CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
PICTORIAL GRAPHS
II
TRANSFORMATIONS FOR UNDERACHIEVERS IN GRADE 10
A project submitted in partial satisfaction of the
requirements for the degree of Master of Arts in
Secondary Education
by
Eileen Frances Parker, CSJ
January, 19 76
California State University, Northridge
December, 1975
ii
TABLE OF CONTENTS
Page
Chapter
Title Page
i
ii
Approval Page
iii
Table of Contents
1.
2.
Abstract
v
INTRODUCTION
1
Origin
1
Purpose
1
Definitions
2
Identification of Underachiever
2
Assumptions
2
1'vlethod
4
RELATED LITERATURE
8
Curriculum Experiments
8
Influence of
8
NCT~
Discovery Learning
9
Curriculum Changes
10
Transformational Geometry
11
Concrete Aids
11
Characteristics of Underachiever
12
Teacher Expectations
13
Time Needs
13
References
15
iii
Page
Appendices
A
B
c
D
PRELIMINARY MATERIALS
17
Teacher Introduction
17
Student Introduction
19
Scoring Form
21
Teacher References
23
Student· References
25
UNIT 1
27
Teacher Commentary
27
Worksheets
30
Quiz 1
39
UNIT 2
42
Teacher Commentary
42
Worksheets
47
Quiz 2
68
Summary of Graphing Procedures
70
SAMPLE DESIGN
73
iv
ABSTRACT
PICTORIAL GRAPHS
TRANSFORMATIONS FOR UNDERACHIEVERS IN GRADE 10
by
Eileen Frances Parker, CSJ
Master of Arts in Secondary Education
January, 19 76
This project provides two units of work on selected
mathematical transformations on the plane for underachieving students in grade 10.
Each unit is activity oriented
with problems presented on worksheets.
Using the heuristic
method, students recall previously learned facts about
graphing points and lines on a rectangular coordinate
system.
They transfer these ideas to graphing on coordin-
ate axes that are altered in size and shape.
Concepts
associated with transformations on the plane are introduced
intuitively by means of pictorial graphs.
A score below the 30th percentile on the Comprehensive
Tests of Basic Skills (CTBS) indicates an underachiever in
mathematics.
Emotional and attitudinal factors contribute
to underachievement; hence effort is made to influence
v
affective as well as cognitive functioning.
A notebook
requirement supplies practice in organizing work, following
directions, coping with sequential events and completing
tasks.
The teacher does not teach in any traditional sense of
the word but instead provides guidance and support for the
student.
A commentary for the teacher contains behaviorial
objectives, ideas for additional explanations of the material and problems for average students.
A solution key is
provided for the worksheets and quiz in each unit.
Addi-
tional reference material for the teacher and student is
suggested.
vi
Chapter 1
INTRODUCTION
Origin
This project
pre~cnts
some basic mathematical ideas
about transformations in a setting designed to stimulate
interest in discovery and a desire to learn.
twofold.
Its origin is
Some ideas in the units originated with Sister
Helen Vincent Oswald, CSJ, and were discussed while she and
the author attended a National Science Foundation Summer
Institute in 1973.
We noted an apparent lack of materials
that gave underachievers a pleasant and meaningful experlence with mathematics.
Students rejected anything they
suspected was prepared for a lower grade level.
It was
obvious that repetition of basic mathematical ideas was
needed by many students.
In order to be successful, they
needed reinforcement in basic skills.
Further impetus to the project came from Swadener
(1974) who suggested the introduction of concepts associ-
ated with transformations on the plane at an intuitive
level for the purpose of enhancing student understanding
of mathematics in general.
The author is indebted to him
for several ideas that were incorporated into this project.
Purpose
The purpose of the project is to provide two units of
work on selected mathematical transformations on the plane
for underachieving and/or low ability students in grade 10.
1
2
Concepts are introduced on a nonrigorous, intuitive level;
effort is made to influence affective as well as cognitive
functioning.
Definitions
Underachiever:
one who lacks the mathematical skill
and competence acceptable for his grade level.
Transformation:
a one-to-one correspondence between
the points of a plane and the same or other points of the
plane,
Identification of Underachiever
Comprehensive Tests of Basic Skills (CTBS) (McGraw
Hill, 1973) was the testing instrument.
Students with
scores below the 30th percentile arc at least one grade
level below their peers in mathematical achievement,
Corn-
bined with previous grades in mathematics, measurements of
reading ability, and an evaluation of grades earned in
other academic subjects a decision was made by mathematics
teachers that a student was an underachiever.
The following assumptions about underachievement in
mathematics are fundamental to the units:
1.
Underachievers have experienced failure.
2.
Underachievers almost always have a poor selfimage.
3.
Teachers need not understand specific reasons for
a student's past failure.
3
4.
Development of competence in one
~rea
is a desir-
able goal.
5.
Different motivational forces within each unique
person can be influenced.
By definition, the underachiever has not been successful in mathematical endeavors.
He is often unable to do
fundamental operations with whole numbers, fractions and
decimals.
A past history of frustration and failure in arithmetic conditions a student against any further work in mathe-
-
matics;
To enhance his self-image, every sign of success
or even good intent should receive a reward so that the
fear, anxiety and frustration of past failures can be relieved,
Immediate positive reinforcement helps to create
a success-oriented environment in whicl1 the underachiever
can begin to gain self-confidence.
An understanding of the reasons why a given student
failed is not essential for the person who subsequently
assumes the role of teacher.
Knowledge of tl1e causes of
past failure might be helpful in diagnosing a student's
problems with mathematics, but the teacher must help many
students overcome the accumulation of past unsuccessful
experiences.
This is best accomplished by developing a
student's competence in at least one area.
Competency in even one area is a desirable goal in
itself because it fosters cognitive growth.
stud~nt
Moreover, the
gains self-confidence in his ability to learn new
4
skills.
Both outcomes are satisfying and rewarding 1n and
for themselves.
No matter what characteristits underachievers possess
in common, they are not alike.
Each is a person with a
unique set of strengths and weaknesses.
Similarly, the
motivational forces within each person differ as do the
conditions that optimize attention, performance and learning. ·It is the teacher's job to alter the deficient
affective and cognitive functioning of these students and
thus be of real service to them.
Method
This project contains preliminary materials for
teacher and students and two units.
and C.)
(See Appendices A, B,
Each unit is activity-oriented with problems pre-
sented on worksheets.
discovery learning.
The basic instructional
strat~gy
is
Employing the heuristic method,
students
1.
recall the vocabulary and method of graphing
points and given linear equations on a rectangular
coordinate system;
2.
graph the same linear equations on altered axes;
3.
analyze changes in polygons under transformations
on rectangular and altered coordinate axes.
The word "transformation" is not used.
Students have
some trouble understanding this function because they
visualize domains and ranges as geometric sets of points.
Concepts associated with transformations on the plane such
5
as reflections, translations, rotations and symmetry are
introduced on an intuitive
l~vel
by pictorial graphs.
The
procedure involves the variation of three elements on the
standard rectangular coordinate system:
1.
units of measure on the axes;
2.
measure of the angle between the axes;
3.
axes are curvilinear rather than rectilinear.
Students may devise their own solutions to the
problems and possibly experience what Polya (1954) described:
Your problem may be modest; but if it challenges your
curiosity and brings into play your inventive faculties,
and if you solve it by your own means, you may experience
the tension and enjoy the triumph of discovery. Such
experience at a susceptible age may create a taste for
mental work and leave their imprint on mind and character
for a lifetime.
(p. v)
Materials are provided to assist the teacher in presenting the unit.
These include
1.
behaviorial objectives;
2.
teacher commentary;
3.
solution key for worksheets and quizzes;
4.
supplementary references;
5.
scoring sheet.
Materials for the student include
I.
general directions;
2.
supplementary references;
3.
worksheets;
4.
quizzes.
6
Each student must keep a notebook.
Fulfillment of
this requirement provides experience in the skills of organizing work, following directions, coping with sequential
events, and completing tasks.
In its final form the note-
book will
1.
contain all directions, worksheets, quizzes and
scoring sheet;
2.
be graded on completeness and amount of work done
on each page;
3.
determine approximately two-thirds of the final
grade.
A quiz is given at the end of each unit.
The points
earned on the quizzes are approximately one-third of the
final grade.
The teacher's main function is to create an atmosphere
that fosters good affective and cognitive functioning in
the students.
Teacher materials are designed to assist the
teacher achieve this goal.
