are involved?

4. The school board of a certain high school recently
ruled that its athletic program must be self supporting.

What are the mathematical problems involved?

5. The student council of a certain high school wishes to

sponsor a magazine sale to raise funds for the purchase
of shades for the windows in the gymnasium so that
movies can be shown during the daytime. What mathe-

matical problems are involved?

26

6. Mr. John T. Siebert has decided to paint his house this

7.

8.

spring. What are the mathematical problems involved?

my. Henry Thompson runs a small commercial cannery. He
wishes to place his order now for the paper which will
be required for labels for the coming season. What is

the mathematical problem involved?

One of the requirements for a certain rank in the
Girl Scouts used to be to make an accurate construc-
tion of the American flag. What are the mathematical

problems involved?

27

9. Our government recently constructed a large office
building in Washington. They wished to cut the cost
of a square foot of space to a minimum. They also
wished the distance between the offices to be such
as would provide the swiftest passage between them.
For engineering reasons straight walls were desired.

What mathematical problems were involved?

28

UNIT TWO

TO FIND THE HIDDFW ASSUKPTIOUS IN ARGUMEHTS

DIRECTIONS. Back of the statement in each of the

 

following exercises there are hidden assumptions. In
some cases there may be only one, in others there are
several. State clearly in the space below the assump-
tion or assumptions upon which each exercise is based.

1.

2.

5.

4.

5.

Lines AB and CD will not meet no matter how far they

are produced. Therefore the lines are parallel.

If two parallel lines are cut by a transversal, the

interior angles on the same side of transversal are

supplementary but not equal.

In a triangle the sum of the squares of two sides is

equal to the square of the third.

The sum of the interior angles of a pentagon is 540

degrees.

Each interior angle of a convex hexagon contains 120

degrees.

29

 

 

E.
6. 79 M/ .8 EF is a transversal cutting
C. Aé/// 0 lines AB and CD at M and N.
/ Therefore LBUM -...- LDMF.
F'
‘7. 8 Given'b ABC with the bisector of L. B

meeting A0 at D. We can prove that
bAABD 32’ BBC because two right tri-

 

angles are congruent if a leg and an

acute angle of one is equal to a leg and an acute angle

of the other.

8. Eventually, why not now? Gold Medal Flour.

9. We buy only furniture manufactured in Grand Rapids.

 

10‘ REDUCE 3t05PoundsaWeek
Yet EAT New!

30

11.

. BehimLY

WHAT IS SHE CALLING YOU-

 

12.

15.

GET RID OF FAT...PITCH IN, HELP WINI

  

'_.".... '
'Og"¥. :::;‘;'

overs

 

"-‘OQ‘.’O
‘wtfihcnsg

‘WA 6 <
@% W /@%M (95%; /

‘She rain ’I ivp’Ils Eta,
in; {M st‘ldrsinsher;
’ i , mammar-

LEE.»-

51

 

.0. '
"¢Is:,a.
’p

 

 

 

 

 

 

-~'- -.-. I,"
_.‘ ,
0‘
,5 .
‘ /
' .
D A u
t
.O
f
U.-
D
r-v‘.
..s
‘I
.--.".
|_.'.o
_r:,.'
‘ J. 2,.
v... . 9
.7 r3. ,. Y. ’. '.'
-" . ' ' L " A;
' ' l“;‘ .
...... J,;.’.r ‘ Titeob §$§;.

 

Ms; B. Is slim. She works all day, is s .4 Mrs. 3. Is "tion.-{She JWIYIMIQ‘T‘: H.

Nurse.’ s Aid‘e st 'n'iint‘. Like many smart handy. f0"; Iiét“ f‘unjly'. ‘Slie dknowsiahu‘

'wom‘eni she keeps in trim and stays slim' ’ ‘ tempting whoIe gain Bmdsviitfififis
'- (h: ‘emm‘e R’mp W‘, J, ‘1 . i ’ ~ ‘ n‘ . ' ‘ :rye “IVOE'HIMYQ We,“ m§u€ an 5:

firm] RnKtispIréduang plan tLEéL}. ' iésfiooflférthem,‘soom’t’yr : H;
cqmesincnspncsdy -t'to-es I'm ’7

  

I

vmslly overweight: Ad ees Ry-Krisp, 21

Chsglfietboufl SWe,St.Louis Mb: -' ., ~ to serve-sscrsckeri‘t’ofissfi 532;}.
.0-‘.-,. ,.._- .1‘ '..,_... _ _ “magnate o‘“. ..“
toteflilb~v.'.~.~_x.t -_ - avsvl.‘, NW 7""'- ‘i"‘ ’
;‘::'*'_""“ , 1- ~"‘TfT;‘:‘.‘_f:.3:21!) :r‘

52

UNIT THREE

TO DISTIHGUISH BETWEEN IMPORTANT AND IRRBLEVAHT FACTOR

DIRECTIOEfi.

IN ANY SITUATION

In each exercise there is a general

statement or question followed by four particular state-

ments. Part of these are important and part are irrele-

vant to the problem or question. If you feel a statement
is important, circle the "IMP"; if you feel the statement
is irrelevant, circle the "IRR."

1. AB bisects 1. CAD, and A0 : AD. Prove AACB E AABD.

mm mm 1.
IMP IRE 2.
IMP IRR z.
in IR 4.

C.

 

I)

If two sides and the included angle of one
triangle are respectively equal to two
sides and the included angle of another
triangle, the triangles are congruent.
Parts of a bisected angle are equal.
If two angles and the included side of one
triangle are respectively equal to two
angles and the included side of another
triangle, the triangles are congruent.

LEAD = 3'? degrees.

II. AB and BD are line segments which bisect each other
at C. Prove AB H DE.

IMP

IMP
IMP

IRR

IRR
IRR

l.

2.

 

 

.3 .5

Corresponding parts of congruent figures
are equal.
If two parallel lines are cut by a trans-

versal, the corresponding angles are equal.

5. Parts of a bisected line segment are equal.

4.

If two sides and the included angle of one
triangle are respectively equal to two
sides and the included angle of another

triangle, the triangles are congruent.

III. AB is a diameter of circle 0. ProveLABC = é-L.AOC

I‘D
“.4.

ms?

IMP

IRR

IRR

1.

2.

3.

4.

 

In any circle a central angle is equal numeri-
cally in degrees to its intercepted arc.

In any circle an inscribed angle is equal
numerically in degrees to fi-its intercepted
arc.

A0 = Z".

L.O : 50 degrees.

54

IV. AB and CD are diameters of circle 0. Prove that
quadrilateral ACBD is a parallelogram.

’9 CL

 

 

 

 

D E?

IMP IRR l. l—AOO is supplementary to L—DQA.

IMP IRR 2. L.A00 : L_DOB.

IMP IRR 3. If the diagonals of a quadrilateral bisect
each other, the figure is a parallelogram.

IMP IRR 4. A square is a parallelogram which has two

adjacent sides equal and one angle a right

angle.

V. FG, GD, and DE are tangent to circle 0 at F, C, and B
respectively. Prove that GD : FG + DE.

  

 

Ga
IMP IRR 1. Two tangents to a circle from a point

GD

outside a circle are equal.

IMP IRR. 2. An.angle formed by two tangents, intersecting
outside a circle, is equal numerically in

degrees to one-half the difference of the

intercepted arcs.

VI 0

VII.

55

IMP IRR 3. If equals are added to equals, the
results are equal.
IMP IRR 4. If equals are subtracted from equals,

the results are equal.

Should the Junior class sell Christmas cards this year?

IMP IRR l. In\past years at our high school money has
been.made through the sale of Christmas
cards.

IMP IRR 2. The Christmas cards come in boxes.

IMP IRR 8. A Christmas card salesman was at our
school last week.

IMP IRR 4. Due to the war emergency, the Juniors
should substitute a bond selling campaign
or some activity directly relating to the

war effort.

Should I complete high school?
IMP IRR 1. High school students advance further than

those without a high school education.
IMP IRR 2. Lloyd Smith graduated from high school

and everybody knows how dumb he is.

IMP IRR 3. My mother needs my help and I could get
a Job if I dropped out of school.

IMP IRR 4. I like our school yells.

36

VIII. How much money should I spend on clothes?
IMP IRR l. The cost of clothing has increased greatly
in the last two years.
IMP IRR 2. My Job requires me to make a good appear-
ance at all times.

IMP IRR 3. The new hats are certainly odd this season.

IMP IR 4. Our government is asking us to buy only

what is necessary.

57

UHIT FOUR
TO RECOGNIZE DEPENDENCIES AND RELATIONSHIPS
PART ONE

DIRECTIO§§. In each of the exercises below there are
two figures. If the figures are equal (but not congruent)
circle the "E", if the figures are similar (but not congruent)
circle the "S", if the figures are congruent circle the "C",

if the figures possess none of these relationships circle
the "N". Decide each case on the basis of the evidence

given rather than on the basis of the appearance of the
figures. (Note: A. will be used to indicate a right angle.)

/\

E S O N 1.

 

 

E S C H 2.

 

 

 

‘ u

E S C N 5.
5/!

[I

 

E S C N 4.

.I
5.
5"
0.
‘1’
3 "
:5 II
;\\\\\\\\;>5 ‘,

 

ESCNB.

 

 

38

90°

 

1+”

 

EBONY.

5|!

ESCNB.

 

3601519.

 

 

 

 

 

 

 

 

 

a"

 

 

 

 

 

39

 

 

 

 

 

3" ‘30

PART TWO
DIRECTIQES. In each exercise below underline the

answer which will make the statement correct.

1.

If the altitude of a triangle is doubled, the area is

(%, 8, 1/3, 4) times the area of the original triangle.

2.

3.

4.

5.

6.

7.

If AB equals B0 in triangle ABC, and other parts of the
triangle are allowed to vary, as.» A increases L-B will
(increase, decrease) and.a.c will (increase, decrease).
The value of an angle inscribed in a circle depends upon
(the size of the circle, its intercepted arc, the length
of its sides).

A straight line is related to a line segment as a circle
is to (a chord, a diameter, a radius, an are, a center).
Circle 0 has a radius of 5". If the radius is doubled,
the circumference of the second circle is (2, i, 4, %,
5, l/B, 25, l/25) times the circumference of original
circle.

If the radius of a circle with 3" radius is doubled,

the area of the second circle is (2, s, g», 1/3, 6, 1/6,
l/4, 4) times the area of the original circle.

If L, A and L- B are supplementary, as L. A increases 1., B

(increases, decreases).

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

40

The sum of the angles of any regular polygon depends on
(the size of the polygon, the number of sides, the radius
of the inscribed circle).

If in AABO, side AB is 5", AC is 6", and BC is 4", the
largest angle is (L.A,L-B,£_C) and the smallest angle
18 (LA, LB, (.0).

A straight angle is related to a right angle as a semi-
circle is to a (circle, arc, quadrant, circumference).
If each side of a square is doubled, the area of the
second square is (e, 2, q, 4, 1/8, 8) times the
original square.

Ifth andt—B are vertical angles, as..A increasesL.B

(increases, decreases).

If two lines are parallel to a third, they are (oblique,
parallel, perpendicular) to each other.

If in a given circle a chord increases in length, its
distance from the center (increases, decreases).

If the base and altitude of a triangle are both doubled,
the area of the second triangle is (é, %, 2, 5, 4, 8, 16)
times the area of the original triangle.

The area of a circle depends on its (radius, location,
chords, degrees).

If the central angle of a circle increases, its inter-
cepted arc (increases, decreases).

The radius of a circle is 4". If the radius is doubled,
the diameter of the new circle is (2, 4, g, %) times

the diameter of the original circle.

