Geometry rich Coordinates, leacher EDRS PRICE: ABSTRACT *Analytic GeometryCCErrimulum

448 Pages·2011·12.68 MB·English

ID 160 454.
SE 02
103
,
.
AUTHOR
Others
Haag, V.
An
H.
rich
TITLE
Geometry
Coordinates, leacher
Commemtary,
Revised fditiof.
I, 'Unit 49.
INSTITUTION
Calif. School Mathematics Study
- Stanford Univ
Group.
.v.
.
National Science Founditicr, Washington, I.C.
SPONS. AGENCY
PUB DATE
65
NOTE
448p.; For: related dooupents, see SY 025 1C1-104
EDRS PRICE:
MF-40.83 HC- $23.43 -Plus Postage.
*Analytic GeometryCCErrimulum; *Geometry
DESCRIPTORS
*Instruction; Mathematic S Educatidin; Secc
ary
EdmcftionT-*Setoridary SChoerfftthematics; *Tea
Guides
/
*School MathematicsStudy Group
IDENTIFIERS
ABSTRACT
This is part one of a two-paft manual for teachers
using SMSG high
hool text materials. The commentary is crgaiized
Ic
into four parts. The first part contains an introduction and a short,
section on estimates of class time needed to cover each chapter. The
second or main part consists of a chapter-by-chapter comae Lary on
part
is a colleciicn of essays on toPi
the text. The thmird
that
cannot conveniently be dealt with in the main part of the o
mentary
in connection with a particular passage, The fourth part contains
answers to Illustrative Text Items and.the solutions to the problems.(
Chapter topcis include: introduction tc forial geometry; sets,
points, lines, and planes; distance and coordinate systems, angles;
congruence; parallelism; and similarity. (NN)
V
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SENTOFFICIAL NATIONAL INSTITUTE OE
EDUCAT'ON POSITION OR
USERS GE THE ERIC SYSTEM
LU
GEOMETRY WITH
COORDINATES
PART I
SCHOOL MATHEMATICS STUDY
GROUP
eom try
with Coordinates
Teacher's Commentary, Part I
REVISED EDITION
,Prepared under the supervi
a Panel consisting of:
Franklin and Marshall College
V. FL Haag
Cineinnati Board of Education
Mildred Keiffer
South Eugene High School,
Oscar Schaaf
Eugene, Oregon
Montclair State College,
M. A. Sobel
Uppe-r'Montclair, New Jersey
Thomas Carr Howe High School,
Marie
Indianapolis, Indiana
Amherst College
A. B. Willcox
Q
Stanford, California
Distributed for the School Mathematics Study Group
by A. C. Vroman, Inc., 367 Pasadena Avenue, Pasadena California
.Financial support for School Mathematics
Study Group has been provided by the
National Science Foundation.
Permission to make verbatim use of material in this
book must be secured frbrn the Director of'SMSG.
Such permission will be 'granted except in unusual
circumstances. Publicui6ns incorporating SMSG
materials mist include both an acknowledgment of
the SMSG copyright (Yale University or Stanford
University, as the case may be) and a disclaimer of
SMSG endorsement. Exclusive license will not be
grant
save in exceptibnal circumstances, and then
oA by specific action of the Advisory Board
of SMSG.
1965 by The Board of Trustees
of the Leland Stanford junior Un v
All rights reserved.
Printed in the United States of America.
4
Contents
-INTRODUCTION.
WORD ABOUT THE =PROBLEM SETS
USING THE TIME AVAILABLE
1=
INTRODUCTION TO FORMAL GEOMETRY
Chapter 1.
11
SETS, POINTS, LINES,,AND PLANES
Chapter 2.
35
SYSTEMS
-Chapter 3- _DISTABCEAND COORDINATE
ANGLES
Chapter 4.
.
.
93
CONGRUENCE .
Chapter 5.
.
. .
.
. .
.
.
.
.
.
.
.
113
PARIVILELISM
Chapter 6.
.
.
.
.
.
.
.
.
.
.
129
SIMILARITY .
Chapter 7. .
.
.
.
.
.
ACH
TALKS
1 4
FACTS AND THEORIES
.
.
.
.
. .
.
.
. .
.
.
.
149
EQUALITY, CONGRUENCE, AND EQUIVALENCE
.