Specifically, the teacher is to
1.
give general directions;
2.
describe grading procedures and notebook requirements;
3.
explain material that is troublesome;
4.
provide supplementary problems for more advanced
students;
S.
administer and score quizzes;
6.
assign a final grade.
7
This description of the teacher's function clearly
identifies someone who furnishes guidance and support to
the students.
The units are designed to enable below-
average students to learn on their own but frequently they
require additional motivation as they do their work.
The
teacher is expected to provide this service and thus encourage them to do something creative with what they have
learned.
(See Appendix D.)
The worksheets show both typewritten and handwritten
materials.
Because it is desirable to have a solution key,
two kinds of typeface are used: standard denotes all student materials and script indicates solutions.
Handwritten
materials consist of solid and broken lines: solid denote
all student materials and broken illustrate solutions.
Chapter 2
RELATED LITERATURE
The review of literature related to this project was
limited to the following topics: curriculum experiments,
influence of NCTM, discovery learning, curriculum changes,
transformational geometry, concrete aids, characteristics
of the underachiever, teacher expectations, and time needs.
Pikaart and Wilson's (1972) complaint that "research
information about the learner in mathematics is insufficient" (p. 26) is no longer completely true.
Recently,
studies have been completed that provide information for
the identification and teaching of underachievers.
~iculum
Experiments
Curricular experiments have been carried out at leading universities throughout the country.
SMSG, UICSM,
SSMCIS are the alphabetical designations of three, widely
known and respected mathematics programs which have labored
diligently to make mathematics education learnable and
useful.
Generally, the new mathematics programs were aimed
at the above average student.
Only in the last decade have
textbook authors and publishers recognized and responded to
the needs of slow children and youth.
Influence of NCTM
At the April 1966 meeting of the Yearbook Planning
Committee of the National Council of Teachers of
~lathemat­
ics (NCTM) it was agreed to recommend the publication of a
8
9
yearbook on the slow learner in mathematics.
By this time
it was recognized that all members of society need greater
mathematical competency to be able to perform even simple
tasks in an era marked by phenomenal tecl1nological advance.
Interest focused on those who had trouble learning generally and among these, are those who have trouble learning
mathematics.
The Thirty-fifth Yearbook (1972) entitled
The slow learner in mathematics, was devoted entirely to a
--------------~~.
i
-
~d
~
consideration of the slow learner.
~cknowledged
The editorial panel
that the main purpose of the book was to pro-
vide ideas for teaching these students and not to ''prescribe specific content in mathematics for all slow
learners'' (p. ix).
To encourage active involvement of the
student in the learning process, over two hundred references were listed for
~igned
materials~
aids and activities de-
to make learning possible and enjoyable.
This only
pointed up the usefulness of a project such as this one.
Discovery Learning
Psychologists and educationalists have been interested
in one of the methods of learning known as ''discovery
learning''.
Also called heuristic, the method relies on the
strategy of teaching students a technique of thinking that
emphasizes the steps involved in plausible reasoning from
a given problem to a solution.
In a recent article,
Hughes (1974) reviewed the origins and development of heuristic.
After brief mention of Socrates and nineteenth
.LV
century writers, he outlined significant contributions and
conclusions of theoreticians and researchers who have investigated discovery learning.
He reported agreement that
heuristic works but that questions about its use and usefulness have still to be answered.
In his book How to solve it (1945), George Polya made
a major contribution to the study of problem solving by
using heuristic.
His principal aim was to teach a method
of reasoning which can be used to solve more than one problem.
He expressed his approach in a few questions:
1s the unknown?
What are the data?
Do you knmv a related problem?"
techniques of
thinking~
"What
What is the condition?
(p. 7)
Because he taught
his emphasis was on the means of
solving, rather than the solution of the problem.
Ct;!_ricull~
One of the curricular changes in mathematics has been
the emphasis on transformations.
Usiskin (1974) reported
investigations conducted in Europe to determine how the
transformation concept can be developed at an intuitive
level and how it can be formalized to give an axiomatic
approach to geometry.
Before 1970 material on transforma-
tions was included in few geometry textbooks published in
the United States.
For instance, Kelly
& Ladd
(1965)
pioneered a textbook in which transformations are given
strong attention.
Usiskin (1974) drew attention to a lack
of interest in transformational approaches in this country
because of confusion about the nature of motion in geometry
11
and the language used to describe it; for instance, distance-preserving transformations are called isometrics,
rigid motions, or just motions.
Phillips
& Zwoycr
Dienes
& Golding
(1967),
(1970) and Dennis (1969) published mater-
ials that can be used by elementary school children.
Transformational Geometry
The concepts of transformation and invariance are considered to be rudimentary notions that pervade all of mathematics and science (Dienes
& Golding,
1967).
Mathematical
properties from the various branches of geometry can be
described in terms of transformations.
These may be repre-
sented through several types of manipulative activities;
for example, geometric m-odels, geoboards, pattern blocks,
and mirror cards.
Transformational geometry topics may be approached
through figure drawings.
Such an approach is consistent
with a general consensus among psychologists and educators
that, at a very basic level, knowledge can be gained
through personal involvement and interaction (Williford,
1972).
Piaget
& Inhelder
(1956) described an action compo-
nent of cognitive functioning that is built upon sensorimotor actions but goes beyond them.
Concrete Aids
In a recent study, Kuhfittig (1974) investigated the
effect of using concrete materials as opposed to more
abstract training procedures to produce higher achievement
on specific training tasks.
He observed that "the fact
that a student is ultimately expected to function on the
abstract level does not imply that he must also be taught
on this level."
(p. 107)
His main conclusion was that low
ability students benefit from aids more than high ability
students in mastering abstract skills.
c~'
Gawronski (1972) reported success in teaching a geometry course to students "characterized as a group who did
not have a great deal of success in mathematics."
(p. 283)
The inductive approach was used but the predominant characteristic of the course was initial presentation at a
concrete leve 1.
Characteristics of the Underachiever
The emotional and attitudinal factors which contribute
to underachievement were summarized by Callahan and Robinson (1973):
Underachievers tend to exhibit such characteristics as
nervousness, insecurity, instability and immaturity. They
seem to lack motivation, application and purpose in their
study and systematic procedures for doing work. Typically,
they do not function well under pressure. They seem to be
conditioned to anticipate failure and defeat and generally
have poor attitudes towards subjects, the school and individual teachers.
(IIamza, 1951; Wellington and Wellington,
1965; Ross, 1962, and Fine, 1967) F~ctors in the students'
family environment also seem to be an influence in producing underachievers. Low income, limited education of
parents, large family, lack of facilities and attitude in
the home which encourages study, lack companionship between
parents and children, limited reading or conversation in
the home and limited social interaction with adults outside
the larger family unit are cited as being prevalent circumstances in the horne environment of underachievers.
(Dale
and Griffith, 1965; Ross, 1962)
(p. 578)
Schultz (1972) maintained that underachievers are
deficient in affective as well as cognitive functioning.
13
In his opinion, the one most common characteristic shared
by underachievers is a poor self-image with regard to mathematics.
Poor reading ability is often a problem for underachievers.
Jacobs (1974) wrote a geometry textbook with
this in mind and explained that he had tried to present the
material in a way that would help students improve their
reading skills.
He criticized many mathematics books as
being "either so dull or so condescending that many students do not want to read them."
(p. vii)
!.~~~~ Exp~~
Rosenthal and Jacobson (1968) declared they had demonstrated that the teacher's expectations about students may
be an extremely important influence on the affective functioning of the student.
While teacher expectancy of
learner performance is of interest, much of the research is
equivocal; for example, the failure of Jose
& Cody
(1971)
to replicate the much quoted finding of the Rosenthal and
Jacobson (1968) study.
A recent study (Moore, Means,
&
Gagne, 1973) has shown that the function of expectancy
statements in learning is to change a student's perception
of the difficulty of a task thereby stimulating his effort.
Time Needs
The School Mathematics Study Group (SMSG) investigated
the effects of allowing below average students twice as
much time as average students to complete a learning task.
Herriot (1967) reported that ninth grade below average
14
students who had been allowed two years to complete an
algebra course scored higher on a standard test than did
corresponding above average control students who spent one
year on the same course.
Hence, Begle (1971) suggested
that the amount of instruction time in certain basic skills
be directly proportional to the ability of the student.
Callahan
& Robinson
(1973) conjectured that underachievers
in mathematics require significantly more time to master
learning tasks.