19.

20.

21.

22.

41

If two lines are perpendicular to a third line, they
are (oblique, parallel, perpendicular) to each other.
As the number of sides of a regular polygon increases,
the value of each interior angle (increases, decreases).
If the opposite angles of a quadrilateral are equal,

the opposite sides are (perpendicular, equal, parallel),
and the consecutive angles are (equal, complementary,

supplementary).

In A ABC, t- C is a right angle. As LA increases, 4.. B

(increases, decreases).

UNIT FIVE

DIRECTIONS.

 

match these.

PART ONE

If two straight
lines intersect
the'vertical

angles are equal.

Base angles of
an isosceles
triangle are

0 qual e

If two parallel 5.
lines are cut by
a transversal,
the alternate in- 4.
terior angles are

equal 0

Corresponding
parts of congru-
ent figures are

equal.

 

B
k.
a D

R
o
e
F

TO ASSOCIATE GENERAL STATEMENTS WITH SPECIFIC INSTANCES

In the left column are general statements;
in the right column are Specific instances.
In the blank at the left, place the number
corresponding to the statement in the right column.

Yen are to

AC bisects
LABD. There-
fore L.BAC a

LOAD.

LABC and 1- DBE
are right angles.
Therefore L.ABD
: L-CBEe

In the above figure of exer-

cise 2,

LABC :

 

. 6'
F< if
C’ D

z_DBE.

In the figures
at the left,L.C

45

. Complements of 5. 'AB and CD

the same or q are straight

equal angles N\\\\\\ lines inter-
w- ° 4 9
are equal. \\\\\\ secting at 0.
D

LAOC : LDOBe

 

 

 

 

 

____. If two angles 6. 7n 7,:
of one tri- . / /
angle are equal 72' "8 a. ‘ c D at u fiF
respectively to
two angles of
another triangle,
the third angles AC and DF are straight lines.
are equal. (—1 : 1.3. Therefore.L.2 = L.4.

___, Central angles 7. 3 If AB : BC.
which intercept L.A : 1.0.
equal arcs are 7; Ac,
equal.

___. Angles inscribed 8. B
in the same seg- ' ' E
ment of a circle 1) 4C; D AF
are equal.

If LA: in and 1.3:). E,
L. C :- ,__,F.

____. If two parallel
lines are cut 9. \\ a is parallel
by a transver- : A; to b.
sal, the alter- l--1 : L.2.

nate exterior

angles are equal.

___, Parts of a

bisected

angle are

equal.

Supplements
of the same
or equal
angles are

equal.

In the same
circle or
equal circles
inscribed
angles inter-
cepting the
same or equal

arcs are equal.

If two parallel
lines are cut
by a transver-
sal, the cer-
responding
angles are.

equal.

10.

11.

12.

13.

 

 

44

a is parallel
to b.

1.1 : (—20

a is parallel
to b.

L1: (.20

AB is equal
and parallel
to CD in
quadrilateral.
Therefore

LB:LDe

AB and CD

are equal
arcs of

circle 0.

TherefOre

..L1 = [’20

. Opposite angles 14,

of a parallelo-

gram are equal.

. All right 15.

angles are

 

45
In diagram

to the left
1: 20

AC is an

arc of a

equal. fl‘ ocircle.

ABC_ - ADC.

PART TWO

. At most American 1,

high schools stup
dents receive the

-same rights, privi-
leges, and honors 2.
regardless of
their race.

. Most of the
workers in war
plants have been. 5.
very well paid.

. The high school
youth of America
are doing their 4.
part toward win-

ning the war.

Wayne County is meeting its
quota of 6,500 pints of blood
per week to the Red Cross
Blood Bank.

It was unbelievable that

Mr. Gene Diedrich, Super-
intendent of the Lapeer
Methodist Sunday School,
should commit such a crime.
Of course Mr. Dwight Corbin,
being a patriotic man, does
not complain about his
income tax.

Leroy Bell, a negro student
at Williamsburg, is captain
of the debate team and star

forward on the basketball team.

The President of
the United States
deserves the re-
spect of the
American people.
A few Americans
complain bitterly
over certain
shortages in

. consumer goods.
The public ex-
pects good con-
duct of a church
worker.

Honesty is the
best policy.
Citizens on the
home front are
contributing
greatly to the
war effort.

A good citizen
is always able
to accept res-
ponsibility and
to contribute

his share.

5.

6.

7.

8.

46

According to Supt. m. L.
McCoy, "Wayne High School
students are annually
supplying more than one
million man hours, abso—
lutely Vital and essential
to the rapid winning of
the war."

Ho, David, you shouldn‘t
keep the purse you found
in the hall. Turn it in

to the high school principal.
George Bowers, worker at
Willow Run, has a good
income..

The local press chastised
the citizens for booing in
the local theater when the
president‘s picture flashed
on the screen. And this was
done even though it was a

campaign year and the paper
was of the opposite party.

9. Utica has passed a law

requiring all youth between
17 and 18 to be off the streets
before 10:00 p.m.

47

Many communities 10. Mrs. Ashley became very
have been passing angry when she could buy
laws hoping to no soap powder in Plymouth
curb Juvenile last Wednesday.

delinquency.

48

UNIT SIX
TO UNDERSTaND THE NATURE OF PROOF

DIRECTIONS. Answer each Question completely and

carefully in the space below.

1.

2.

3.

4.

5.

Is an assumption necessarily true? Explain.

State three geometric statements which are accepted

without proof.

State three geometric statements which are proven.

How might indirect reasoning be used to determine

what hour George has geometry?

Are the conclusions reached by indirect reasoning

always valid? Explain.

49

6. How might inductive reasoning be used to prove that
all members of the Oxford High School Band own their

instruments?

7. Are the conclusions reached by inductive reasoning

always valid? Explain.

8. How might deductive reasoning be used to prove that
John Thomas, a graduate of Dearbcrn High School,
completed a course in American history?

9. Are the conclusions reached by deductive reasoning
always valid? Explain.

10. How might analogy be used to prove that Dr. Irwin
Woodbury can help Mrs. Floyd Winter?

11. Are the conclusions reached by analogy always valid?
Explain.

12.

13.

50

State the converse: "If two angles are equal, their

complements are equal."

State the inverse: "If it is good weather, we will

have a picnic."

51

UNIT SEVEN

TO DISTINGUISH BETYEEN CORRECT AND INCORRECT REASONING

DIRECTIQNS. In each exercise you are to accept the

assumptions. You are not to decide whether the assumptions
are true, or whether the conclusions are true. Yen are
merely to decide whether the reasoning used is valid. If
the reasoning is valid, circle the "V"; if the reasoning

is not valid circle the "N". The conclusions are under-
lined to distinguish them from the assumptions.

V N 1. Only Boy Scouts marched in the parade. Dick

Jackson marched in the parade. Dicngacksgn

 

was a qufiScout.

V N 2. "Biz" Trout is pitching at Brigg‘s Stadium today.
I saw "Diz" Trout today. Therefore I was at
31153' s St adium.

V N 3. Geometry classes are held in rooms 111 and 114.
John has a class in 114. Therefore John takes
geometry.

V N 4. A state capitol should be located in the central
part of a state. Boston is the capitol of
Massachusetts. Therefore Boston is located in
the central part of the state.

V N 6. Dresses of good quality are expensive. This
dress is expensive. Therefore this dress is
of_goodgqpali:y.

V N 6. Rochester is north of Detroit. Oxford is north
of Rochester. Therefore Oxford is_northfcf Detroit.

V N 7. If a man lives in Aster, he lives in Stithero.

If a man lives in Stithero, he lives_in Aster.

N

8.

9.

10.

11.

12.

13.

14.

15.

52

If a man lives in Aster, he lives in Stithero.
If a man does net live in Aster, he does not
live in Stitherg.

If a man lives in Astar, he lives in Stithero.
If a man_does not livg in Stithero, he does not
live in Astgg.

If one has an Elgin watch, he has a good time

piece. John's watch is not an Elgin. Therefdge

 

John's watch is not a good timepiece.

 

All classes in 208 are Chemistry classes. George
Cushing has a class in 208. George Cushing_is
enrolled in Chgmistgg.

 

There are five members in the Smith family. Mrs.'
Smith is 42 years old; Donald Smith is 10; Mar-
garet Smith is 16; and.Mr. Smith is 44. '33
conclude that_there is no_one in the Smith_
familygunder ninefyears of egg.

All high school graduates are well educated. 'All
well educated people speak good English. Therefore

 

all high school graduates speak good English.
Oxydol makes clothes whiter than any other soap

'powder. Mrs. Jones has the whitest clothes on

the street. Therefore Mrs. Jones uses Oxydgl.

There are three schools in Oxford. Washington

School was built in 1926. Monroe School was
built in 1931. Lincoln School was built in

1938. All schools in Oxford hays been built

in the last_twenty;years.

V N

is.

17.

18.

19.

53

Dick lives one block from Bob. Bob lives two

blocks from Jarvis. Therefore Dick lives three

 

blocks from Jarvis.

 

All non-resident students must pay a tuition to
attend Romeo High School. Julius Hunt is a non-
resident student of Romeo High School. Therefore

 

Juliungunt must pay tuition.

Jack Stover played football and broke his arm.
lng_playqfootbalqu will break mygarm.

The Morenci High School closes Friday, June 18th.
The Juniors and Seniors are having a picnic the
last Monday of school. The freshmen are having
their Picnic the last Tuesday of school, while
the sophomores are having their picnic the last

 

Wednesday of the school year. Therefore all thfl,
glasses_in our high school are having a picnic

the last;week of school.

 

V N 20. Margaret is older than Helen. Hazel is younger

than Helen. Geraldine is older than Margaret.

Geraldine is older_than Helen.

54

UNIT EIGHT
TO DRAW CONCLUSIONS AND IDENTIFY THE TYPES OF THINKING USED

DIRECTIONS. In each of the following exercises there
are a number of given facts from which you are to draw a

conclusion if possible. Underneath the exercise to the

left check the type of reasoning you used in reaching

your conclusion--inductive, deductive, or indirect. Then
write your conclusion in the space to the right. If no
conclusion is possible from the given facts, indicate
this by a check to the left and leave the space to the
right blank.

I. Students at Lenoy High School having a B average in
any subject, and having a 90% attendance in the class
are excused from the semester examination in that
class. Mildred received 3 A's and 2 B's for her
five monthly marks in geometry this semester.

Inductive Conclusion if any
Deductive

Indirect

No conclusion

II. There are no new teachers this year in the Jefferson

School at Roxenburg. There are no new teachers this
year in the Franklin School at Roxenburg. These are
the only two schools in Boxenburg.

Inductive Conclusion if any

Deductive

Indirect

___ No conclusion

III. Horace, a member of the Camera Club, is 15 years of ago.

Alice, a member of the Camera Club, is 14 years of age.

Benjamin, a member of the Camera Club, is 16 years of age.

55

Herbert, a member of the Camera Club, is 16 years of age.

Inductive Conclusion if any
Deductive
Indirect

No conclusion

IV. The Tigers defeated the Indians 8 to 5. The Indians

V.

VI.

defeated the Senators 5 to 2. The Tigers play the
Senators next week.

Inductive Conclusion if any
Deductive

Indirect

No Conclusion

If x = 2, y a 8. If’x : 3, y u 12. If x = 4, y = 16.
If’x - 5, y u 20. (Limit conclusion, if any, to these
four examples.)
___.Induotive Conclusion if any
____ Deductive
,___ Indirect
__ No conclusion
I am going to the "Sweater Swing" Friday. Which one of
my sweaters shall I wear--the white, green, red, or blue
one? The white one is soiled and it is too late to clean
it. The red has become stretched out of shape. The
green is the wrong shade to go with the plaid skirt I
am wearing.
.___ Inductive Conclusion if any
Deductive
Indirect

No conclusion

56

VII. Marjorie Phillips attends Sunday School in Roxenburg.

VIII.