.
.
157
.
.
THE CONCEPT OF CONGRUENCE
0
0
.
187
SOME REMARKS ON STUDIES II
4.
.
.
.
WHICli SIMILARITY
A DEVELOPMENT IN
5.
191
PRECEDES CONGRUENCE
O7
INTRODUCTION TO NON-EUCLIDEAN GEOMETRY .
.
6.
231
ANSWERS TO ILLUSTRATIVE TEST ITEMS
.
. .
.
.
*
.
.
*
255
ANSWERS TO PROBLEMS
t-
INTRODUCTION
The text that you are about tcrteach from Is the resqlt
experienced hl-gh school teachers and-
, of collaboration between
This'commentary is designed' to-
-university mathematicians.
help you in several ways:
To help you in lesson planning,
1.
To explain why we believe: At_is worthwhile to make
2.
certain changes from the traditional treatment.
to problems
To save you, work by giving
in the problem sets as well as a suggested method
of solution to all but the simplest problems,
To provide additional background inforMation.
4.
You can see from this list that our object in writing this
to help you to present our treatment of geometry
commentary
as effectively as possible.
At this point you may ask "If the text is so different
and/or difficult that it needs an elaborate comMentary,'what
Our
is going to be the students' reaction to the text?"
of geometry
Any formal treatment
answer is as follows:
would be new to the student and remarks An the text
commenting on why we didn't treat this or that matter in the
After all, he
traditional way would be meaningless to him.
Also, it
does not know what the traditional treatment is.
is natural for us,to equate being different with being
It may well be that it is difficult for us
difficult.
However, we feel certain
because it seems so different.
that once you become accustomed to our terminology and
treatment, you will find your geometric insight sharpened
and the increased clarity obtained well worth the break
We also feel certain that
from the traditional treatment.
clearly expressed thought in precise language cannot help
but evoke real intellectual enjoyment on the student's part
as well as making the subject easier for him to grasp.
The commentary is- organi=zed into four parts.
The first
part, in a.cillition-Ito
tt'le introducion yoll
are n4,4 reading,
contains A short 4ection on estimates of class
needed
time
io cover each chapter.
At they beginning, of', the printdd
solutpnsjor some of the problem sets,.comments
regarding
problems in the-sets are made.
The second or main part consists of
a chapter by chap
chapter
commentary on the text organized as follows:
1.
Remarks that apply generally to the chapter.
)_ 2.
Paragraph by pYragraph comments
as appropqate.
3
Suggested ,supplementary problems that could be
used.
either for additional assignments
or possibly as
test auestions: -These are called illustrative'
Test Items.
The third part
,entitled "Talks to Teachers.
s -a'- collection of essays on topics that cannot conveniently
be dealt with in the main part of the
commentaryinrconnection
with' a7particular passage.
The essays inclvde some of the
most important
t of the commentary.
They
b
referred to hereafte' in this manual simply
as Talks.
The fourth par
I
contains, first, the answers to the
Illustrative Test Ito
and, second, the solutions to the
problems..
One of our goals is the development of
analytic geometry
land-inhand with synthetic geometry,
to emphasize that both
are deductive systems and that it is useful
to have more than
one mode of attack in solving prablms.
We begin our
development synthetically, but we
are naturally led to a
consideration of coordinate systeMs through
our heavy reliance
on the real number system (just
as In the earlier SMSG
Geometry).
However,'we do not make full dse of
coordinate
systems until the student has
a good command of the method
of synthetic proof, acquired
through considerable practice.
thent that was
Obviously, we wanted to choose a cleVel
best froM the pedagogic point of view as well as sound
Equally obvjous was the fact that there
mathematically.
was no handy formula fo guide us in making such a choice.
The followidg are some of the factors that we considered in
coming to our decision.
Foremost from the mathematical point of view was the
desire to choose as p6stulates a small number of independent
strategically placed" 1: the body of-geometry
statements
,
so that every area of interest couldbe reached by a short
At the same time, we wanted to have the
deductive chain.