15
References
Begle, E. G. Time devoted to instruction and student
achievement. Educational Studies in Mathematics, 1971,
!·
220-224.
Callahan, L. G. & Robinson, M. L. Task-analysis procedures
in mathematics instruction of achievers and underachievers. School Science and Mathematics, 1973, 73,
578-584.
----
Dennis, R. Informal geometry through symmetry.
Arithmetic Teacher, 1969, ~. 433-436.
The
Dienes, z. P. & Golding, E. W. Geometry throush transformations. New York: Herder and Heraer, 19'57.
Gawronski, J, D. An informal approach to high school
geometry. School Science and fv!athematics, 1972, .zl,
281-283.
~---
Herriot, S. T. The slow learner project: the secondary
school 'slow learner' in mathematics. School Mathematics
Study Gro~_ReE~"Jumber_2,, Stanford UnTvers""ity-,~I9157. ~Hughes, B. B. Heuristic teaching in mathematics. Educa!1-on=a]_ Studi~s in Mathern~, 1974, ~, 291-299.
Jacobs,. H. R•.A teache~_£uide =t..?
Franc1.sco: 1'v~eman,1Yiir.
gecm~,tnr.L•
San
Jose, J. & Cody, J. Teacher-pupil interaction as it relates
to attempted changes in teacher expectancy of academic
ability and achievement. American Educational Research
Jo~.~1al,
1971,
~'
39-50.
Kelly, P, J. & Ladd, N. E. Geomet~v.
Foresman and Company, 196J.
Chicago: Scott,
Kuhfittig, P. K. The relative effectiveness of concrete
aids in discovery learning. School Science and Mathe~tic~,
1974,
~'
104-108.
Moore, J. W., Means, V. M., & Gagne, E. D. Expectancy
statements in meaningful classroom learning. Paper
presented at the annual meeting of the American Educational Research Association, Honolulu, April, 1973,
(ERIC number ED 083 347).
16
National Council of Teachers of Mathematics. The slowlearner in mathematics,
(35th yearbook) liJaslnngton,
D. C.: Nat1onal Council of Teachers of Mathematics, 1972.
Phillips, J. M. & Zwoyer, R, E. Motion geometry.
(Books 1,
2, 3, and 4), Sacramento, Calif.: Cal1forn1a State
Department of Education, 1970.
Pikaart, L. & Wilson, J. W. The research literature. In
National Council of Teachers of Mathematics, The slowlearner in mathematics. (35th yearbook) \1/ashington;-D. C.: NaiTona1 Counc:il of Teachers of t-.1athematics, 1972.
Piaget, J. & Inhelder, B. The child' ~__s:Q.~~-IJtion of
London: Routledge and KegarPaul, Tif5b-:----
spac~.
Polya, G. How to solve it. Princeton: Princeton University
Press, l!r~na-edition 1957).
Rosenthal, R. & Jacobson, L. Pygmalion in the classroom.
New York: Holt, Rinehart & Wii1Ston, fg()s.
----~Schultz, R. W. Characteristics and needs of the slow
learner. In National Council of Teachers of Mathematics,
The slow~J.earner in mathematics.
(35th yearbook) Washingforl,~:~: Nationaf"""'CO'i:illCiT of Teachers of Mathematics, 1972.
Swadener, M. Pictures, graphs, and transformations a
distorted view of plane figures for middle grades.
~ri ~~!E~~!~' 1974, ~~ 383-389.
The
Usiskin, Z, Transformations in high school geometry before
1970. The Mathematics Teacher, 1974, £2., 353-360.
Williford, J. A study of transformational geometry
instruction in the primary grades. Journal for Research
in Mathematics Education, 1972, ~~ 260-27L
=
Appendix A
PRELH1INARY MATERIALS
Teacher Introduction
These two units of work are a study of some selected
mathematical transformations.
They are designed for under-
achieving students in grade 10.
mation" is not used.
The actual word "transfor-
Following the heuristic method,
students perform a series of tasks and are led to discover
tlteir own correct conclusions.
Students receive worksheets
on which they graph points and lines on rectangular and
altered coordinate axes and then analyze the resultant
changes in a picture.
The teacher does not really "teach" this work, but is
to direct activities so that learning is possible.
Speci-
fically, the teacher describes what is to be done and how
success will be judged.
Materials provided with the units
include:
1.
behaviorial objectives;
2.
teaching commentary;
3.
quizzes;
4.
solutions for worksheet and quiz problems;
5.
quiz and notebook scoring form;
6.
supplementary teacher references;
7.
supplementary student references.
Unit goals are explained to the students.
They like
to know the purpose of their work, grading procedures and
general directions.
General directions are written once
17
18
because this helps create a feeling that the unit is connected and not just a series of pages to be worked.
Also,
this placement of directions helps the student develop
study skills; he must remember directions were given and
recall where to find them.
Students keep all materials in a notebook.
grading procedure the pages are not corrected.
In the
It is
supposed that by circulating in the classroom the teacher
has already corrected errors.
It is the theory of the
author that it is important that underachievers learn responsibility; hence, the notebook is judged by completeness
and not by correctness of work.
Students can achieve
highly if the daily work has been done.
Teacher References is a list of additional material
and explanations that may be useful and interesting to the
teacher.
For the student, supplementary reading and dis-
cussions of related topics can be found in the books listed
under Student References.
19
Student Introduction
In these units we shall be working with graphs, but
not in the same way as you did in Algebra 1.
Instead we
shall explore some new ideas and experiment with different
kinds of graphs.
Problem Number
What it is about
1.1
- 1.2
Graphing refresher
1 .. 3
- 1.29
Differences in a figure plotted on
cartesian, slant, and curvey graphs
2.1
- 2.4
Changes in the size of units on the axes
affect the size and shape of figures
plotted on the graph
2.5
- 2.19
Multiplication and division of coordinates changes the size and shape of a
figure
2.20 - 2.39
Addition and subtraction of coordinates
changes the size and shape of a figure
When you have learned about these changes in graphs
you will begin to understand how the "crazy" mirrors in a
funhouse work.
You will also understand how designs can be
made by working with the coordinates of points.
If you
create an original design of your own you will demonstrate
that you have learned your lesson well.
The books listed
under Student References will give you some help.
Your
teacher can tell you where these books can be found.
Unit materials must be kept in order in a notebook.
The problems are written on worksheets.
You are expected
20
to work all problems and answer all questions.
There will
be a quiz at the end of unit 1 and another at the end of
unit 2.
Points earned will be recorded on a
scori~g
form.
Your scoring form is very important because it will be the
only record of the points you have earned.
20 points
Unit 1 quiz
25 points
Unit 2 qu1z
99 points
Notebook
Possible for everything
144 points
Notebooks will be collected and the points earned will
be determined by the completeness of your work.
Pages
should be in the following order:
1.
Student Introduction
2.
Student References
3.
Scoring Form
4.
Worksheets for Unit 1
5.
Quiz for Unit 1
6.
Worksheets for Unit 2
7•
Quiz for Unit 2
8.
Summary of graphing procedures.
Be sure you understand what you are expected to do
before you begin to work.
any questions.
Ask your teacher now if you have
21
Scoring Form
Name
Problem
Number
Page
Page Included Page Included Page Included
Completed
Missing Not Worked
Partly Worked
Points
1
0
2
3
1.1 -1.2
1.3 -1.4
1.5 -1.7
1.8 -1.10
-1.11-1.14
1.15-1.17
~~~~
-~---""
1.18-1.21.
·-
-
~-
1.22-1.25
1. 26-1..29
__,...__.,...>IQ:
2.1 -2.2
~--
2.3 -2.4
...
~..,..,.._..
~._....
2.5 -2.7
2.8 -2.10
-2.11-2.13
2.14-2.16
2.17
2.18-2.19
2.20-2.21
2.22-2.25
2.26
2.27-2.33
-
--~
--=!111:!:
"--·
-"'>~--.....,.,.,.
......,...,-'I!S:EltLA
=--
--
22
Scoring Form
Page 2
Name
Problem
Number
Points
Page
Page Included Page Included Page Included
Missing Not Worked
Partly Worked
Completed
0
1
3
2
2.34
2.35-2.39
2.40-2.42
2.43-2.45
=r.-=-=_
2.46-2.51
2.52-2.53
2.54-2.S5
~
-
"n:ut
2.56-2.57
-
--'"""
_
.........