There are three Sunday Schools. The Methodist Sunday
School is at 9:30; the Baptist at 10:00; and the
Congregational at 12:00. Only the Baptist and Congre-
gational churches are across the bridge on the south
side of town. Marjorie told me yesterday that she

was sometimes delayed by the traffic on the south side

bridge, but she was always home for a twelve o'clock

dinner.
___ Inductive Conclusion if any
____ Deductive
Indirect

____No conclusion
There are five boys in the Tiger Patrol-~Floyd, William,
Jack, Robert, and Preston. Floyd is a first class scout.
William is a first class scout. Preston is a first
class scout. Jack is a first class scout. Robert
is a first class scout.
____ Inductive . Conclusion if any
Deductive
Indirect

No conclusion

--—

IX. All Juniors and Seniors at Wayne High School are

required to take physical fitness. Marilyn Tucker
is a Junior at Wayne High School.
Inductive Conclusion if any

Deductive

57

Indirect

No conclusion

X. All the clerks at Kem's Hardware are paid 75 cents per

XI.

XII.

XIII.

hour. John Jones clerks at Ham's Hardware.
___ Inductive Conclusion if any
Deductive
Indirect
___ No conclusion
a and b are two straight lines lying in a plane.
a does not intersect b.
Inductive Conclusion if any
Deductive
Indirect

—*

___.No conclusion
No married women teach at Ahmes High School. Bernice
Henderson teaches at Ahmes High School.
Inductive Conclusion if any
Deductive
Indirect
____No conclusion
There are three Sunday Schools in Hoxenburg. The
Congregational Sunday School is holding its Christmas
program Friday evening, December 23rd. The Methodist
Sunday School is having a Christmas entertainment on
December 23rd at eight o'clock. The Baptist pregram

for Christmas is the Friday before Christmas at 7:30 p.m.

58

Inductive Conclusion if any

Deductive
Indirect

___.No conclusion
XIV. Gerald Bailey takes geometry second hour. There are
three geometry classes the second hour. George Thompson
says he is not in his class which is taught by Mr. Pen-
dill. Miss Norcross told me Gerald was not in her \
class. The other class is taught by Mr. Keith.
____Inductive Conclusion if any
____ Deductive
__ Indirect
___,No conclusion
XV. A pair of Gold Cross shoes costs $6.50. I have Just
spent $6.50.
____Inductive Conclusion if any
Deductive
____Indirect
___'No conclusion
XVI. Edward did not earn more money than Jim last week.
Jim did not earn more money than Edward last week.
____ Inductive Conclusion if any
Deductive
Indirect

No conclusion

59

XVII. Allan takes mechanical drawing. Yesterday I looked in
Mr. Kennedy's class and he was not there. I asked Mr.
Severance and he said Allan was not in any of his
classes. The only other mechanical drawing class
is taught by Mr. Ennis.

Inductive Conclusion if any
Deductive
Indirect
____No conclusion
XVIII. All Juniors at Wayne High School are required to take
American history. Lucille Wilson is a Junior at Wayne
High School.
____ Inductive Conclusion if any
___,Deductive
.___ Indirect
_____ No conclusion
XIX. The Freshman class of Jonesville High School gave $26
to the Red Cross. The Sophomore class gave $35 to the
Red Cross. The Junior and Senior classes each gave
350 to the Red Cross.
____ Inductive Conclusion if any
Deductive
Indirect

No conclusion

XX. All members of our football team are invited to a
banquet to be given by the Rotary Club. Tom Nyberg

is a member of our football team.

Inductive
Deductive
Indirect

No conclusion

Conclusion if any

60

61

UEIT NIKE
TO BE CONSISTEET IN HIS THIHKING

DIRECTIOHS. The way to score well on this test is to

be strictly honest. This is a free country and you have a
right to your Opinions. Each statement below is such that
some people honestly believe it is true, and others equally
sincere believe it false. If you believe a statement true,
circle the "T"; if you believe it false, circle the "F".

T F l.

Negroes should be prevented from migrating into
northern communities more rapidly than they can
be readily assimilated.

High school students should be allowed complete
freedom in choosing whatever subjects they be-
lieve will be the most beneficial.

or course, the allies should have bombed Rome.
It was a military target.

President Roosevelt should have a fourth term
if it is the will of the American people.
During war such frills as interscholastic high
school athletics should be abolished to make
room for more essential training, as physical
fitness and preinduction courses.

Americans should gladly share their food with
people of other countries who are harder hit

by the war than.we.

Since men are drafted into our armed forces, it
is only fair for labor also to be drafted.

Our young peoPIe should have the same right to

stay out evenings as the adults in the community.

'3

"El

*§

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

62

Rationing of some kind is the best method for
dividing scarce goods in war time.

Our school assembly programs should be continued
because they are cultural.

The allies should not have bombed Rome because of
its religious and cultural heritage.
Interscholastic high school sports should be con-
tinued during war time to keep up civilian morale.
A curfew should be established for all youths

between 10 and 18 to decrease Juvenile delinquency.

‘Our constitution should be amended to limit the

term of president to two terms.

Our school assemblies should be abolished for the
students are often restless and inattentive.
Certain subjects should be required of every high
school graduate.

Even in time of war, an American citizen should
have the right to work where and when he wishes.
Rationing should be abolished as it leads to

the black market and other corrupt practices.
Negroes should be permitted to live wherever
they wish.

Why should Uncle Sam be looked upon as Santa

Claus? We should feed our own civilians and

soldiers, and other countries should do the same.

63

UNIT TEN
TO EXAMIHE STATEMERTS CRITICALLY TO SEE IF ARGUMEDTS ARE
SOUND AND BASED ON ACCEPTED FACTS
DIRECTIONS. Below are samples of everyday reasoning.

Discuss. State whether or not the argument is correct.
Give reason for your conclusion. Make complete explanation.
1. "0h," exclaimed.Mrs. Davis, "I feel Just terrible: The

doctor said to take one of these tablets every two hours.

I'll take two every hour instead. I want to get well

as quickly as possible."

2. The Senior class of Lake Orion High School had a perfect
attendance last Thursday. Mary Henderson has been en-
rolled as a senior at Lake Orion High School for the
last three months; so she must have been in school

Thursday.

5. A patriotic lady writing to the Detroit News Letter Box
shortly after the rationing of canned goods complained,
"Surely the government is making a mistake in rationing

canned goods. How can I and my neighbors continue to give

such a large number of tin cans to the scrap metal drive?"

64

4. _

L , .
. ALBANY, N. Y.—(U.P.l—It was
Ibound to happen.

A bus driver, who stopped at the
end of the line before starting his
'next trip, stepped off the vehiele to
smoke a cigaret.

A woman and a man got. on the
,bus. Soon he heard an argument
in progress. I

“What is wrongr lady?" he asked.
I “I put three dimes in the coin
box and he put, in only one," she
.replied vehemently as she stared
at the man seated across the aisle.

“The fare," the driver explained,
“is one dime.”

“But," said the woman, “it clearly
says on the outside of the bus,
“Seats 30; stands 10.’ That means
if you want a seat it eosts 30 cents,
standing room 10 cents. And he,"
pointing her finger at the male pas-
senger, “is sitting."

“Lady,” explained the bus driver,
“that refers to the capacity of the
.bus, not the fare."

 

 

5. Students seldom do good work the day before the Christmas

holidays. Therefore school should be closed one day

earlier.

6. Brown's Variety Store started advertising four months
ago in the Radcliffe Gazette. Its sales this month are
20% higher than they were four months ago. Advertising
in the Radcliffe Gazette surely pays well.

65

7. Help your body to eliminate the natural way. Buy

Serutan. Read backwards, it spells "nature's".

8. "Oh, my dear," said Mrs. Grant to Mrs. Forbush, "we
must hurry and buy our winter's supply of soap before

the hoarders get it all."

9. I killed dear old Tabby last Wednesday evening. That
very night my brother died in Texas. I shall never
kill a pet again.

10. The following appeared as a part of an editorial
in "The State Journal" (Lansing newspaper) on July
6’ 1944.

An old game has been revived and is being
used to show that the war will end in 1944. You
take any or the principal war leaders, add his
birth year, his age, the year in which he took
office and the number of years in office and the .
. answer is 3888. Dividing the total byhtwo. the
answer is, 1944;.

66

11. John laid the four kings of a standard deck of playing

cards on the table before me. I picked up one-—it was
the king of spades. I turned over the second; it was
the king of diamonds. I looked at another; it was the

king of hearts. So without looking at the fourth card
I knew it was the king of clubs.

12. I know it to be a fact that Samuel Washington, a negro,
attacked a white woman. Therefore all negroes in our

community are bad citizens.

13. Mr. Cruden asserted, "Every good citizen should invest
at least 10% of his earnings in war bonds. I am a good
citizen, but I don't believe I'll need to invest much

money in war bonds because so many others are doing it.”

14. fields is a student at Plymouth High School. She is
.not a freshman. She is not a Senior. Therefore she

is a Junior.

67

15. Do not vote for Grimes for Senator. He is a Lutheran.

16. Peeple who doubt these statements are dumb. But you,
my friends, are intelligent so you will believe what

I have to say.

17. Shortly after the race riot in Detroit in the summer
of 1943 the following editorial appeared in the

Detroit News.

 

68

"r Truth often fails to catch up with falsehood,”
but it is our hope at least to give it a chance..
The News yesterday printed the sequel to this:
:story. The white rioter in the picture was:
found to be one George Miller, a characteré
with a police record including larceny and:
[counterfeiting He was arrested, identified by!
One of the two patrolman in the picture, tried
a f sentenced to 90 days for assault.
he patrolman, Paul Gyslvic of Central'
‘Station, explained why Miller was not arrested
at the time: Both officers were fully occupied
with assisting the Negro riot victim, who had.
been stabbed and would have fallen, had they
let go of him. ,
This unfortunate picture has contributed to‘
.‘an impression, especially outside Detroit, that'
police conduct in theirigts was marked; by,

partiality.
18. I know John Norris is a Boy Scout because he is twelve

 

 

years of age and Boy Scouts must be at least twelve

years old.

19. I had forgotten whether O .1f d orfr r, so for an
answer on my test I did not know whether to use 7"
or 14". I put down 7". When my paper was returned,
it was marked wrong. Therefore I know that 14" was

the correct answer.

20. When Mrs. Emery Tienken's neighbors have a party she
cannot sleep. She told me this morning that she slept
scarcely a wink all last night. Her neighbors must

have been having another one of their noisy parties.

69

UHIT ELEVEN

TO HAVE A SCIEHTIFIC SYSTEK FOR SOLVIKG HIS OWE PROBLERS

Below is an outline of steps for problem solving which

may be helpful to you.

1. State your problem.

2. Define necessary terms.

5. State any assumptions you use.

4. Collect data (facts, figures, details), read,
conduct interviews.

5. Investigate similarities to other problems.

6. Study all information secured.

7. Formulate possible solutions.

8. Give your conclusion and evidence to support it.

9. Test conclusion. -

I. For this exercise your problem is "The Attendance Problem

II.

in our Local Schools." State definitely how you would
study this problem in an effort to irprove conditions.

The outline above will be helpful. YOu do not actually
need to conduct the investigation, but indicate concretely
what you would do, and how you would do it. Place work
on another sheet.

