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i NSF/SED-82027

U.S. DEPARTMENT OF EDUCATIONNATIONAL INSTITUTE OF EDUCATION

EDUCATIONAL I3ESOURCES INFORMATIONIaNTER (ERIC)

This document has been reprOduced asr.oceived from the person or, organizationoriginating it

Minor changes have been made to improvereproduction Quality

Points of view or opinions stated in this document do not ncessarily represent official NIEpositean or policy

N.

;

Final Report

Application of Computer Graphicsto Graphing in Algebra and Trigonometry,

National Science Foundation

National Institute,of Education

SED 80-12447

January 1, 1981 L'December 31, _1982

.i

"P'ERMISSION TO REPRODUCE THIS IMATERIAL HAS BtEN GRANTED BY

-PeOu'eja_ 5a.6-6-

.F

a

..

,TO.THE EDUCATIONAL RE,SOURCESINFORMATION CENTER (ERIC):'

,

4

NSF/SED-82027

U.S. DEPARTMENT OF EDUCATIONNATIONAL INSTITUTE OF EDUCATION

EDUCATIONAL VSOURCES INFORMATIONCENTER (ERIC)

This document has been ripro'ducad asieceived from the person or/ organizationonginating

Minor changes have tsan made to improvereproduction quality

Points of view or opinions stated in this document do not necessarily represent official NIEposition Or policy

Final Report

Application of Computer Graphicsto Graphing in Algebra and Trigonometry

National Science Foundation

National Institute,of Education

SED 80-12447

January 1, 1981 '-'December 31,,1982

"P.ERMISSION TO REPRODUCE THISMATERIAL HAS BtEN GRANTED BY

Pelf 4,:c,,:a_ 13a1/4-

TOTHE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC).".

This project was designed to improve-the graphing competency of students enrolled

in the elementary algebra, intermediate algebra and trigonometry courses at Virginia

Commonwealth University.. The immediate goal of the project was to design and imple-

ment computer graphics programs (cIlled "graphing lessons") that would be available

for use in the three courses mentioned above. The software package was to be inter-_

active and would give the student control over the events occurring during a lesson.

The grant provided funds for purchasing three APple II Plus computers which were

initially used to create the'lessons and are now available to the students. Students

enrolled in the three mathematics courses are assumed to have no special knowledge of

or skills in using the computer. Indeed, the lessont are written so that the presence

of the colnputer will not hinder the student's progress or.cause the student additional

anxiety. It is hoped, in fact, that use of these graphing lessons will decrase stu-

dent anxiety and improve graphing skills. The lessons give positive reinforcement and

get the student actively involved in the process%

The student interacts with the computer *by' typing in answers or otherwise reply-

ing questions, plotting point's and taking quizzes at appropriate places in the

lesson. The lesSons are written in Pascal and, again, they are designed so that no

computer knowledge is required of the student; once a student is seated and the machine

4 -

turned on by an attendant, he or she, with reasonable care, should be able to work

through the lesson unaided. However, In the event of difficulty, an attendaht will

be on hand to assist.

The lessons are intehded to be supplemental in nature and should be used by the

student only after.having been introduced to the topics in the classroom. By gsing

the.computer the student has the opportunity to see grAphs drawn quickly and accu-.

rately - something that is generally difficult for an,instructor to do with only

chalk and blackboard. Thus, the graphing lessons reinforce the concepts covered in

the classroom and, as a result, the stuBent's graphing skills-are expected to improve.

3

4

,

.As previously mentioned, the lessons are intended to be supplementary. When an

instructor completes a formal lecture on each topic, the student'g attention is called-

to the fact that a graphing lesson is available for that topic. The instructor empha-

sizes that the lesson is instructive, easy to use, and will enhance the,student's

understanding of the lecture. ,

In keeping with the objective of providing a friendly, nonthreatening learning

/experience, the Apple computers are kept in a comfortable room with informal surround-

ings. An attendant is present to turn_on the machines and generally help students

begin each lesson. As noted, once the machine is turned on it should be possible for

a student to work through,each lesson unaided. However, an attendant is present if/

any difficulty - computer-related or mathematical - arises.