=no
··'-·~
of pa.ges
@
0 points
of pages
@
1 point
of pages
@
2 points
of pages
@
3 points
Total
Summary of points earned:
out of 99
Quiz 1
out of 20 points
Quiz 2
out of 25 points
Notebook
out of 99 points
Total
out of 144 points
23
Teacher References
Bell, E. T. The queen of the sciences.
Stechert & Co., lg3~; 13~ pages.
New York: G. E.
Chapter V (pp. 58-72) on transformations contains a
brief historical background and a short, readable outline
of the evolution of the concepts of the mathematics of
groups and invariants.
Coxeter, J. S. M. Introduction to geometry.
J 0 hn wi 1 e y & son s;J. n c. -;-~~r0br,-'4ir3~p"a g"e s .
New York:
Chapter 3 "Isometry in the Euclidean Plane" (pp. 3949) is a discussion of why a rotation or a translation can
be achieved as a continuous motjon while a reflection cannot.
The seven possible ways to repeat a pattern on an
endless strip are presented.
Lieber, L. R. Human values and science 5 art and mathematics • New Yo rk'TI:;:':--"·rr.-:JOrl: o11rcorrlrra11 y , J ii.'C7-;J~0"'6T;--~
149 pages.
Chapter 4 is a cleverly written discussion about invariants under a transformation.
Some of the examples
would be meaningful to students.
Polya, G. How to solve it.
Princeton, New Jersey:
Princetoi1 UnTversTtyrress, 1945, 204 pages.
This is a valuable explanation of the heuristic method
of teaching and its application to problem-solving.
It
contains excellent and essential informatiori for anyone
using the heuristic method in a classroom situation.
Swadener, M. Pictures, graphs, and transformations: a distorted view of plane figures for middle grade~. The
Arithmetic Teacher.
1974, .,U-_, 383-389.
24
This is a discussion of how pictorial graphs may be
used to introduce concepts associated with transformations
of the plane on an intuitive level that further enhances
student understanding of mathematics in general.
The pro-
cedure suggested is very similar to that used in these
units.
Usiskin,
1970.
z.
Transformations in high school geometry before
Teacher. 1974, ~~ 353-360.
Th~~atics
This is a useful historical introduction and extensive
bibliography for the teacher.
Yaglom, I. M. Geometric transformations.
(trans. from the
Russian by Allen~ShielCISj, Ne\7 Yorl(·:-Random House, Inc.,
1962• 133 pages.
Distance-preserving transformations are considered and
the reader is simply and directly introduced to some important group theoretic concepts.
In the short basic text the
author has successfully striven for simplicity and clearness rather than for rigor and logical exactness.
25
Student References
Adler, I. The new mathematics.
1958.
New York: John Day Company,
The representation of ordered pairs as arrows is carefully and clearly explained and various transformations of
the plane are discussed in a simple, non-rigorous way.
This is good material for a student who is bored with
classwork and wants to investigate practical applications.
Bergamini, D.
~~·
New York: Time Inc., 1963.
This is a colorful pictorial essay on topology that
provides excellent, thought-provoking supplementary reading
material.
Contains material on ordered pairs and quadrilaterals
that would help students who need drill in plotting points
and working with graphs.
Hogben, L. Mathematics for the million.
Norton & ~Y ,-~nr'!r:----h---
New York: W. W.
Chapter IX contains interesting historical background
material and many useful examples of how graphs are used in
daily living.
Applications of curvilinear coordinates are
discussed.
Kasner, E. & Newman, J. Mathematics and the imagination.
New York-: Simon and Scnuster, -I940.
Contains interesting anecdotes about historical persons and various problems they encountered that actually
were propositions of topology.
Very readable.
26
Phillips, J. ~1. & Zwoyer, R. E. tvlotion teometry.
(Books 1,
2, 3, 4). Sacramento, Californ1a: Ca 1fornia State
Department of Education, 1970.
Excellent supplementary material for students who
express interest or need additional work to understand
transformational concepts.
The material is very readable
and can be handled by the student with little assistance.
Roper, s. ~er and pencil geometrr.
Publicat1ons, 1970.
Chicago: Franklin
Good discussion of area that extends the work done in
Unit 2.
Appendix B
UNIT 1
Teacher Commentary
The purpose of this commentary is to provide ideas on
how to motivate students and how to help them enjoy this
work.
It includes behaviorial objectives, teaching proce-
dures, suggestions for illustrative models, and supplementary problems for quicker students.
Although problems can be assigned for homework, the
worksheets should be done in class.
circul~te
The teacher should
throughout the room and help those who need it.
Behaviorial objectives for this unit are:
1.
Given a list of terms related to a rectangular
coordinate system, the student
2.
a.
defines each term in writing,
b.
illustrates each term on a grid.
Given a set of points on a rectangular coordinate
system, the student
a.
3.
reads and records the coordinates.
Given the coordinates of a set of points, the
student
a.
plots the corresponding points and labels
them.
4.
Given a linear equation and values of one of the
variables, the student
a.
computes corresponding values of the other
variable,
27
28
b.
plots the points and draws the line on
cartesian, slant and curvey graphs,
c.
analyzes the relationship of the line's to each
other.
Explanatory comments appear below, to be used as the
teacher sees fit.
Problem
Number
1.1
Comment
I always tell the students that they probably
know this already and that the refresher lets
me know that everyone is starting out with the
same set of facts.
1.4
Check carefully to be sure the first coordinate is plotted horizontally and the second is
plotted vertically.
1.10
Some may have difficulty with the w6rd
"relationship."
It is well to explain it
carefully.
1.11
Students enjoy working with the slant and
curvey graphs.
Intuitively, they seem to know
exactly what to do.
A model for a slant graph
can be made by threading straws together with
a cardboard support across the top and bottom.
29
Any stretchable material can be used as a
model for the curvey graph.
1. 22
Notice the position of x and y in the second
equation.
Call this to the attention of
students and caution them to keep this change
in mind when computing values for the table.
Quiz
The quiz was designed so that students would
get correct answers.
Expect high scores.
The
students know the material, so why not a good
mark?
30
Worksheets
Graphing Refresher
1.1
Write a definition for each word.
Informal definitions
Horizontal
are acceptable.
Vertical
Ordered pair
x coordinate
y coordinate
Coordinate axes
Parallel lines
-----------------·----------~---
Perpendicular lines
Origin
--------~--~--~--------
Variable
1.2
Illustrate the terms in the vocabulary.
-I---'
1
ny co rr
~ct
i lu st rat i01
'1-S
( cc ~pt ab ~e.
31
1.3
Given the coordinate system at the
right, determine the coordinates
of each labeled point.
A ( 3, 2)
E (-3, 1)
B (2, -2)
F (-3,-2)
c
G (-1-1/2,4)
A
E
( 2, 1)
D (-1,1)
D
c
H (0.,-2)
B
H
F
r-- 1---
."'
G
1.4
Given the coordinate system at the
right, plot and label the points
corresponding to these ordered
D
pairs.
A ( 3, -1)
B
(0,2)
c (5,0)
B
E (-4,-1)
c
F (0 ,O)
F
G (-3,2)
E
D (-2,3)
H
G
H (4,1 1/2)
A
32
1.5
Complete the tables for each equation.
a)
n-:-+-_:~1-~-:-+-1-_--:-
y=2x -1
-2
b)
y=x +4
2
1.6
Write the values from the above tables as ordered pairs.
a)
y=2x -1
y=x +4
b)
( 2" 3)
(2,6)
(1,1)
(1,5)
(0,-1)
( 0 .• 4)
(-1,-3)
(-1,3)
(-2 .. -5)
(~2,1;2)
1.7
Graph each set of ordered pairs and draw the line.
each line.
'
ul
/b:
/
/
/
!/
I
/
I
/
.I
..
l
/
~
/
I
/
/
/
J
I
Label
33
1.8
Complete the tables.
0
X
1st equation
y=x +2
y
I I I
y=x
4
0
1
2
-1
0
1
2
I -1
0
1
2
-1
-3
--2
-1
X
3rd equation
y=x -3
y
I
-2
I
1
3
I
y
I
2
X
2nd equation
-1
2
1
0
-2
I
-2
-2
I I
-4
-5
1.9
Graph the equations in 1. 8.
,,...
/
1/_
/
/
I
/
/
Cartesian Graph
/
/
/
~
I/
/
J
/
I/
/
/
/
/
/
/
/
/
"
I
/
/
"
1.10
What relationship do the lines have to each other?
they are earattel and they do not intersect
34
1.11
Graph the equations in 1.8.