While the preceding exercise was theory, this is to be
actual practice. Choose any local or school problem

(as recreation, curfew ruling, school assemblies, movies).
Do not merely tell what to do, but actually do it. The
above outline will help. Study the problem, and invas-
tigate it. Decide on the best solution. Hand in a
careful report stating step by step what you have done,
and include the actual definitions, assumptions, and

data. Give your conclusion and evidence to support it.

If possible, test solution; if not, indicate how it

could be done.

70

BEHAVIOR PATTERH FOR OBJECTIVE TWO

TO DEVELOP THE ABILITY AED DESIRE TO USE LANGUAGE CLEARLY

AND PRECISELY

Breaking this general objective into concrete changes

in.pupil behavior, at the end of a course in plane geometry

a student should be able:

1.

2.

5.

4.

5.

6.

7.

8.

9.

to select the significant words and phrases in
any statement that need defining,

to recognize the properties of a good definition,
to define words clearly,

to understand written and spoken language more
readily,

to be precise and exact in statements,

to give directions which can be easily followed,
to seek a genuine understanding of definitions
not possible by memorization,

to understand the importance of a single word in
making a correct statement, and

to appreciate the importance of using language
clearly and precisely.

71

UNIT TWELVE
TO SELECT THE SIGNIFICANT WORDS AND PHRASES IN ANY
STATEMENT THAT NEED DEFINING
DIRECTIONS. Underline the words in each statement
which need a particular definition to limit their meaning
in the given context. The last ten examples are actual
clippings from magazines and new3papers.

1. Tom and I have said goodbye to frills for the duration.

2. The student council of Lincoln High School adOpted the
following resolution: "Resolved that all students with
good standings should be excused from final examinations."

3. The diagonals of a regular pentagon divide each other
in extreme and.mean ratio.

4. waste can't be tolerated.

5. School letters should be awarded for outstanding

V achievement in school.

6. Two triangles are similar if their corresponding sides
are proportional.

7. Our high school gymnasium is used free by school groups,
community groups pay a fee, and other groups may not
use it at all.

8. On a recent ration list, sausage has the following point
values per pound-~dry sausage 9, semi-dry sausage 7,
group 1 fresh smoked and cooked sausage 6, group 2
fresh smoked and cooked sausage 5, group 5 fresh smoked
and cooked sausage 4, group 4 fresh cooked and smoked

sausage 3.

72

9. On the same ration list, processed butter was valued at

4 points per pound, and all other butter at 10 points

per pound.
10. The area of a semicircle on the hypotenuse of a right
triangle equals the sum of the areas of the semicircles

on the legs.

11,

 

12.

NCHANTING is the word for Carolee
Arnold! Whether she’s gracing a social
function in Washington, where her
father served in Congress for 8 years,
or whether she’s getting right down to
earth on one of the family’s mid-west farms—
her artless, chiseled beauty is captivating. Her
pale gold hair is like comsilk, her eyes azure
blue, her complexion wild-rose sweet!

 

13.

 
  

:Socwlogtsts Want to Know
i What Makes Zoot Suiter Zoot.

14.

 

‘75

15. amalgamaemaami
16.

.1. Ann You A Nansen?
Ln ‘7 . V , 7 ._ ,r “a, i. a

1‘7.

 

18.

DON’I,'T_R|-!ST CHEAP SUWESL

19.

20.

Fatima 6f" st. 'Clai,
Pledge Loyalty

 
 
      

Finer Pleasure—
PLUS
M Protection!

. - v-‘n. .,

T

n-I'.‘

;,

 

 

 

’74

75

UNIT THIRTEEN
TO RECOGNIZE THE PROPERTIES OF A GOOD DEFINITION

DIRECTIONS. A good definition should possess the
following propErties:
l. The term should be placed in its class. .
2. It should be distinguished from all other members
of its class.
5. The term should be defined in words simpler than
the term being defined.
4. The definition should be concise.
If the definition possesses these proPerties of a good
definition, circle the "+"; if it oes not possess these
properties, circle the "-". At the right side and follow-
ing each definition are numbers corresponding to the four
properties of a good definition as 118 ed above. In each
exercise in which ou circle the "-", also circle the
number (or numbers corresponding to the property (or
properties) of a good definition lacking in the given
exercise.

 

+ — 1. A triangle is a three-sided polygon.

l 2 5 4
+ - 2. Geometry is a subject taught in our high school.

1 2 5 4

+ - 3. Interest is where money is paid for the use of

money. 1 2 3 4
+ - 4. A stream of electrons passing through a conductor

is an electric current. 1 2 5 4
+ - 5. A parallelogram is a geometric figure which has

its opposite sides equal and parallel, its oppo-

site angles equal, and its consecutive angles

supplementary. l 2 3 5
+ — 6. A sentence is a group of words expressing a

complete thought. 1 2 5 4
t - 7. Democracy is a form of government.

1 2 Z 4

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

76

Similar triangles are triangles which are similar.

1 2 3 4
Circles in the same plane with the same center are
concentric circles.

l 2 5 4
A ballad is a form of poetry.

1 2 5 4
Centrifugal force is where all parts of a
whirling body go outward.

l 2 3 4
Cohesion is the sticking together of the same
kind of molecules.

1 2 5 4
A parallelogram is a geometric figure.

1 2 3 4
An adjective is a word which modifies a noun

or pronoun. 1 2 3 4

A triangle is a three sided polygon all of whose
sides are straight having three vertices and
three angles whose sum is 180 degrees.

, 1 2 5 4'
A straight line which intersects a circle in

two points is a accent.

I 2 5 4

A socialized recitation is a recitation which

is socialized.
l 2 3 4

+

+

+

- 18. A violin is a musical instrument.

1 2 3 4
- 19. An angle is when two lines meet at a point.
1 2 5 4
- 20. Friction is the resistance to movement of one
object upon the surface of another.

1 2 3 4

77

78

UNIT FOURTEEN
TO DEFIEE WORDS CLEARLY

DIRECTIONS. Certain words are underlined in each of

the followin statements. Define these words in any man-
ner that wil give the statements one definite meaning.
It is not necessary to recall the local situation or
remember anything you have read. There is more than one
possible answer in each case. But remember the words
must be defined so as to give each statement an exact
meaning.

I. Honor students at our school are excused from required

study halls.

1. Honor students

II. Extra gasoline should be given to all families not

 

within walkinggdistance of their victory gardens.
2. Walking distance

3. Victory gardens

III. According to the Wayne High School salary schedule,
teachers who are the sole support of a family receive
$200 extra each year.

4. Sole support of a family

IV.

V.

VI.

VII.

79

Criminals should lose their citizenship.
5. Criminals

The sophomoreg held a class meetigg Friday.

 

6. Sophomores

7. Class meeting

The Strand Theater ad reads, "Evening rates - adults
$.40; kiddies $.11."
80 Adlllts

9. Kiddies

Qualified members of thefootball team_will be
awarded with the school letter.
10. Qualified members of the football team

so

VIII. Last evening's headline in the Detroit News read,

"Roosevelt issues warning to neutrals."

 

11. Bentrals

IX. Each school club must have a snonsor.

M

12. School club

13. Sponsor

X. Mr. Charles Brown believes that the residences of
Negroes should be restricted to certain sections
of our city.

140 Negroes

81
UEIT FIFTEEE

TO UEDERSTAHD WEITTEI ADD SPOKEN ENGLISH MORE READILY

DIRECTIONS. Each of the following exercises involves

a construction. Follow the directions exactly and place
your construction in the space provided below.

 

I. 1. Draw a circle with center at O and any convenient

radius.

2. Draw any diameter AB.

5. Determine C, the midpoint of A0.

4. Determine D, the midpoint of B0.

5. With 0 as a center and 00 as a radius, draw a
semicircle‘OA.

6. With D as a center and CD as a radius, draw a
semicircle OB, such that semicircle DA and

semicircle OE are on opposite sides of AB.

II. 1. Draw a line segment KY.
2. With.X as center and 2" as radius, strike an arc
intersecting XY at Z.
5. With X as vertex and one side XZ construct an

angle of 45 degrees. Label the other side XT.

4.

5.

6.
7.

8.

III. 1.
2.
3.

4.

82

With X as vertex and one side XZ, construct another

angle of 45 degrees on the Opposite side of 22.
Label the other side XS.

With.X as center and a radius of 1%", strike an
arc intersecting KT at W.

Draw NZ.

With X as center and a radius of 1%", strike an
arc intersecting XS at V.

Draw VZ.

Construct a rectangle.
At one vertex place the tenth letter of the alphabet.
At a consecutive vertex place the fifth letter from

the end of the alphabet.
Place the letter K at the vertex which is opposite
the vertex designated in step 2. With K as center

and segment between K and vertex designated in step

3 as radius,construct a circle.

83

5. At the unlabeled Vertex draw three concentric circles
such that one circle intersects the rectangle at two
points, one circle intersects the rectangle at one
point, and the other circle intersects the rectangle

at no points.

84

UEIT SIXTEAT
TO BE PRECISE .ED EXACT E STATEMENTS

DIRECTIONS. After each number there are five suggested
ways of finishing the statement given. More than one may

be partly right. Yen are to select the BEST one. Draw a
ring around the letter to the right which corresponds to
the best answer.
1. An obtuse angle is

A) an angle which is not a right angle.

B) an angle greater than a right angle.

Cl an angle less than a straight angle.

D) an angle which is not acute.

E) an angle more than a right angle and less than a

straight angle. Answer.....A B C D E

2. An equilateral triangle does not have
A) three angles.

B) one obtuse angle.

0) three equal sides.

D) two acute angles.

E) three vertices.

Answer.....A B C D E
5. A tangent

A) touches a circle at one point.

B) is a straight line which intersects a circle at
at least one point.

C) is a line which intersects a circle at two and
only two points.

D) is a straight line which touches a circle at one

and only one point.

4.

5.

85

E) is a straight line passing through the center of a
circle.

Answer....A B C D E

Two triangles are not necessarily congruent if

A) three sides of one are reapectively equal to three
sides of the other.

B) two sides and the included angle of one are respec-
tively equal to two sides and the included angle
of the other.

C) two sides and an angle Of one are respectively
equal to two sides and an angle of the other.

D) two angles and a side of one are respectively equal
to two angles and a side of the other.

E) two angles and the included side Of one are respec-
tively equal to two angles and the included side of
the other.

Answer....A B C D E

A parallelogram is

A) a figure having opposite sides parallel.

B) a quadrilateral with all sides parallel.

C) a quadrilateral with all sides equal and the
Opposite sides parallel.

D) a quadrilateral with the Opposite sides parallel.

E) a quadrilateral having two sides parallel.

AnswereeeeA B C D E

6.

7.

8.

86

If two

A)

B)

c)

D)

E)

parallel lines are cut by a transversal, the alter-
nate interior angles are equal.

lines are out by a transversal the alternate interior
angles are equal.

lines are parallel, the alternate interior angles

are equal.

lines are cut by a transversal, the interior angles
are equal.

lines are cut by a transversal, the alternate angles
are equal.

Answer 0 e 0 0A B C D E

The locus Of points equidiStant from the extremities

of a line segment

A)
B)
c)
D)

E)

is perpendicular to the segment.

is parallel to the segment.

intersects the segment obliquely.

bisects the segment.

is the perpendicular bisector of the segment.
Answer....A B C D E

If two parallel lines are cut obliquely by a transversal,

two angles are not equal if

A)
B)

c)
D)

they are alternate interior angles.

they are interior angles on the same side of
transversal.

they are alternate exterior angles.

they are vertical angles.

87

E) they are corresponding angles.
Answer....A B C D E

9. An acute triangle has

A) all sides equal.

B) three acute angles.