Each lesson is written so that the student has the option, at various times,4

of continuing the lesson, stopping altogether, or redoing a portion of the lesson.

The lessons which are currently 'available for use are listed below.

List of, Graphing Lessons

5,

1. Introduction to Use of Keyboard

2. Linear Equations

(ac ) equations of the form Ax + By + C = 0

(b) equations of the form' y =nix + b

3. What's My Line. (Game)

4. Inverse Functipns

5. Quadratic.Equation y=Ax2+ Bx + C

. ,

(a) effect of A

(b) effect of B

(c) effect of C ,

(d) experiment with choices of A, B, C.

6. The Parabola Game

7. Exponential Functions (Part,l) ..

,

.,

.1

,

4

44

ate>

,

In 'tddition, specifications have been written for lessons on: ,

1. Exponential Functions (Part 2)

2. Logarithm Functions

3. The Trigonometric Functions (six lessons)

(a) Sine Function

114-Cine Function *(c) Tpngent Function

(d) Secant Function

(e) Cosecant Function

OD Cotangent Function

Writing computer code has proven lo be a very time-consuming task and by far

most of the time 1fas been devoted to this phase of the project.

The completed lesons are currently being tested by students

mathematics laboratory. Students in the remedial algebra and college algebra courses

are invited to use the lessons at anytime but preferably at the same time they study

the corresponding material in the textbook. Data is being gathered to determine the

4frequency of use, the reaction towards the machine and the students' general reaction

to the graphs lessons. A questionnaire is administered at the completion of 2ce lesson.

in our audio-tutorial

Following the testing dt the'ijessons the project members plan to write nstructor

and users guides and hope to be able to distribute the software package throu

agency such as CONDiJIT.

h some

Description of the 6raphing Lessons

Introduciion to Use of Keyboard

The student learns how to enter numerical data and learns the proper keys to

use to respond to the true-false and yes-no questions. The lesson contains enough.77 '

examples and enough sample questions so that by the time the student has worked

through it he or she should feel at ease'using the machine.

2. Linear Equations

The student is introduced to

The lesson emphasizes graphically

-"\

The student actively participates

equations of4the form Ax +By+ C.= 0 and y=mx + b.

the geometric meaning of slope as rise over run.

in the lesson 'hy using the paddles to plot points.

-":

3. What's My Line

This is a game which tests the student's comprehension of Lesson 2. The com-

puter draws a line and the student is asked to supply the values for m and b in

the equation y=mx+ b. A record is kept of the student's attempts When the student's

answers are typed. At the end the student's.score is printed. The level of diffi-

culty gradually increases and the student may quit at any time.

4. Inverse Functions

The student is introduced to the concept of an inverse f.Ictioñ via the

geometric notion of reflection in the line x y. The paddles ar used to move

points (x, y) to (y, x) .

5. Quadratic Equations yz.-Ax2+ Bx+,C (4 separate lessons)

(a) Effect of A The effect of the coefficient A is iiudied with B and C fixed.

(b) Effect of B The effect of B is Audied by.varying B while keeping A and

fixed.

(c) Effect of C The effect of C-is examined by keeping A and B fixed.

(d) Experimelft with Choices of A, B, C The student sees in this lesson the

ombined effect of all three coefficients.

6. The Tarabola Game

The computer draws parabolas and the student is asked to supply the correct

TUof A,.B, and C. A record is kept of a student's attempts and a score is

printed when the correct answers are typed in. Students may continue playing as

long as they wish or quit at any time.

7. The Exponentials Function (Part 1)

The graphs of y=a?( are examined for various valties of a greater than 1.

The stOdent actively participates in plotting points and when enough points have

been plotted they are connected by the computer to draw a smooth Curve:,

Description of the Specifications

for the Remaining Graphing Lessons

The specifications for the following graphing lessons have been written, but at

this time ttle coding is incomplete.