Slant Graph
1.12
l\'hat is the relationship between these li;1es?
Is it the
do not intersect
1.13
Graph the equations in 1.8.
Curvey Graph
1.14
What is the relationship between these lines?
changed?
lines not varallel -
Has anything
lines not straiqht -
lines do not appear to intersect
35
1.15
Complete the tables.
X
1st equation
y=x +2
y
I
X
2nd equation
y=l/2x +2
y
I
y=-x +2
y
~~
·~\
1
-1
2
3
1
0
2
-2
I
2
X
3rd equation
0
I
I
3
HB
1
I
1
0
1
2
2
1
0
I
4
-4
4
0
-1
-2
3
4
1.16
Graph the equations in 1.15.
/
"
I'-
--1-·
Cartesian Graph
....
.......
/
/
I/
..,
/
•"-...... ·/
/
/
i/
-
/
' I"-
""'.
/
1.17
What relationship do the lines have to each other?
they intersect at (0,2)
. .v
~
-li'
"
.
36
1.18
Graph equations in 1.15.
Slant Graph
1.19
What is the relationship between these lines?
change?
Did anything stay the same?
Did anything
lines intersect at
1. 20
Graph equations in 1.15.
Curvey Graph
1. 21
Is the relationship still there?
Did anything stay the same?
(OJ2) -
Did anything change?
the lines still intersect at
the lines are now curved and not straight
37
1. 23
Examine the two equations carefully!
How do they differ?
~E.JL..E.f ansiL,ers ,aC:~l?ta,bZe ---~---
1. 24
Graph the equations in 1.22.
"~
'
'
"-""""
1 -1-·
Cartesian Graph
l
rv
1. 25
What relationship do the lines have to each other?
axes?
To the
lines are perpendicular - lines intersect at
(2~3)
-each line is parallel to one of the axes"
38
1. 26
Graph the equations in 1.22.
Slant Graph
1. 27
Is the relationship still the same?
Did anything change?
Did anything stay the same? .,...J;J!!f!S :!21._.P.erpen4j_cJ;<lar - - · - - lines I}_,ti:J:..} in t!:.!_'.J!_e c_! =at ( ~.LJ }_~ l~f '} e s
f!.!iiL.E. aral:J~
l. 28
Graph the equations in 1.22.
Curvey Graph
1. 29
Is the relationship still there?
Did anything stay the same?
Did anything change?
lines are not straight -
lines not paralLeL to one of the axes - Lines still
intersect at (2,3)
39
Unit - Pictorial Graphs
Quiz 1
Name
page 1
Show what the figure in Column 1 looks like when drawn on
the axes in Column 2.
Column 1
Column 2
1)
r-7
D
I
,!_ ___!
· - - - - t - - ·---~
/
I
I
(
I
3)
D
r --,
I
I
40
Unit - Pictorial Graphs
Quiz 1
Name
page 2
Show what the figure in Column 1 looks like when drawn on
the axes in Column 2.
Column 1
Column 2
4)
I
5)
\
\"
I \
I
\.
'\
...
-'\
l
... -"'
6)
1-"
I
41
Unit - Pictorial Graphs
Quiz 1
Name
page 3
Questions 7 - 10.
Sketch what this
figure looks like when drawn on the
sets of axes below.
7)
8)
,,
'-
1
/''
/
1/
''
9)
\
--J~
\
\
\
)'. . .
I
&-····=I
'-
', -;-_~
10)
j\
I
I
A
I
\
\
"
I . \
\
'\
I
I
I
I
4
"'
I
Appendix C
UNIT 2
Teacher Commenta!2'
You may have graded the worksheets in Unit 1 before
going on to this unit.
of the students.
If so, call this to the attention
They seem to be more highly motivated
when they know they have already achieved success in
Unit l.
Behaviorial objectives for this unit are
1.
Given a set of ordered pairs the student
a.
plots the points on a graph whose axes are
marked off in units of different sizes;
b.
connects the points to form a figure;
c.
identifies and analyzes changes in the figure
caused by the size of the units on the axes.
2.
Given coordinates of four points symmetric to the
origin of a rectangular coordinate system, the
student
a.
multiplies the coordinates of selected points
by a given number, plots the new coordinates
and draws the figure;
b.
recognizes that opposite sides of a rectangle
are parallel;
c.
recognizes that adjacent sides of a rectangle
are perpendicular;
d.
makes arithmetic changes in the coordinates of
the given points and then draws the figure
42
43
determined by the changed coordinates;
e.
analyzes changes in the coordinates and determines what kind of change will produce a
certain kind of quadrilateral.
3.
Given cartesian, slant, and curvey coordinate
systems and a set of coordinates the student
a.
plots the points;
b.
connects the points and draws the figure;
c.
adds a number to each coordinate of each point
and draws the resulting figure;
d.
multiplies each coordinate of each point by a
given number and draws the resulting figure;
e.
compares two figures and notes any changes.
The problems in this unit give the teacher an opportunity to observe students proficiency in basic computation.
Deficiency in these skills should be noted and drill given
at an appropriate time.
Comments about specific problems
appear below.
Problem
Number
2. 1
Comment
Most students get the idea immediately if you
tell them·to "connect the dots."
Be sure to
watch them devise short cuts on the next three
graphs.
You may be
amaze~
at how fast they
learn what is going to happen.
Some students
show real insight during this exercise.
44
2.20
Some students have trouble understanding the
directions for this exercise.
It might be
well to find the coordinates for Figure 2.20a
with the entire class working together.
Once
they understand, students have no difficulty.
. 2.23
The phrase "located in reference to" may need
to be explained.
2.34
Students seem able to
generaliz~
quickly.
For
more advanced students you may like to pursue
the following.
1.
Graph x=y.
2.
Increase x by 2 (a horizontal shift to the
right).
3.
Then
x~y
+2.
But the equation in step 2 can be rewritten as y=x -2.
(This .shows a vertical
shift down)
4.
Question:
Does this give us one shift or
two?
You can also justify that (x - 2)
2
+
(y) 2 = s2
represents a circle of radius 5 shifted horizontally 2 units to the right of the origin.
1.
x2 + y2 = 52
2.
x2 = s2 - y2
3.
X
=
±
/s2 -
circle at origin, radius 5
y2
equation solved for x
4.
X
=
±
I
s2 -y2 +2
circle of step 1 is now
shifted 2 units to the
right horizontally.
-2 = ±
I
s2 -y2
5.
X
6.
(x
- 2)2 = s2 -y2
7.
(x
-
2) 2
+
-
y2 = s2 but this is the same
circle given 1n step 4.
The figures are translated slightly but
because the student is answering questions
about the size of the figure, he does not seem
to be bothered by the horizontal or vertical
shift.
2.45
Area may be obtained simply by counting
squares.
Allow any reasonable estimates; pre-
cision is not an issue here.
For generality,
it might be better to use some figure other
than a square but the underachiever does not
need complications.
The following figure might be used in work
with above average students.
They can see
that all vertical and/or horizontal measurements are directly proportional to the original figure.
-
f\
.Multiply first
~
coordinate by 2.
1-- ~
"'
'\
~
~~
1\
\
"" ~
"
\
'
\
\
-
-- -~ """
\
I'-.
~'-..
r
--
46
2.54
With slow students you may wish to omit the
discussion of area when both horizontal and
vertical distances are changed.
Again, the square is the simplest figure to
use.
However, it would be better to use a
more general figure for better students.
47
Worksheets
Points
1.
(1,4)
6.
(-1,-6)
11.
(-2,4)
2.
(1,-4)
7.
(-2,-7)
12.
(0,7)
3.
(2,-5)
8.
(-2,-5)
13.
(2,4)
~~
4.
(2,-7)
9.
5.
(1,-6)
10.
(-1,-4)
14.
(1,4)
(-1,4)
2.1
2.2
Plot the points and
Plot the points and
connect them in order.
connect them in order.
~~~
~~
-
/I .\
--
.-
-
/
J
I'
I
'
--
/
/
"'- .
\
j' """
~
'r-
'
-
~
/
/
if
/
''
',
..
~·
,. .....
..... ...
/
.
-
p
.......
.........
r-...._
48
2.3
2.4
Plot the points and
Plot the points and
connect them in order
connect them in order.