Cl one obtuse and two acute angles.

D) one acute angle.

E) one right and two acute angles.

Answer....A B C D B
10. A rectangle is not always

A) a square.

B) a parallelogram.

C) a plane figure.

D) a polygon.

E) a quadrilateral.

Answer....A B C D E
11. A circle

A) is a closed curved line.

B) is a closed curve lying in a plane.

C) is a closed curved line which lies in a plane and ,
all points of which are equidistant from a point
within it called a center.

D) is a curved line, lying in a plane, all points
of which are equidistant from a fixed point.

E) is a closed line all points of which are equi-
distant from another point.

Answer. 0 0 0A B C D E

88

12. If two lines lie in the same plane

13.

14.

Al
Bl
C)
D)
E)

they are parallel.

they are parallel or perpendicular.

they are perpendicular.

they intersect.

they are parallel or they intersect.
Answer....A B C D E

A square is not a

A)
B)
c)
D)

In

a)

Bl

c)

D)

a)

polygon.
trapezoid.
rectangle.
quadrilateral.
parallelogram.
Answer....A B C D E
any right triangle
the hypotenuse is equal to the sum of the legs.

the square of the hypotenuse is equal to the sum
of the squares of the legs.

the square of the hypotenuse is equal to the
difference of the squares of the legs. _

the square of the hypotenuse is equal to the sum
of the logs.

the hypotenuse is equal to the difference of the

squares of the legs.

Answer....A B C D E

89

15. If a line segment

A)

B)

c)

D)

E)

intersects two sides of a triangle, it divides these
sides into segments which are proportional.

is parallel to one side of a triangle, it divides the
other two sides into segments which are proportional.
is parallel to one side of a triangle and intersects
the other two sides, it divides these sides into
segments which are proportional.

is parallel to one side of a triangle and intersects
the other two sides, it divides these sides into
equal segments.

is perpendicular to one side of a triangle and
intersects the other two sides, it divides these
sides into segments which are proportional.

Answer....A B C D E

16. A parallelogram does not necessarily have

17.

A)
B)
c)
D)

E)

An
A)

B)
c)
D)

Opposite angles equal.
opposite sides parallel.
opposite sides equal.
opposite sides equal and parallel.
diagonals equal.
Answer....A B C D E
exterior angle of a polygon is
any angle outside of the polygon.

formed.by two sides.
formed by two sides extended.

formed by a side of a polygon and an adjacent side

extended.

E)

18. An

90

formed by a side of a polygon and another side

extended.
Answer....A B C D E
isosceles triangle does not always have
three equal sides.
three vertices.
two equal angles.
two acute angles.
three angles.

Answer. 0 0 0A B C D E

19. A quadrilateral can never be a

A)
B)
C)
13)

.LJ

TP)

square.
rhombus.
hexagon.
parallelogram.
rectangle.
Answer....A B C D E

20. Adjacent angles are angles

A)
B)

c)

which have a common side.

which are next to each other.

which have a common vertex and a common side lying
between them.

whose sum is 90 degrees.

whose sum is 180 degrees.

AnswercoooA B C D E

91
UNIT SEVEHTEEH

TO GIVE DIRECTIONS WHICH CAN BE EASILY FOLLOWED

DIRECTIONS. This test consists of two exercises. A
simple construction is illustrated for each. Ybu are to
write the directions for each construction step by step
on a separate sheet of paper. Then hand each set of direc-
tions to two students--one who has never studied geometry,
and one who has studied geometry but is not in this class.
If they have trouble, revise your directions and try
again. Yen are to hand in your sheets of directions
and the constructions by the students you asked to
cooPerate with you.

 

1.
/j?771_\\\\\\
\\\\,

fl

V

\\K

\
\\

 

 

2.

 

 

 

 

 

92

UNIT EIGEEEEN
TO SEEK A GENUINE UNmrESTAImING OF DEFINITIONS ITOT POSSIBLE
BY mnaIZAEION
DIRECTIONS. If the statement is true, place a circle

around "T"; if the statement is false, place a circle
around "F".

T F 1. A square is always a rectangle.

T F 2. A rectangle is always a square.

T F 3. A rectangle is always a quadrilateral.
T F 4. A square is always a polygon.

T F 6. A parallelogram is always a polygon.

T F 6. A trapezoid is always a quadrilateral.
T F 7. A parallelogram is always a quadrilateral.
T F 8. A rhombus is always a parallelogram.

T F 9. A parallelogram is always a square.

T F 10. A square is always a quadrilateral.

T F 11. A parallelogram is always a rectangle.
T F 12. A square is always a parallelogram.

T F 15. A polygon is always a quadrilateral.

T F 14. A rhombus is always a quadrilateral.

T F 15. A quadrilateral is always a square.

T F 16. A quadrilateral is always a rectangle.
T F 17. A rhombus is always a rectangle.

T F 18. A trapezoid is always a parallelogram.
T F 19. A quadrilateral is always a polygon.

T F 20. A rectangle is always a parallelogram.

93
UNIT HIHETE

TO UNDERSTAND THE IMPORTANCE OF A SINGLE WORD IN MAKING A

CORRECT STATEI'A'EET

DIRECTIONS. Each of the following statements is

 

incorrect in the present form. But each statement can be
made correct byhthe insertion of a single word. Place a

"A

" between e two words where the inserted word belongs,

and write the required word directly above it.

1.

2.
3.

4.
5.

6.

7.

8.

9.

10.

11.

12.

15.

If two lines are out by a transversal, the corresponding
angles are equal.

Corresponding parts of triangles are equal.

The sum of the interior angles of any polygon of n

sides is (n - 2) straight angles.

The sides of a parallelogram are equal.

A central angle of a circle is equal numerically in
degrees to its arc.

Lines intercept proportional segments on all transversals.
The acute angles of a triangle are complementary.

If two straight lines intersect, the angles are equal.
If two lines are cut by a transversal so that a pair

of interior angles are equal, the lines are parallel.
The area of a triangle is equal to the product of its
base and altitude.

The sum of the angles of any convex polygon, formed by
extending the sides in succession equals 560 degrees.

A line perpendicular to a radius at its extremity is
tangent to the circle.

An angle in a circle is equal numerically in degrees

to one-half its intercepted arc.

94

14. An exterior angle of a triangle is greater than either

interior angle.

15. Every point on the bisector of a segment is equidistant
from the ends of the segment.

16. The angles of a parallelogram are supplementary.

95
UNIT TWENTY

TO APPRECIATE THE IMPORTANCE OF USING LANGUAGE CLEARLY

AND PRECISELY

DIRECTIONS. Answer each question carefully and

 

completely. Use your own ideas.

I. One of the objectives for the study of geometry is to

II.

III.

develop the ability and desire to use language clearly
and precisely. List fields of work and phases of living

-in which you think this objective particularly inportant.

Certain legislators of one of our states grew tired of

different approximations given pi and so prOposed a bill
which would set its value permanently at 5 1/7. What
is your reaction to this legislation?

"Juvenile delinquency in Stratsburg has increased in

the preportion of three to two in the past year,"
stated Judge Brown. What evidence do you find to

'show that he was unfamiliar with mathematical terms?

96

1v, The clipping at the left
'r 18 already apparent that the word is an excerpt from "The
“Fascist” will be one of the hardest-
worked words in the Presidential cam- ll 8V7 Yorker" . What Prob...

paign. Henry Wallace called some peo-
ple Fascists in a speech, and next day

up jumped Harrison Spangler, the Re- ' 19m dOeS th is pres ent?
publican, to remark that if there were

any Fascists in this country you would HOW 1 s it connect 65. with
find them in the New Deal’s palace

guard. It is getting so a Fascist is a man _
who_.votes the other way. Persons who the Obj GCtive we are c 011

vote your way, of course, continue to
be "right-minded people." ; m“ “3 sider 1118?

V. What properties do you think an income tax blank and

accompanying directions should possess?

VI. The following is a part of an article appearing in the
July, 1943, copy of "Common Sense". The article is

9‘7

entitled "What is a Puerto Rican?" and is simply signed

"Corporal." Read the following carefully so you.will

be able to comment on it.

I work in a filler replacement pool at a staging area
in southern Virginia. The pool consists of four com-
' panics. All Negroes are segregated into one company.

My job is classification; I am the non-com in charge of
A a section in the pool Adjutant’s Office, which is respon-

sible for classifying men according to the military duties
1 for which their background and training best qualify

them. As part of this work, we handle the Soldier’s Quali-

fication Card, his basic personnel record. Included on i

these cards is a notation of race: white, Negro, Indian,

Eskimo, and so forth. .

One day, while thumbing through the cards of the

N egroec in the colored company, I came across one on

which the notation was not “Negro” but “Puerto Rican."

Now, my dealings with Puerto Ricane have been

to my acquaintance with one of the Puerto Rican boys

in my own, white, company. My ethnic knowledge of

them was limited to a belief—apparently well-founded—
f that they are in varying degrees a mixture of white, In-
‘ dian, and Negro.

Anyway, I yanked this card from the file and showed
it to my Officer, the Adjutant of the pool, inquiring guile-
lessly whether a Puerto Rican was indeed a Negro. My
lieutenant, a liberal New York attorney, was perplexed.
Segregation is oflicially part of Army policy, but is a
Puerto Rican a Negro? He didn’t know, so very unhap-
pily, he called in the Executive Oflicer of the Pool. This
oficer, a cavalry lieutenant, was firm about his beliefs.
Whites must go with whites, Negroes with Negroes. “It’s
very simple," he said. But is a Puerto Rican a Negro?
The Executive Officer didn’t know. He called in the
pool Commander. The three officers discussed the mat-
ter and got nowhere. Finally, the Executive Officer de-
cided to rely on scientific authority. “Let’s call in the pool
surgeon,” he said. So the pool surgeon was summoned
and the delicate situation explained to him. He could give
them no scientific satisfaction, but he did make a sound
suggestion. “Let’s find out what the man looks like." A
capital idea. So they called in the Puerto Rican’s company .
commander. They explained the situation and asked,
"What does Private R . . . look like ?” The company
i commander thought a moment and said, “Well, he’s as
t dark as most of the other boys, but . . . ” a thought hit
T_Nm;flms@§3wfihMEummdl.

I-

 

98

By this time the officers had become disconcerted, and
aware of how foolish their predicament was. But the mat-
ter had to be settled. Finally, the Executive Ofiicer
thought of checking the man’s military record to find out
with what kind of outfits he had been connected. The
records showed that he had been inducted in Harlem and
sent to a Negro outfit. So that question was disposed of.

But one more question, this time from the pool Com-
mander. “Is he happy where he is ?” The company oom—
mander said that he believed so. “Well, then, leave him
where he is unless he complains.” ,

And so, on the side of the street where I live, a Puerto
Rican is classified as “white” and lives in our barracks.
On the other side of the street, where I work, a Puerto
Rican is classified as “N egro” and is assigned to a colored
company. One looks just as dark as the other.

What connection do you see between this article and the
objective we are considering? What difficulties does
it present? What point does it illustrate? Give any

personal comment on the article you care to make.

VII. What are the dangers which come from a careless use

of words?

99

VIII. At the 1942 Kiwanis Convention held in Saginaw,
Michigan, Father Cairnes of Monroe stated, "The
common denominator of religion is zero," Is the
mathematical use of terms in this statement

correct? Give an explanation of your answer.

100

BEHAVIOR PATTERN FOR OBJECTI n NUMBER THREE

TO APPRECIATE BETT‘; THE WORLD IE WHICH WE LIVE

Interpreting this objective concretely in terms of

pupil behavior, at the end of the course in plane geometry

the student should be expected:

10
2.