Sometimes, unfortunately,
postulates "intuitively reasonable".
thesetum-desi-Des seemed -to-be-tn-conflitt-
,
In addition there was the choice of technical language.
o
s desirable to use as technical words
On the one hand it
--those that have strong connectiOn_with_colloquial usage and
On the other hand we d_
not
our intuitive geometric ideas.
want to have our mathematical language so clone to colloquial
usage that:there' was danger that the wrong one of several
In the interest of good
possible meanings might be taken.
teaching of mathematics we can hardly overemphasize the
pedagogical importance of reinforcing for your students the
as given in the text by consistently
strict meaning of ter
using them only in th ir technical sense.
In connection with the aim to provide additional back=
Firs
groUnd i formation there are several things to be said.
ous that, in a textbook at this level, many
it is o
We have cut
ons have to be logically incomplete.
disc
some corners, expecting the students' intuition to take over,
Often, in fact,
and we believe this is as it should be.
it would take considerable argument with most studentsto
convince them that they had indeed depended on their intuition.
On the other hand, some students may see this dependence and
ask how the argument-may be revised to avoid such a question.
The running commentary is designed to help you when this -
As a further aid you should have available a copy
happens.
of "Studies in Mathematics," Volume
1,1,, Euclidean Geometry
Based on Ruler and Protractor Axioms, by
C. W. Curtis,
P. H. Bails, and R.
Walker.
It contains, expecially in the
-first chapters, much material that
could-have been put _in the
Talks.
It, also contains detailed proofs of bb.sic
theorems
that are not mentioned in the text.
The properties stated
in these theorems
andare
are intilitively obvious
generally accepted by students
without comment. A completely
logical developetent of geometry
must, nevertheless, contain
proofs of these theorems, and
so they are includeaere for
whatever use you wish to, make of
them.
When we refer to the
reference volume,. we will speak,of it
as "Studies
II."
Inasmuch as "Studies
I1" 'was_speoifiLalj
ritten_for
previous SMSG Geometry text, they G
edition, as opposed fo
this which is the
GW
edition, there is a short
essay in
the Talks establishing the necessary
connection between the
-terminology-usedintUdIeS'1I-,11
and ours.
Some teachers may enjoy referring to
a lighter
presentation of some geometric idea.
To them we suggest
"Studies in Mathematics," Volume
Concepts of Informal
V,
Geometry.
Information about the books can be obtained
from
School Mathematics Study-q-
up, Cedar Hall, Stanford University,
Stanford, California.
i
Although we felt it unwise tmake
our text logic2lly
complete in its theorems and
we did attempt to give a
,
completo faun
ion of postulates and definitions.
On such
a foundation a student can build
as elaborate and complete
a structure as his capabilities permit,
with the help of his
teacher and of supplementary reading.
Obviously you will like
some features of this text better
than others.
In any case, we hopethat
you, will teach it
following the presntation of the
text and taking into
conSideration priorities and emphases
suggested in this
Commentary.
Suggestions for improving the text
are invited;
send them to School Mathematics Stpdy
Grou_--Cedar Hall,
Stanford University, Stanford, California.
A WORD ABOUT PROBLEM SETS
are:
problem sets
The fundamental purposes of the
comprehension and
To reinforce the students'
t 1.
techniques being
appreciation of the ideas and
developed;
to recall
To develop the Students' ability
2.
understoo
accurately and wally concepts once
To ektance the students' `power to apply
effectively
combination& of concepts and techniques
to the solution of problems;
problems and
_enge the students to analyze
To
readTlyapparent.; a-
probe for solutions, not-
students' thinking
Occasionally, to orient the
introduced but to be
toward a concept not yet
ext material.
aeveloped in the succeedin
will commence with
general rule,-each problem set
A:
"drill home" the
exercises designed to
a number of simple
the problems
After these, the difficulty' of
basic concepts.
order of appearance, the-
will increase, roughly with their
challenge-type
two, as a general rule, being
last one
Nproblems.
icable, however desirable, for every
It is hardly pra-
"proper"
contain identifiably, a
o
probleM set to be, or
We have attempted
given student.
homework assignment for a
to allow the
and enough of the
to provide "good" problems,
Teachers are urged torfread
teacher somefreefom of choice.
with
and possibly their solUtions,
the problems of each set,
situations:.
homework assignments to fit
a view to tailoring
in a set, perhaps explora-
There may be certain problems
into succeeding sections, which
tory in nature and leading
student.
recommend assigning to every
the authors strongly
meaning
"starred," the asterisk
Such probleMs are
"Do not omit."