//\
J\
I
•
\
I
/
\
i/
-
,,
p
I
I
\
~-
\I
__I
I
1-\·,
'
I'
I
\
\
~__._
I
,
__
-\
·-- ---
·-
-
---t-·
-r-- i -
-
-
1---
~
-- l---· r-- i-·
-
-·- f---
···-
1--· >
~
=· ~- --- ---<
-]~
--
~~
~-~~f'
-----
-
'--
/
~/
1\
~
\
I
'
v
/
v. .
1'-
'
v"
' ",
'
',
-·~ ~~;.
49
2.5
Plot the points A(-2,-2), B(2,-2), C(2,2) and D(-2,2).
Connect the points A, B, C and D in that order.
~
~
-
1 ..
~
--
1
-
Figure 2.5
2.6
In Figure 2.5, what is the relationship between the
2. 7
In Figure 2.5, what is the relationship between the
adjacent sides?
In the next set of exercises you will make changes in
the coordinates of points A, Bt C and D and draw new
figures.
Perform the operation indicated to get the new
coordinates.
Write down the new coordinates.
Points will be named A', B', C' and D'.
of the new figure A'B'C'D' and record it.
The new
Estimate the area
50
)
2.8
Multiply the
X
coordinate of
D'
C'
/
A and C by 2 •
I
,
i/
I
A'
(-4,-2)
B' (2,-2)
I
C' ( 4 J 2)
D' (-2, 2)
Estimated area of A'B'C'D' is
I
,,7
II'
'18 ,
A
24
li.
Figure 2.8
-
2.9
.
Multiply the x coordinate of
D
n and D by 2.
-f-.--
(!! ,-....-!
A'
(-2,-2)
C'
(2.,2)
B'
(4,-2)
D'
(~4.,2)
Estimated area of A'B'C'D' is
,_
1\
c'
[\
1--
\
eL\ ----
·--F--
!~
I
-1--~ ! r -
24
--
\
\
\-
B'
figure 2.9
2.10
___
l... "' c·-,-
D.r t./
Multiply the y coordinate of
A and C by 2.
'
-·
A'
(-2,~4)
B'
(2,-2)
./
Estimated area of A'B'C'D' is
_,
24
.
v"' ~
/
AI
Figure 2.10
51
-....
D'
2.11
.1'-lul tiply the y coordinate of
I
1',
.....
~"--
.... C'
B and D by 2 •
A' (-2, -2)
C' (2, 2)
B' (2,-4)
D'
(-2,4)
Estimated area of A'B'C'D' is
......
A' "' ~ ....
24
""'~ ....... 8,
"'
Figure 2.11
2.12
1'<
Multiply the
A, B,
c
X
coordinate of
D'
~-
and D by 2.
C'
"""'-
--
A'
(-4_,2)
C' ( 4, 2)
B'
(4.,~2)
D'
~-~ f . -
(-4,2)
Estimated area of J\'B'C'D' is
32
.
A'
=r
·-
-~
-- .......
,___ ~
~-
i·~
B'
'--
-
-
Figure 2.12
2.13
D'
Multiply the y coordinate of
~,
~·.
A, B, c and D by 2.
-~
'
A'
(~2,~4)
B' (2,-4)
.
C' ( 2 J 4)
D' (-2, 4)
E$timated area of A'B'C'D' is
32
A'
Figure 2.13
~
,
52
-. C'
2.14
Multiply the x coordinate and
D'
,.
II
{
(-4,,-4)
B'
(4,-4)
C'
II
(4,4)
I
I
D'
-
I
(-4, 4)
I
Estimated area of A'B'C'D' is
32
-
I,_.
~
,
-
I
I
A'
-
I
1
I
the y coordinate of A and C by 2 •
I
~,..-
~
.
It
j.;-
lB'
-
Figure 2,14
2.15
Multiply the
D'
X
-- - - --- f---. ~ - ~- ~
coordinate and
- ........
\
the y coordinate of B and D by 2 •
-- ---\
'I\
(~2,-2)
C' (2, 2)
B'
(4.,~4)
D' (-4,4)
~-·'"i·
--= -~-
\
..
\
Estimated area of A'B'C'D' is
32
~'--·
G'
j
·-r - It
'
I
A'
-
F'----
..._
\
A,
-
---·-
-.:: .... ,
...
1--1---
~-
.-..,.
._\
\
---- - -1 -
-~
\
+- -
\
=+-,~--
'?/
P'"~
--.,
C'
-
Figure 2.15
2.16
;.
Multiply the x coordinate and
~,
-~
the y coordinate of A, B,
c
and D
~!--1
~-
r--=.. r-:~
by 2.
A' (-4,-4)
C' ( 4, 4)
B' ( 4,- 4)
D' (-4,4)
Estimated area of A'B'C'D' is
~
64
lA'
B'
Figure 2,16
53
2.17
In Figure 2.5, AB and CD are opposite sides and BC and
DA are opposite sides.
Opposite sides AB and BC are
parallel to each other and opposite sides BC and DA are
also parallel to each other.
Sides AB and BC are adjacent sides and are perpendicular to each other.
Sides CD and DA are adjacent sides and
are also perpendicular to each other.
Examine the figures you have just drawn.
If the sides
are either parallel or perpendicular, record this relationship on the table.
Leave a blank if the sides are neither
parallel nor perpendicular.
Relationship between
Figure
number
Opposite sides
____
2.8
-~~~a_Lf:!!.l
2.9
--~!'.5!} ~-f!~!:_-~.
2.10
~~~
2.11
__,PE.!!.!:J:_e__~--
2.12
~l~el
2.13
-
2. 14
Adjacent sides
eara~le l: - -
. ee r> pJ:.!J:i! i c.!:!:} a,!..__
yer>pen..9jp3 lar>
..l?.,a:._aJ.._~
2.15
----12.f!!' a l l e l
2.16
parallel
p_er>pendicular>
54
2.18
Do you think that some of your estimates about the
size of the area of A'B'C'D' in Figures 2.8 through 2.16
are better than others?
~-variety
lies
Why?
of answers is possible.
2.19
Examine the figure for which you think you got the
best estimate of its area.
What is the relationship
What is the relationship between the adjacent sides?
_..---J2~~!.::f!1:~-~~·-~·
tions or
dra~
Can you make any observa-
any conclusions from these facts?
Formula for the area of a sauare (rectangle) applies,
~n~--...~--~.,.---.r.!<V"&,..C..'O•__,V.~.=---~·>""""'OL~.;r-=~...-""-~"'~~-')OOL.-w;;c.~~~~~-~'"""'''''"''"'=ao«>:<o:>:,,..,.-~,.,..-n.~......,..~
55
[;
II
"
.......
I'
....
I
\
!./
f',
/
r--., !(",
I
Iii
r--
"
\
i',
D"
2.20b
Figure 2.20
I
~'........ lc
.......
I
r--....., C'
I
~ r-..._
\
'-....
-...........
J
I
Figure
B'
B
"'
Figure
...........
1\
\
\
~
\
1
L1.
v
\
1',
\
........
.....
_,
\
........
D'
D
2.20
Figure 2.20a
Figure
2.20a
Figure 2.20b
A (-3,1)
AI
( 5 J 1)
A" (-8_, 1)
B (-1,4)
B'
(? !J 4)
B" (-6 ,4)
c ( l, 3)
c•
n•
(9
C" (-4, 3)
D (2,-2)
jl
3)
(10_,-2)
D" (-3,-2)
2.20
Make changes in the coordinates of points A, B, C and
D according to these instructions and record as ordered
pairs.
To get Figure 2.20a:
add 8 to the first coordinate of each
point.
add 8 to the second coordinate of
each point.
To get Figure 2.20b:
subtract 5 from the first coordinate
of each point.
subtract 5 from the second coordinate
of each point.
56
2.21
Plot the points for each figure on the graph.
the points in order.
Connect
Label each figure 2.20, 2.20a or
2.20b.
2.22
Do you think the three figures differ in size?
shape?
no
no
2.23
Where is Figure 2.20a located in reference to
Figure 2 • 2 0 ? _ _L~r> t!:~~!L..!:!?:!.f:l~E-!J!. e._!.f:g '3:.!_.21..___.
Fiaure 2.20.
'""""""""'"~-"'-~~~~~.....no.->~..,~".--·~----...,.-~~~~........~~~
2.24
Where is Figure 2.20b located in reference to
Figure 2.20?