3.
4o

6.
7.

to possess a knowledge of the history of geometry,
to be alert to the geometric properties of objects
about him,

to be familiar with common geometric applications,

to recognize geometric forms wherever found,

to perform simple geometric constructions,

to create geometric designs for life situations, and

to realize how basic geometry really is to our

civilization.

101

UEIT TWEETYFONE
TO POSSESS A HE WLEDGE OF THE HISTORY OF GEOMETRY

DIRECTIOHS. In each statement there are one or more

 

sets offparentheses. In each set underline the word or
phrase which would make the statement true.

1.

2.

3.

4.
5.

6.

7.

8.

9.

The thinking of the Egyptians in geometry differed

from the Greeks in that the Egyptians did (more, less)
abstracting and generalizing.-

Pythagoras was a (Greek, Egyptian, Roman, Chinese)
philoSOpher who lived in the sixth century (A.D., 3.0.)
Non-Euclidean forms of geometry were advanced by (Thales,
Lobachewski, Bolyai, Plato, Hippocrates, Riemann).
Egyptian geometry was based on (experiment, reasoning).
During the Middle Ages the (deductive, inductive)
method of thinking flourished.

Geometry is called "the gift of the File" because

(of its many angles and curves, its mouth is triangular,
by overflowing its banks it forced man to find ways of
measuring land, all the great geometricians lived on
its banks.)

Over the door of (Gaues's, Plato's, Euclid's, Eudoxus')
academy was written, "Let no one destitute of geometry
enter here."

The word Geometry comes from two (Egyption, Greek,

Roman) words.

The meaning of these words was (logic and reason, to measure

the earth, to draw angles, shadows of the pyramids).

10.

11.

12.

15.

14.

15.

16.

17.

18.

19.

102

The Greeks (failed, succeeded) in improving upon the
Egyptian method of thinking.

Pythagoras became the central figure of a secret
association called (Pythagoreans, Geometers, Mathema-
ticians) whose emblem was (a parallelogram, a hexagon,
a six-pointed star).

(Riemann, Plato, Archimedes, Euclid) is reputed to have
said, "God eternally geometrizes."

(Plato, Euclid) was born before (Plato, Euclid).

The oldest record of man's development and use of
geometry is found in (The Elements, Rhind Papyrus,
Ahmes Papyrus, The Whetstone of Witte).

(Aristotle, Thales, Euclid, Herodotus) is called "the
father of Greek geometry" because he began to make of
geometry 8 deductive science.

(Thales, Plato, Euclid, Aristotle, Archimedes) was the
author of the "Elements".

The principal use made of geometry by the ancient
Egyptians was in the field of (law, scholastic training,
mensuration).

The three famous problems of antiquity were (squaring
the circle, trisecting the circle, cubing the circle,
duplication of cube, duplication of angle, bisection
of the angle, trisection of the angle.)

Euclidean geometry is (the only form, one of several

forms) of plane geometry.

105

20. Euclid's contribution to Geometry was largely

(discovery and original proof of propositions,

collection and organization of propositions).

104

UNIT TWENTYhTWO
TO BE ALERT TO THE GEOMETRIO PROPERTIES OF OBJECTS ABOUT HIM

DIRECTIONS. Answer each question in the blank ( )
provided below the question.

1. Below are six symmetrical objects. In each case the
object possesses point symmetry, line symmetry, or
both point and line symmetry. Place the type of

symmetry each possesses before the corresponding

number.

1?\8 \\
*Afi'. ”#2“ .\
~ . ”y? el". ‘ ’9-
71%? H”fn»ln 'flnrr' g
T«y§fifiiw€t «M? “if" is. .,

I
.‘n

. . “sign,” '~ I ‘L '-
AWF113y‘giafifo‘

 

105

Answer

 

 

 

 

O‘U‘fitflNH

 

II. List three capital letters which are good illustrations
of parallel lines cut by a transversal.

Answer 7

 

III. If a coin rolled along the edge of a rule, what is the

locus of the center of the coin?
Answer 10

IV. What is the path traced by the end of a minute hand
of a clock during an hour?p
Answer _il, i 11
V. What is the locus of a point on the handle of a
pencil sharpener?
Answer 12
VI. What vegetable when sliced illustrates concentric
circles?
Answer 13

VII. Give an example in nature of a hexagon.

Answer 14

VIII.

X.

XI.

XII.

XIII.

XIV.

106

What kind of an angle do the hands of a watch make
at 5:30?

 

 

Answer I 15
At 7:00?
Answer ,__16
, What frequently found on machines are good examples

of tangents?

Answer __ 17

In what sport does some of the equipment represent

concentric circles?

 

 

 

 

Answer __W W 18
What kind of lines do a pair of railroad tracks
illustrate?

Answer pfiw 19
Are all lines perpendicular to a given vertical line
horizontal?

Answer 20
Are all lines perpendicular to a given horizontal line
vertical? ‘

Answer _w21
What example in industry can you give of tangent

circles?

Answer 22

What kind of a line is formed when a piece of paper

is folded in a sharp crease?

Answer 23

107

UNIT TWENTY-Twins
TO BE FAMILIAR WITH COMMON GEOMETRIC APPLICATIONS
DIRECTIONS. Each example is a practical illustration
of the use of geometry. Fill in the blanks of each exercise.

1. A plan of a bedroom is shown below.

 

 

 

Fire.
D
e
_. e
’e
clout Door
) ....__J

 

 

Scale 1" 8'

The width of the room door is feet.
The width of the oldest door is feet.
The length of the entire room is feet.
2. A surveyor knows the longest side of a triangular
' lot lies Opposite the angle of the lot.
5. . If one point is directly
I. north of another point,

do they have the same
latitude?

 

If one point is directly

west of another point, do

 

they have the same latitude?

 

 

108

Does an arc degree of latitude vary in length in
different places on the earth? . Does an
arc degree of longitude? .

4. A ship in distress gives its position as 135°W. longi-
tude and 4Com, latitude. Place "x" for ship's position

on map above.

5. Two plane mirrors KY and

Y2 are set at right angles,
and the angle of incidence
( L-i) equals 35 degrees.
Then the value of the

angle of reflection (4. r)

 

0. would be degrees.

6. I wish to take a photograph of an object 20 feet high
so that the image will be 4 inches high. I am using

a 6 inch focal length. Therefore I must stand
feet from the object.

7. A mariner's compass is
divided into 32 equal
angles of'____ degrees.
An angle of ____ degrees
lies between a fire E.

by N. of observer and a

 

 

forest tower S. W. by S.

109

8 ” i.\~,'.{-‘y‘.‘,’,“‘ . . K- \
’ an ‘.-.- ,
t ‘ .

.9 \\‘ .' ‘\-"-‘,’l 'L .....
' u,_.. _‘I'

    

Area of partial eclipse .
Moon’s orbit)" /
Earth’s cone-shaped shadow

Geometry is used in study of eclipses.

 

lines indicate the extent of the eclipse.
9. A major tried in the diatonic scale consists of any
three notes whose vibrations have the ratio 4:5:6.
If the first note is 288, the other two are
and vibrations.
10. Below are two doors. Of the two doors, door

is the more rigid.

 

 

 

11. A carpenter is copying a triangle. He has made two
angles equal to the two angles of the cepy, and one
side equal to one side of copy. Does he necessarily

have an accurate copy?

 

12. The directions and
magnitudes of two
forces acting from

 

the same point are

represented to scale'

by AB and AD. Their

15.

14.

15.

16.

110

combined effect in magnitude and direction is equivalent

to , which is called the force.

 

 

 

 

 

A diagonal piece was

nailed on to this gate

to make it fie
A __ ft. board is needed to put a diagonal brace
on a box cover 5 ft. by 4 ft.; a ft. board

 

is needed to put a brace on a gate 5 ft. by 12 ft.

A man rows with a force of 5 miles an hour at right

angles to the current of the stream which is 2 miles
an hour. The actual speed of the boat will be
miles per hour.

Two tones are harmonious if the ratio of the frequencies

of their vibrations may be expressed by two of the simple

numbers 1, 2, 3, 4, 5, or 6.

 

Middle C D E F G A B O

 

 

Ratio of
vibrations 1
compared with
middle 0

9/8 5/4 4/5 3/2 5/3

 

 

 

 

 

 

 

 

 

 

Do 0 and E chord? . A and B? .

111

17. Will 50 feet of fence enclose a larger area when in

the shape of a square, an equilateral triangle, or

a circle? .

18. Two pulleys are 25 inches apart, and each has a radius
of 3 inches. It will take a belt inches long
to go around the two pulleys.

 

 

 

112

UNIT TWENTYAFOUR
TO RECOGNIZE GEOMETRIC FORMS WHEREVER FOUND
DIRECTIONS. Each example contains an illustration.
In the space below write what you see in the illustration

which is geometric. Be complete.
1.

 

2.

 

113

 

  

.w"

gaseous" Lumenm Cuuscu'

  

OF THE HEDEEMER,
FT. WAYNE. IND.

 

 

 

4.

 

 

114

 

115

 

HBEDEFEHIJKLM
NflPflRSTUVWXYZ

   

10.

11.

 

‘ in

1.

2.

5.

4.

5.

6.

116

UITIT THIRTY-FIVE
TO PBJBORTI SIHPLB GEOMETRIC COYSTRUCTIOYS

DIRECTIONS. Make the construction for each exercise
the sapce below.

Construct an angle at A equal to angle B.

'3 3'

Construct a six-pointed star.

 

Construct a perpendicular to AB at C; also a perpen-
dicular to AB through D.
(D

79 I *r 3
Cr

Construct a line parallel to AB through C,

:W.\\‘\\\\\\‘\\\‘\;f;\\‘
.8

Construct a right triangle with legs 1%" and 2".

 

Divide AB into five equal parts.

79

 

117

7. Construct a circle passing through points A, B, and C.
fin

C/

8. Construct a tangent to circle 0 at A; also two tangents

to circle 0 passing through B.

’3

9. Construct the mean proportional between m and n.

'WW 77

10. Construct a parallelogram with one angle 60 degrees,

and two adjacent sides 3" and 2".

118

UNIT TWENTYQSIK
TO CREATE GEOMETRIC DESIGNS FOR LIFE SITUATIONS

DIRECTIQN§. Ybu are to make three original geometric
designs of practical use. The following list is suggestive

only.

 

1. Eng.

2. Bedspread.

5. Quilt.

4. Emblem.

5. Trademark.

6. Label.

7. Flower garden,

8. Landscaping.
9. Book covers.
10. Clothing.
11. Stained glass window.
12. Lunch cloth.
13. Building.
14. Monument.
15. Upholstering.
16. Border.
17. Set of dishes.
18. Wall paper.
19. Cloth.

20. Linoleum.

119
UHIT TWENTYASEVEH
TO REALIZE HOW BASIC GEOMETRY REALLY IS TO OUR CIVILIZATION

DIRECTION§. Express your own thoughts clearly and in
detail. Be original and be complete.

 

I. 1. Why is the triangle important to carpentry?

2. Why is the circle important to transportation?

5. Why is the tangent important to industry?

4. Why is area important to the housewife and farmer?

5. Why is similarity important to map making and

scale drawings?

6. Why is prOportion important to chemistry, physics,

and engineering?

7. Why is the Pythagorean Theorem important to designers,

plumbers, and mechanics?

120

8. Why are polygons important to pattern designers?

II. How is geometry related to each of the following fields?
1. Art.

2. Nature.

30 Drafting.

4. Architecture.

5. Engineering..

60 war?