8. The Exponential Function (Part 2)

-9. The.Logarithm Function10. :The Sine Function11. _The Cosine Function12. The Tangent Function13. The Cosecant Function14. The Secant Function15. The Cotangent Function

ci

- The specifications for Lesson 8 call for examination of the graphs o y= ax

cur values of a between 0 and 1. When completed, it should be very similar to

Lesson 7. Lesson 9 specifications would produce_a_le.sson which_would introduce

the student first to the graph of y= log2x. This would be achieved by having the

studInt find y values for given x vallies.and then having the machine plot the points

and draw the graph: Next, y=loglox would be treated in a similar fashion, and ,r"

fiL ly the connection between the'logarithm and exponential function.would,be

exp'-rr-d using the inverse function concepts alreaf discussed in.Lesson 4.

TL: specifications for the six trigonometric functions are basically the same,

so only those for tile sine function will be described. The student is first given

the equation y=sin(x) and a table of values for x. The x values given are x= 0,

ff/2, Tr, 3ff/2 and 2ff. The student is asked to type in the-correct values for y.

It is then noted that additional points are needed for an accurate graph'so the

x values ¶112, ff/8, Tr/6, ir/SL.Tr/4, ff/3 and asked to uSe a pocket calculator to find

the y values. If the student does not have a pockei calculator, he or ,,she is

instructed to press return and the y values will be displayed. A similar process

'is used for the intervals W2, n], 3n/2] and [3n/2: 2n] so a complete graph

of one pull sine wave is obtained.

A.

.4.

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,

,

APPENDIX I

SAMPLES OF INPUT AND OUTPUT

,

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The following pages represent a sample ofthe interactions between the student and the computer.

The displays are taken from a ublesson, a qui,z and a game.

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APPENDIX II

ABSTRACTS OF PAPERS PRESENTED

.

o

' ,

4

,

_

32

.:,

-

e

\

a?'

Scanning on the Fly:An Approach to the Us.er Interface*

'Steven R. SeidelMathematical Sciences DePartmentVirginia Commonwealth University

Richmond, Virginia 23284

ABSTRACT

Educational software must provide a "friendly" user interface in

order to prevent student frustration. A recurring source of such frus-

tration is the task of responding to prompts issued by the computer.Scanning on the fly reduces the difficulty of this task. It prevents

the possibility of errors of form in student responses by accepting onlykeystrokes that lead to correctly formatted responses. This approach

has theadvantage that all text displayed by the computer pertains tothe content of the lesson and none to the technical use of the computer.4Its implementation is based on the notion of a finite state autamaton,yielding reliable and easily modifiable software. Scanning on the fly

has been used in the development of microcomputer-based instructionalsoftware for college algebra and trigonometry. Applications to other

CAI courseware are discussed, students' reactions are summarized, and

a Pascal implementation is given.

JTo appear in Computers and Education

.*This work 'partially supported by NSF-NIE Grant SED 80-l2447.

a

33

. -

Plotting Curlies on Low Resolution Graphics Systems*

Steven R.' Seidel

Mathematical Sciences DepartmentVirginia Commonwealth University

rRichmond, Virginia 23284

ABSTRACT

Standard techniques for plotting a curveo.on low resolution graphics

systems represent the curve as a collection of straight line segments or

y unevenly spaced dots. Neither of these representations,is appealing from

a pedagogical standpoint. Both make it difficult, for_the student to relate

his experience on the computeeto the material presented in the classroom

and in the textbook because neither bears much resemblance to the usual

types of drawings made by,the instructor and the illustrator. A technique

for plotting low resolution curves is Presented that overcomes this diffi-

culty by taking advantage of the mind's propensity for fil,ling in missing

visual information. Curves are plotted as collections of evenly spaced

points or short line segme4s. The spacing between successive points.on

each curve is kept constant, to within one pixel, even as the slope of the

curve changes. By choosing a spacing between the.points of as little As

four or five pixels a "natueal looking" curve results; the eye is-encour-

L aged to fill in the missing information and the underlying limitations of

sc n resolution are slibeessfully hidden. This plotting teChnique has

hP'n used in the development of microcomputer-based instructional sciftware

fn. .allege algebra and trigonometry. Applications to CAI courseware are

disiussed, students' reactions are summarized, and a Pascal implementation

jis given.

Proceedings of the 21st-Southeast Region ACM Conference, April 1983

*This work partially.supported by N5F-NIE Grant SED 80-'12447.

34