Fiaur>e 2.20b is 5 units to the left of
~..---='~-~~"-""""''~-~-~~=~..,...,~~-»Gn~~-~-~~
2.25
Suppose the directions had said to add 2 units to the
first coordinate to get Figure 2.20a.
In that case, where
do you think Figure 2.20a would be located in reference to
Figure 2.20?
__!ig~~~20a ~?uld b!~ts
riaht of Figure 2.20.
~
to the
------------------------------------------------
57
Figure
2.26
Figure
2.26a
A (- 3' 1)
A' (-3, 3)
B (-1,4)
B'
c
C' ( 1, 5)
(1, 3)
D (2,-2)
'
B ,, '
(-1,6)
A"
(~·.3_,-2)
Bn (-1,1)
C" ( 1, 0)
D" (2,-5)
13"'
C"'
"_,
A' lf'
C''
'
111 I .......... ......
.........
L_
Figure
2.26c
I
Fi ~u e ~. ' 6c
II
D' ( 2, 0)
A"
......
1
I
Figure
2.26b
~
I
.
.., rl/
(0 .t 6)
I
v
·" ['--......
( 2 .t 9)
II
';v
ll
I ' N
t-:-,f
.....
......
-J-- .
vr-
D"
b
\ Dr
=~ f?'\ll:l>
•
p·. •.gu tre 2 • 26
1
I' \
f;.,J
i--- i--·
1--
D
-- -·
r---
~---
--- 1--
\
"" ~'-_.....,
- t-·
I
F" ou tre 2. t~-(
·~:"""'
I
-
- 1--·
\\
_\
r--,
\
I'....._
\
00
...........
....
l'
..c ·~\
r--......
I
A' !""'-,
D"' (5 J .3)
C' 1',
1\\
R' l' ..
---t
ll
( 4 _,B)
\
~
F: ..illJrre
- - 2. 2 6...! l;;;-- -D"
-=~~~1-Jf
2.26
Make changes in the coordinates of points A, B, C and
D according to these instructions and record as ordered
pairs.
To get Figure 2.26a:
Use the same first coordinates.
add 2 to the second coordinates.
To get Figure 2.26b:
use the same first coordinates.
subtract 3 from each of the second
coordinates.
To get Figure 2.26c:
add 3 to each first coordinate.
add 5 to each second coordinate.
58
2.27
Plot the points for each figure on the grid.
2.28
Do the figures differ 1n size?
no
shape?
no
2.29
Where is Figure 2.26a located in reference to
Figure 2.26? ~~p__2___u_n_i_t_s_.~·-=~--------------------·------------2.30
1'vhere is Figure 2. 26
Figure 2.26?
located in reference to
dovm 3 units.
~~~---~~~--
2.31
Where is Figure 2.26c located in reference to
2.32
Suppose you were asked to make Figure 2.26c by
subtracting 4 from the first coordinate of each point of
Figure 2.26 and adding 7 to each of the second coordinates.
Then where would Figure 2.26c be located in reference to
2.33
What must be done to the coordinates of the points of
a figure if the figure is to be moved
to the right 8 units
down 4 units
add 8 to the first coordinate
subtract 4 from the second coordinate
to the left 3 units
subtract 3 [rom the first coordinate
59
F g.
3~
2
f
lD'
/
\
\
/
\ I//
\
T
A //
• z4
1 ig.
',
b
1\
;
'
AL
/
...-
\
I
C'
'//
\
/
I
\
I\
I
!"\.
\
B'
\
~
P·i
.
\
-
2. ~41
f)"
- -
\
/ \
'
L
-
c
/
1'
!--=
=
-
T
-x- /
- - - 1-- ---- -- f--\ L,
- __
\
' I/
\
/
A' [\ ~
'
f-.--
,
/
\
/1-'
. C---
I
'
V"
I/
/
--
E
- - ~-
+--- f---·
C"
!-1---
-
Figure 2.34
- - -- - · -
·--- -
Figure 2.34a
'--
Figure 2.34b
A (0 ,4)
A'
( 4" 8)
A" (-5,-1)
B ( 2 '1)
B' (6, 5)
B" (-3,-4)
c
C'
C" (-1,-2)
( 4 J 3)
D (3,7)
( 8 J 7)
D' (7,11)
D"
(-2J2)
2.34
Make changes in the coordinates of points A, B, C and
D according to the following instructions.
To get Figure 2.34a:
add 4 to each coordinate of each
point.
Record the act of new
coordinates.
60
To get Figure 2.34b:
subtract 5 from each coordinate of
each point.
Record the set of new
coordinates.
2.35
Plot the points for each figure on the graph in 2.34.
Connect the points.
Label each figure.
Name the corner
points.
2. 36
Do the figures differ jn size?
no
shape?. _n_£~=·
2.37
Where is Figure 2.34a located in reference to Figure
2.34?
Fiaure 2.34a is located above and to the riqht
·
_._..,_,.~~~~~~..-~"'·-.......~"""~-=~~'"""~~~""'"'.,..,..~"' ... -~.~=--~~-==-""'"-=-=
2.38
Where is Figure 2.34b located in reference to Figure
2.34?
..
Fiaure 2.34b is located below and to the left
· -..~~"'"---=~~ ... ~-==~=--==<!:o~~'>,._,._'l:'ZA't~'""J-....--'=~- ~~.."c-~.,....~
of Ficrure 2. 34.
~-..:no~~--.........,~-=·~~--~----~-~~·-..,.,.,.~
.,.___
,__..._~·
2.39
Plot the ordered pairs for Figures 2.34, 2.34a and
2.34b on a slant graph and on a curvey graph.
Label the
figures using the letters that match each ordered pair.
61
Slant
Graph
Curvey
Graph
Figure 2.39a
62
2.40
Examine the figures on the slant graph (Figure 2.39).
Are they the same size? _ _}J;...e_s__ shape?
yes
2.41
Examine the figures on the curvey graph (Figure 2.39a).
Are they the same size?
no
shape?
no
2.42
Examine the figuTes you have drawn on the three kinds
of graphs (Figures 2.34, 2.39 and 2.39a).
Do you see any
advantage to using one kind of graph ratheT than another
kind?
Please try to explain.
Rea~dar'
Gravh - eas1:eBt to d:r•az,, - eas1:est to read
·
~...-...~~_....~..,-~~~=-...~-=,.....,..,.,.~.,..,.~_,_,__~_........_.. =-"'....-',~~........_..,.,,.__..~.................--.,..,.-....c."",or·~-.,.....,_,.-""__.."'~
Slant Graph
- a L-it-tle more diff1:cult to Por7( uith
....
Curvey
Graph
..
- difficult to draw - hard to locate
..
~~.-~nr~=cv"Yr.~~-.,.~~~.<.~~o--~~"'~>::h.:O~~""~£'"~-==v.~~~·;.r _.-...-,=:-_..n•~""y,.;~-<>=~.~-""''=-,...-·,.,._..~_-:,r-
~.,.~~
-=~~._~,.."~""~ _,..,_,~~~,-~~~~---=:o-..o-~..,.,.~,~~...,.,._,.._~~~=.'"~...._~-.,.-=~-~=~~~::.:u::--~-=""~-.&-~ru
points - size and shape
of figures
....
....
..........
~~=~.-.. ~~'""-~~~-""·-=~~..,.._...,...:;r:tl~~-~~-""'"~'·~=.,...,....~---=~
=---=~.\!.~..:r-a.ll'>x.t.. ,.,.,"-~"
.....
~,..,.:llii:ti~
63
L'
b
c, (
I\
l.il , I
~
I
A'
/"
F gu re 2 • 43
~
Pi i;ru1 e 2.
··3a
r--- 1 -
2.43
The corner points of Figure 2.43 are A(-4,1), B(-1,1),
C(-1,4) and D(-4p4).
Double the first coordinates and keep
the second coordinates the same.
coTner points for Figure 2.43a.
Figure 2.43a are J\'(=8_,1),
What is the distance
A and B?
3
--~"""-~
A' and B'?
B and C?
-
2.45
-
The corner points for
B'(~2.~~1),
C'(-2_,4)
and D'
(~8,4).
between the following points?
c
and D?
3
~-
6
C' and D'?
-
6
3
D
and A?
-
3
3
D' and A'?
~-
B' and C'?
Use these new points as
What is the area of Figure 2.43?
What is the area of Figure 2.43a?
3
9 s51.
18 sq_.
units
units
64
2.46
Refer to Figures 2.43 and 2.43a and then make this
statement correct by crossing out the words that do not
belong.