7. Surveying.

8. History:

9. Recreation

10. Full enjoyment of living.

122 ,

CONCLUSION

These twenty-seven tests present a great variety -—

not only of areas tested, but also of form and structure.
This variety was neither desired on the one hand, nor was
any attempt made to avoid it on the other. For tests are
made to test objectives, and form is a secondary matter.
In formulating a test, the focus of attention should be
on the purpose for which the test is being made, rather
than any preconceiVed pattern. It is true that some of
the twenty-seven tests are not of the standard objective
types, but can all of the objectives for teaching a course
ever be measured by limiting tests to certain types? Is
the apparent inadequacy of the standard objective type of
tests for measuring certain objectives any reason why we
should cease investigating and conclude that there are
some objectives which cannot be tested?' It is the belief
of the writer that the development of tests for certain
objectives has been impeded by the belief that a good
test must have a certain form. Because of the ease and
accuracy of scoring, objective tests should be used when
possible, but when they are inadequate other methods
should be employed.

Some of the material included in these tests is of
local significance and current interest. This is necessary

if the material is to have real meaning to the high school
pupil.' This will at times necessitate the rewriting of

125

certain sections of the tests, but the value to be gained
from having the material of vital interest is worth the
revision necessary. -

Lest these tests appear to give too much emphasis to
what some may term "non-geometric" material, let us recall
that these tests are for only three of the four proposed
objectives. The fourth objective -- to put new meaning
into arithmetic and algebra, and to lay a foundation for
advanced mathematics, the sciences, and specialized voca-
tional training -- is very important and would be tested
by tests composed of what has been traditionally regarded
as geometrical material. Furthermore these tests are not
intended to indicate the ratio of class time to be spent
upon geometric and "non-geometric" material. It is my
belief that training in traditional geometrical material
will develop abilities which can be used in other fields;
and that the introduction of a limited amount of "non-
geometrical" material will broaden the students' outlook
and insure a greater transfer and application of geometry.

The writer would like to acknowledge the inadequacy
of some of the tests submitted. They are not to be
considered as models, but merely as samples of what can
be done. Yet imperfect as they are, they are to be pre-
ferred to the widespread neglect of attempting in any

manner to test for these objectives.

124

The purpose of this paper is not to devise perfect

tests, but to present a method which can be used to im-
prove mathematics instruction in any classroom.' This
method is -- first, to formulate the chief objectives
for the course; second, to interpret these objectives
in terms of changes in student behavior; and third, to
devise means of testing for each of these changes. And
so to summarize the central idea of this report--formu-

lated objectives will not actually become the classroom

objectives of either students or teachers until tests
can be devised and used for measuring these same

objectives.

125

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Bartoo, C. Grover and Osborn Jesse. Plane Geometr .
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Betz, William. Algebra for Todgy. Boston: Ginn and
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Betz, William. "Mathematics in the High School," 36th Year-
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Birkhoff, David George,and Beatley,Ralph, Basic Geometpy.
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Christofferson, H. 0. "Geometry as a Way of Thinking."
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SchoolgScience and Mathematics, XXXIII (March, 1933), Pp.
293-296.

Glaser, Edward.M. An Experiment in the Development of
Critical Thinking. New Yerk: Bureau of Publications,
see era 0 ege, Columbia University, 1941.
PP. 1X + 212. . .

127

Goodrich, Merton Taylor. "Algebra as a Medium for the
Interpretation and Control of Nature " School Science
and Mathematics, XXXV (January, 1935), 16-232

 

Hall, Elizabeth Loetzer. "Applying Geometric Methods of
Thinking to Life Situations," Mathematics Teacher, 31
(December, 1938), 379-384.

Hamley, Herbert Russell. Relational and Functional Thinking
in Mathematics. New York City: Bureau of Publications;"
Teachersr'ColIege, 1934. Pp. l-8, 111-193.

 

 

Hartung, Maurice L. "Iathematics in Progressive Education,"
Mathematics Teacher, 32 (June, 1939) 265-69.

Hartung, Maurice L. "Some Problems in Evaluation,"
Mathematics Teacher, 31 (April, 1938), 175-182.

 

Hassler, Jasper 0.; and Smith Rolland R. The Teaching of
Secondary;Mathematics. New Ybrk: The Macmillan Company,
1930. Pp. 3-18, 65-101, 105-129.

 

Hawkes, Herbert E.- Luhy, William A.; and Touton C. Frank.
First and Second Year Algebra. Boston: Ginn dud Company,

—;_—_

1934. Pp. Viii + 726.

 

Hogben Lancelot. Mathematics for the Million. New Ybrk:
w. A. Norton and Company, Inc.. 19377“'P§T‘17-39, 115-192.

Joint Commission oijathematical Association of America
and the National Council of Teachers of Mathematics.
Thegglace of Mathematics in Secondary Education.

Pp. 14-51, 162-186. BY—

Kelley Truman L. "Values in High School Algebra and
Their Measurement " Teachers' College Record, XXI '
(May, 1920), 246-90.

Keyser Cassius J. Thinking;about Thinking. New Yerk:
E. f. Dutton and Company, 1926. Pp. 91.

Kimball, Reginald Stevens. "Chats with the Editor,"
Practical Mathematics, 5 (May 1943), i - iv.

Kinney, Lucien B. "Why Teach Mathematics?", Mathematics
Teacher, XXXV (April, 1942). 169—174.

Lazar, Nathan. The Importance of Certain Concepts and Laws
of Logic for_the Studyand Teachi of Geometr . Menafiha:
GEOrgeBantamPublishing Company: I938. Pg. 53.

 

128

Leonhardy, Adele; Joseph Margaret- and McLeary, Ralph D.
New Trend Geometry. New Ybrk: Charles E. Merrill

Company, 1940. Pp. ix f 494.

 

Major, George T. Plane Geometry. New York: Charles
Scribner s Sons, 1938. Pp. xv+280.

 

McCormack, Joseph P. Plane Geometry. New York: D. Appleton
Century 00., 1934. Pp. xiv + 455. ‘

McCormack, Thomas J. Why Do We Study Mathematics? Cedar
Rapids: The Torch Press, 1910. Pp. 1-26.

McKee, Charles Lester. "Historical Material in Secondary
Mathematics," School Science and Mathematics, 37
(May, 1937), 588-596. 'Y"

Minnick J. H. TeachinggMathematics in the Secondary Schools.
New York: Prentice Hall, Inc., I939. Pp. 34-50, 288-506.

Mirick, Gordon R. Plane and Solid Geometry. Evanston:
Row, Peterson and‘Co., 1929. Pp. xv + 462.

Mossman, Edith L. "The Primary Purpose of Training in
Mathematics-~Not a Kit of Tools but a Way of Thinking,"
School Science and Mathematics, 38 (December, 1938),
992-1002.

National Committee on Mathematical Requirements. The
Reorganization of Mathematicsip:8econdary EducEIIon.
The Mathematical Association of America, Inc., 1923.
Pp. 5-18. -

National Council of Teachers of Mathematics. A Source
Book of Mathematical Applications. New york?"sureau
of Publications, Teachers‘ College, Columbia Univer-
sity, 1942. Pp. 145-214.

Nyberg, Joseph A. Exercises in Reasoning. Chicago:
Hyde Park High Sc 00 , 40. g. .

Nyberg, Joseph A. Plane Geometrg. New YOrk: American
300k 00., 1929. PPOFVII + 2

O

Nyberg, Joseph A. "Teaching Geometry and Logic," School
Science andeathematigs, XXXIX (March, 1939), ZUI:259}

Pickett, Hale. "Functional Thinking in Geometry,"
Mathematics Teacher, 33 (February, 1940), 69-72.

Reichgott, David, and Lee R. Spiller. Today's Geometr ,
New York, Prentice-Hal1, Inc., 1938. Pp. vaB+B3 .

129

Sohnell, Darcy H. and Crawford, Mildred. Clear Thinking,
an Approach_§@gggg§iglane Geometpy. New York: Harper
and Brothers, 1938. Pp. xviii + 442. .

Schorling, Ralei h. The Teaching of Mathematics. Ann
Arbor: Ann Ar or Press, 1936. Pp. 24-28.

 

 

Schultze Arthur.
Schools. New Yer

 

Shanholt, Henry H. "A New Deal in Geometry," Mathematics
Teacher, XXIX (February, 1936), 67-74.

Shibbi, J. Recent Developments inrthe Teachinggof Geometry.
New York: Maple Press Company, 1932.

Smith, David ene. "On Improvi Algebra Tests "
Teachers‘ Cowlege Recopd,.XXIV March, 1923), 87-94.

Starr, David W. "Enrichment Materials for First Year
Algebra," Mathematics Teacheg, 32 (February, 1939),
68-77. __1_

Stone, John C., and Mallory Virgil S. New Plane Geometr .
Chicago: Benjamin H. San orn a 00., 1938. 'Pp. xiv + 4 4.

Trillingham, C. C. "A Student Teacher's Point of View,"
School Science and Mathematics, XXXIII (October, 1923)

763-776.
Ulmer, Gilbert. "Teaching Geometry for the Purpose of

Developing Ability to do Logical Thinking," Mathematics
Teacher, XXX (December, 1937), 355-357. , P‘—

Wells, Webster, and Hart, Walter W. Progr essive Plane '
Geometry. Boston: D. C. Heath and 00., 9 . p. ix T 390.

Whitmer, Dana Philip, "Progress and Elementary Algebra
Instruction," Ohio Schools, 15 (January, 37), Pg. 9.

Wren, F. L. "The Concept of Dependence in the Teaching of
Plans Geometry," Mathematics Teacher, 31 (February, 1938)
Pp. 70-74.

Ybung, J. W. A. The Teaching of Mathematics in the Elemen—

tar and Secondar School. New Yerk:‘Logmans,“Green and
Company, 1924. Pg. 9-25.

130

my T 9:33:13

WITH COMMENTS AND EXPLANATIOKS

UNIT ONE

Maximum Score - 27

This test calls for the clear statement of problems
based on given data. If the analysis is excellent, score
each section three points, average two points, poor one

point, and if omitted or entirely incorrect no points.

TNIT TWO
Maximum Score - 39

This is a type of test where there is a variety of
answers, and is rather difficult to score. The scorer
must keep an open mind, and consider all possibilities.
Score each exercise one, two, or three points. Two
points indicates an average acceptable answer, and is
the score which should be given in most cases. Three
points indicates an excellent answer showing insight
and careful statement. One point indicates a poor
answer, but an answer which deserves some credit.

It is impossible to provide a set of answers for
this test. The following is merely a sample of accep-
table answers. _

1. Lines AB and CD are in the same plane.

2. The interior angles are not right angles.

3. The triangle is a right triangle.

4. The pentagon is convex.

5. The hexagon is equiangular.

6.
7.

8.
9.

10.

11.

12.

13.

131

AB and CD are parallel lines.
ED is perpendicular to AC.

Every person will at some time use Gold Medal Flour.

Furniture manufactured in Grand Rapids is of

superior quality.

People like to eat plenty.

Many people wish to reduce.

Women like strong muscular men.

Men wish women to admire them.

Every girl wishes to become engaged.

The girl became engaged because she used Pond's cream.

All who use Pond's cream will become lovely; and all
lovely girls will become engaged.

Every citizen wishes to do his part.

Fat people cannot work as efficiently as slim people.

Any person can reduce by eating Ry-Krisp, and thus

do more to help the war effort.

UNIT THREE

Maximum Score - 32

Score by giving one point for each part correctly

circled. The key below shows the correct answer which

should be circled.

I. 10 ILIP II. 1. IMP III. 1. IMP Iv. 1. IRR
2. IMP 2. 11m 2. IRR 2. IRR
z. IRR s. IMP 5. IRR z. IMP

4. IRR 4. IMP 4. 1M 4. IR

 

V. 1. IMP VI. 1. IMP VII. 1. IMP VIII. 1. IMP
2. IR 2. IMP 2, IRR 2. IMP
3. IMP 3. IRR 3. IMP 3. IRR
4. IR 4. IRR 4. IRR 4. IMP
UNIT FOUR
Maximum Score - 37
In part one for each letter correctly circled one point
is given. In part two for each answer correctly underlined
one point is given.
In.part one the letter indicated below should be circled}
1. C 5. N 9. E
2. C 6. E 10. N
3. C 7. S
4. S 8. N

The correct answers in part two are to be underlined.

They are below.

2

decrease, increase
its intercepted are

an arc
2
4

decreases

the number of sides

B, A
quadrant
4
increases
parallel
decreases
4

radius
increases
2
parallel
increases

equal, parallel,

decreases

supplementary

133

UNIT FIVE
MaximumuScore - 25
One point is given for each correct answer.
Answers in correct order are below.

Part One Part Two

CDNIP-‘Dficn

3...:
01
H
O

l-‘
IDOIHO’ONOQUIQDFb
01

UNIT SIX
Jaximum Score fin§§

The first eleven questions should be scored in the
following manner-~two points for the average correct
answer, three points for a very good answer, and one
point flJr a poor answer which is partly correct. The
last two questions are scored one point each. Answers
to the last two questions follow.

12. If two angles have equal complements, the

angles are equal.

13. If it is not good weather, we will not have
a picnic.

mm: 2mm
WM

One point is given for each letter correctly circled.

The letter which should be circled is given below.

134

V 6. V 11. V 16. N
N 7. N 12. N 17. V
N 8. N 13. V 18. N
N 9. V 14. N 19. V
N 10. N 15. V 20. V

UNIT EIGHT

Maximum Score — 50

One point is given in each exercise that is checked

correctly in the left hand column.

Score two points for

each correctly stated conclusion.

I.
II.

III.
IV.
V.
VI.
VII.

VIII.

IX.
X.
XI.
XII.
XIII.

XIV.

XVI.

XVII.
XVIII.

No conclusion

Inductive .

This year there are no new teachers in the

schools in Roxenburg.

No conclusion

No conclusion

Inductive

y = 4x for examples given.

Indirect

I shall wear my blue sweater.

Indirect

Marjorie Phillips attends the Baptist Sunday School.
Inductive

All the boys in the Tiger Patrol are first class
scouts.

Deductive

Marilyn Tucker is required to take physical fitness.
Deductive

John Jones is paid 75 cents per hour.

Indirect

a and b are parallel lines.

Deductive

Bernice Henderson is not married.

Inductive

All the Sunday Schools in Roxenburg are having
their Christmas programs on Friday, December 23rd.
Indirect

Gerald Bailey takes geometry from Mr. Keith.

No conclusion

Indirect

Edward and Jim earned the same amount of money
last week.

No conclusion

Deductive

Lucille Wilson is required to take American History.

XIX. Inductive
All classes of Jonesville High School contributed

to the Red Cross.
XX. Deductive
Tom Nyberg is invited to a banquet to be given
by the Rotary Club.
UNIT NINE
Maximum Score ".19
This test was not to test for beliefs, but consistency

in thinking. The statements are so paired that if one
statement is true, another cannot be true. Start with a
score of 10 and subtract one point for every impossible
combination of answers. In general for each combination
of numbers below, one should be marked true and one false.
In no case is it correct to have both marked true. In a
few cases marked by a "*" a pupil could be doing consistent
thinking and have both marked false. This is possible but
not probable. The scorer may count these correct, but if
possible it would be better to question these few pupils
personally to ascertain whether or not they were thinking
consistently. The pairs of numbers follow.
19 *
l6

898°?

11
14 *
12

(fithtfiNl-J

38°98?

20
17

13‘*

{DOD-Q03
3°

& 18
10 & 15 *

136

UNIT (”OBI
Maximum Score - 60
In each case give two points for the average correct

answer, three points for an exceptionally good answer, and

one point for a poor answer worthy of some credit.

Numbers 2, 11, and 19 are correct.

UNIT ELEVEN
iaximum Score - 60
p The scorer will have to rely strongly on his own judg-
_ ment for scoring this test. Care should be taken to make
the scoring as objective as possible. In part one give a

maximum of twenty points. Following the nine steps of the

outline give points as follows:

Steps 1, 2, and 3 ................. O - 5 points
Step 4 ooooo........o..ooccoeo..o.. 0 D 5 pOintS
Steps 5 and 6 oeoooooocooeoeoeooooo 0 - 5 p01nt8
Steps 7, 8, and 9 .eoeoooeoooooooo. O - 5 paints

For part two give a maximum of forty points. In regard to

the nine steps of the outline give points as follows:

Steps 1, 2, and 3 ................ O - 10 points
Step 4 0000000000000000000.0000... 0 - 10 POints
Steps 5 and 6 .................... O - 10 points
Steps 7, 8, and 9 ................ O - 10 points

Discrimination as to number of points to be given should

be based on thoroughness and quality of work.

UNIT TWELVE
Maximum Score - 40
The maximum score is obtained by giving two points for

each exercise. Here again Judgment by scorer will have to

137

be used. Those words or phrases first listed should be under-
lined, and those enclosed in parentheses may be underlined.

1. frills (duration)

2. good standings (final examinations)
3. regular pentagon, extreme and mean ratio (diagonals)
4. waste, tolerated
5. outstanding achievement
6. similar, corresponding sides, prOportional
7. school groups, community groups ,
8. dry sausage, semi-dry sausage, groups 1, 2, 3, and 4
9. processed
10. area, semicircle hypotenuse
11. farm workers (deferment)
12. enchanting, artless, chiseled, captivating (azure)
13. sociologists, zoot suiter, zoot
14. children ( asaport visa, re-entry permit)
15. absentees drafted)
16. nibbler
17. cuts, impractical
18. cheap substitutes
19. loyalty (pledge)
20. real protection (finer pleasure)

UNIT THIRTEEN
Maximum Score - 32

Give one point for each symbol or number correctly

circled. The answers which should be circled are given below.

1. + 6. ‘- 11. "' 1 . 169 +
2. ". 2 70 "’, 2 12. * 17. -, 5
3. -, 1 a. -, 5 15. —, 2 18. -, 2
4. + 9. + 14. . 19. -, 1
5. -. 4 IO. -, 2 150 -, 4 20. +

UNIT FOURTEEN
Maximum Score - 28
Give two points for each correct definition. Remember
the definition must be limiting, and any definition which
gives exact meaning to the statement is to be counted as
correct. In general the definitions are to be considered as
right or wrong. At the discretion of the scorer in some few

instances one point might be given for a definition.

138

UNIT FIFTEEN

Maximum Score - 25

 

Give one point.for each step of each exercise correctly

done. Give two points extra if the entire exercise is correct.

1.

 

2.

 

 

139
5.

 

 

 

 

 

UNIT SIXTEEN

Maximum Score — 20
Give one point for each correctly circled letter.

The letters which should be circled are listed below.

10 E 60 A 11. C 16. E
2. B 7. E 12. E 17. D
3. D 8. B 13. B 18. A
4. C 9. B 14. B 19. C
5. D 10. A 15. C 20. 0

UNIT SEVENTHEN
‘Maximum Score - 29
The two exercises are each to be scored on a sliding

scale extending from O to 10 points. The scorer should be

as objective as possible.

140

UNIT EIGHTEEN

Maximum Score - 20

In scoring allow one point for each letter circled

correctly. The letters which should be circled are given

below.

1.
2.
3.
4.
5.

T 6. T 11. F 16. F
F 7. T 12. T 17. F
T 8. T 13. F 18. F
T 9. F 14. T 19. T
T 10. T 15. F 20. T

IIT N NETEEN

Maximum Score - 16

Give one point for each correct insertion. The word

which should be inserted is underlined below.

1.

2.
3.

4.
5.

6.

10.
11.

12.
13.
14.

If two parallel lines are cut by a transversal,

the corresponding angles are equal.

Corresponding parts of congruent triangles are equal.
The sum of the interior angles of any convex polygon
of n sides is (n - 2) straight angles.

The c osite sides of a parallelogram are equal.

A cenTraI angle is equal numericaIly in degrees to
its intercepted arc. ‘

Parallel lines intercept proportional segments on
all transversals. '

The acute angles of a right trian 19 are complementary.
If two straight lines InTersect, he vertical angles
are equal.

If two lines are out by a transversal so that a

pair of alternate interior angles are equal, the
lines are paralIEI.

The area of a triangle is equal to one-half the
product of its base and altitude.

The sum of the exterior angles of any convex polygon,
formed by extending the sides in succession equals
360 degrees.

A line perpendicular to a radius at its outer
extremity is tangent to the circle.

An angle inscribgg in a circle is equal numerically
in degrees to one-half its intercepted arc.

An exterior angle of a triangle is greater than
either opposite interior angle.

 

141

15. Every point on the perpendicular bisector of a
segment is equidistant from theIends of the
segment.

16. The consecutive angles of a parallelogram are
supplementary.

 

 

UNIT TWENTY

Maximum Score - 40

Again the scorer must use his own judgment, and

strive to make scoring as objective as possible. For

each exercise grade on a sliding scale extending from

O to 5 points.

UN I T T1173! T Y-ONE

ngimum Score -,§g

In scoring allow one point for each correct answer

underlined.

1.
2.
3.
4.
5.
6.

7.

8.

9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

19.
20.

The correct answers are given below.

less

Greek, B. C.

Lobachewski, Bolyai, Reimann

experiment

deductive

by overflowing its banks it forced man to
find ways of measuring land

Plato's

Greek

to measure the earth

succeeded

P hagoreans, a six-pointed star

P ato

Plato, Euclid

Rhind Papyrus

Thales

Euclid

mensuration

squaring the circle, duplication of cube,
trisection of the angle.

one of several forms.

collection and organization of propositions

142

UN I T T T1317 T Y— TWO

Maximum Score - 23

Give one point for each correct answer as placed on

the right side. Numbers below correspond to answers and

not exercises.

1.
2.
3.
4.
5.
6.

point and line
line
point
point and line
line
point

7-8—9. E, F, H, N, Z (any three)

10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.

23.

straight line parallel to the edge of a ruler
circle

circle

onions

some snowflakes, honey cells (any one example)
acute .
obtuse

belts

archery

parallel

yes

no

gears

straight line

UNIT TWENTYeTHRHE
ngimum Score - 27

In scoring allow one point for each blank filled in

correctly.

2%" , 2' ’ 16"

largest

no, yes, yes, no

(dot placed on map)

55 degrees

30' 6"

11% degrees, 135 degrees
tangent

$60, 432

yes
AC, resultant

143

13. rigid
l4. 5, l3
15. 5.38
16. yes, no
17. circle
18. 68.8
UNIT TWENTY;FOUR
Maximum Score - 33
Score each average correct answer two points, each
detailed answer three points, and each incomplete answer

one point.

UNIT TNNNTYeFIVE
Eggpmum Score - 50
In each exercise allow three points for the correct
method, and two, one, or no additional points for careful

work and accuracy.

UNIT TI’IHI‘ITY—SIX
Maximum Score - 129

For each of the original designs, grade the work as

Q

follows.
Originality ................. O - 10 points
Accuracy of constructions ... O - 15 points
NeatneSB 00000000000000.0000. O - 5 P01nt8
Beauty of finished product .. O - 10 points

UNIT TWENTY;SEVEN
Maximum Score - 54
Score each average correct answer two points, each

outstanding answer three points, and each poor answer one

point.

  
   

 

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