The
horizontal/ve~t~Ga~
line segments are doubled;
the
be~i~eRtal/vertical
line segments remained the ·
same length.
-
e----.-
'.1 ,
.~,
-- --
~
.
~~~
=±J'i
~~~
-jl--
4~F
<~
r--i-·
-
1----
-- ..
I'L
=~~~-
--V~-t_ -·-~J
'
2.
~
?a
+-~-
f-•.,JK~
'
··-
-
K
--IJ-
--
-
Figure 2,47
2.47
The corfier points of Figure 2.47 are J(2,1), K(S,l),
L(5,4) and M(2,4).
Double the second coordinates and keep
the first coordinates the same.
Use these new points as
corner points for Figure 2.47a.
The corner points for
Figure 2.47a are J'(2.,2), K'(5.,2), 1'(5,8) and f\1'(2,8).
65
2. 4 8
What is the distance between the following points?
J and K?
3
L and M?
3
J' and K'?
3
L' and M'?
3
and L?
3
M and J?
3
K' and L'?
6
?-.'!'
K
and J'?
6
2.49
What is the area of Figure 2.47?
Figure 2.47a?
9
18
2.50
Refer to Figures 2.47 and 2.47a and then make this
statement correct by crossing out the words that do not
belong.
The
ho~iaGntal/vertical
line segments were doubled;
the
horizontal/vo~ti~aJ
line segments remained the
same length.
2.51
To stretch a figure horizontally, multiply the
fir~~
coordinate by some number greater than 1.
2.52
To stretch a figure vertically, multiply the
second
coordinate by some number greater than 1.
2.53
Why do you think you must multiply by a number that is
greater than 1?
(Hint: try multiplying by a number less
than 1 and see what happens.) __ if the number is less
than 1 the figure shrinks in size.
66
D'
C'
c
D
C"
D"
c ·~'
-
A'
B"
A
B
-1----
A,
--
·--
-·-
;-
--
1-~
B'
--'{)'·-
Figure 2.54
2.54
The corner points of Figure 2.54 are A(-3t-3),
B(3P-3), C(3,3) and
D(-3~3).
I\1ultiply by 2 both the first
and second coordinates to get the corner points for
A 1 B ' C ' D' •
A1
(.,
6, 6 ) , B ' ( 6, - 6 ) ,
C ' ( 6 3 6 ) and D' ( - 6 J 6 ) •
2.55
What is the relationship between the length of the
sides of ABCD and A'B'C'D'? __ t!;z_~ _sf:de_s of A'B'C'D' are
twice as Zona as the sides of ABCD.
--------~----~------------------=-~---------------------------2.56
What is the ares of ABCD?
the area of A'B'C'D'?
how this happened?
36 s.!l· units
144 sq. units
Area = side x side.
Can you explain
Both sides
were doubted so the area was muttiptied by 2 two's.
67
2.57
Multiply both the first and second coordinates of the
corner points of ABCD in Figure 2.54 by 1/3 to get the
corner points for A"B"C"D".
A"(-1~-1),
B"(l~-1),
C"(1 3 1) and D"(-1~1).
2.58
In f-igure 2.54, how does the area of ABCD compare to
the area of A"B"C"D"?
68
Unit - Pictorial Graphs
Name
Quiz 2
I
C'
I
/
-, .
D'
-
--
<-:
-1I
I
I
-
~--
c
_,...I
};!~~~
---
- -- ---II2
I
~
~--
C'
B
~,
'r-/
I
II
I
I
7
I
I
I /
/i
I
I
/
-b~-Jf+
I' '
'I.
~
--
_,
J
v
', ,p_·r-f--j
B'
i
\'-.
'-;'A·
-~
I
I
I
I
I
'- ....._I
__":~ L
±
A'
--
Fir,ure 1
1.
What are the coordinates of the corner points of ABCD
in Figure 1 ? A ( 1 , - 1) , J3 ( 3 ~ 4) , C (- 1, 3) and D(- 2, 1 ) •
2.
P.lultiply both the first and second coordinates of the
corner points of ABCD by 2.
graph them.
Record these new points and
Name the new figure A'B'C'D'.
A ' ( 2, - 2 ) , B ' ( 6 _, 8) , C ' ( - 2, 6 ) and D ' (- 4 , 2 ) •
3.
Using the coordinates from problem 2, add 3 to the
first coordinate.
Do not change the second coordinates.
Graph Record these new points and graph them.
new figure A"B"C"D".
A"(5,2 ), B"(9_,B), C"(1,6)
and
D"(-1,;~2).
Name the
69
Unit - Pictorial Graphs
Name
Quiz 2 - Page 2
4.
How did the figure ABCD change when it became A'B'C'D'?
_.1.f!ngt1!.. oLsid~creased and area increased
s.
How did the figure A'B'C'D' change when it became
6.
How is the position of A'B'C'D' related to the position
·ICIOfi"-*~-"'~-:1-U'~~--~--"'-~~-<=<>-"<'~.:=-~~-~~~~~~,~~~
le ''t of A "B 11 C "D 11
7.
ConsideT a figure drawn on a graph.
What happens to
the figure if these changes are made?
a.
Add 6 to the first coordinate of each corner
point.
it moves 6 units to the
right
...
....
~~..,.-=--~,.....,.-="•'''''''li~·~-"'~~~-~
...,......,.-~~ ,-=~~~~~
b.
Subtract 2 from the second coordinate of each
c.
Add 3 to the first coordinate and add 2 to the
units to the riaht and 2 units
uv
..
~-<b~~~~-~~~~~~~-~--"""""= __.,.._~J.:.....-~-"'-"'-"'----~~""'"'
d.
Multiply the second coordinates by 1/2.
e.
Multiply the first coordinates by 3.
i t increases in size horizontally
f.
Multiply both first and second coordinates by 4.
sides are 4 times as long and area is
16 times oreater.
--------------~-------------------------------------~~"~~
70
Summary of Graphing Procedures
Cartesian Graph
Figure 1 is a picture of a
')
~
cartesian coordinate system.
1
The axes meet at right angles
and so are perpendicular to
-
each other.
1
~
2
1
"¥
Figure 1
Units are numbered and counted
along each axis.
The units can be
the same size as 1n Figure 1.
-
-
-
- - -1-
l
----
f - - --·-- f -
--- - - --
-
-
Sometimes the units on one ax1s arc
one size and the units on the other
axis arc a different s1zc.
-
Fiaure
.
(.)
2 is an example.
,_
--
fFf
1
I
~
---
---
--.:;o:;r~
- -
Figure 2
To plot a point on a pair of
axes it is essential to know what
A(3,2)
•
the axes look like and the size of
the units.
In Figure 3 you can see points
Bo
A(3,2) and B(-1,-2) plotted without
(-1,-2)
using any other lines except the
axes.
Figure 3
71
Slant Graph
A slant graph looks like
this.
The coordinate axes meet
at an angle so that the lines
,
Units are numbered and
A(2,3)
counted along the axes the same
way they are in a cartesian
coordinate system.
Examine
I
)
4
the figure to the right.
The angle between the axes can change and so can the
size of the units.
the units are.
Always check the axes to see what size
It is possible that the units may be a
different size on each axis.
any of these.
A slant graph can look like
72
Curvey Graph
On a curvey graph the axes
are curves and not straight lines.
C~·
Units are numbered and counted along the axes the same
way as on the cartesian or slant graph.
point A arc (3,5).
ates of point B?
Do you see this?
point C?
The coordinates of
What are the coordin-
If you said B is (-4,4) and C
is (3,-5) you are reading the curvey graph correctly.
Appendix D
SA1v1PLE DESIGN
73
74
The :<nnple design may be created in the following
manner:
1. Draw a square whose corner points are (1,0),
(0,1), (-1,0),
(0,-1).
Multiply each pair of coordinates
by 2 and draw that figure.
Next multiply the coordinates
of the original square by 4 and draw that figure.
Con-
tinue in this manner, multiplying the coordinates of the
original square by the next positive even integer until
the desired number of squares are drawn.
2. On the same axes, draw a square whose corner
points are (lt1),
(1,-1),
(-1,1),
(-1,-1).
Multiply each
pair of coordinates by 3 and draw that figure.
:Next multi-
ply the coordinates of the first square by 5 and draw that
figure.
Continue in this manner, multiplying the coordi-
nates of the first square by the next positive odd integer
until the desired number of squares are drawn.
3. The basic design may be altered by shading various
parts of the figures.