COMPUTER APPLICATIONSAuthor(s): DON INMAN and DONALD CLYDESource: The Mathematics Teacher, Vol. 74, No. 8, Microcomputers (November 1981), pp. 618-620Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962637 .

Accessed: 13/09/2014 04:13

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 110.146.133.181 on Sat, 13 Sep 2014 04:13:28 AMAll use subject to JSTOR Terms and Conditions

COMPUTER APPLICATIONS

By DON INMAN Dymax Corporation

Menlo Park, CA 94025

and DONALD CLYDE Surefire Software Inc.

Vendor, AR 72683

The teaching of problem solving has been one of our most difficult tasks. Prob lem solving is more than just a single skill. The ability to solve problems quickly and efficiently calls on several elusive skills that

must be combined in an organized manner. The computer is a tool that strengthens the student's skills in problem solving. In fact, it might be said to force the student to use

good problem-solving techniques. When the student uses the computer as a

problem-solving tool, the application be comes the vehicle through which the prob lem is posed. In teaching problem solving, practical applications become as important as the techniques used in solving the prob lems.

Applications should?

be as real-life oriented as possible; be rich enough to provide a wide variety of classroom activities at various levels of

competency; be adaptable to assumptions and sim

plifications to fit many levels of com

petency;

provide for partial solutions that can be end solutions at various competency lev els;

provide for extension of computer use to

noncomputer activities;

provide for the gathering of information and facts;

provide for creative solutions.

Let's take a look at a specific application and discuss some of its features. As you fol low the discussion, you will find that com

puter applications become closely tied to

computer simulations or modeling (see the article in this issue by Nancy Roberts).

A Typical Application The simple tin can lends itself to both

simple and complex problems that can be used for both computer and noncomputer

applications. The idea for this application was taken form articles by Donald Clyde (1977, 1978).

At an elementary level measurements of the linear dimensions of a can could be

given. Surface area and volume can then be completed. Students could study the

changes in area and volume that occur by varying one or more linear dimensions.

Better, each student in the class should

bring one or more cans to class. They could make measurements and use the computer to tabulate areas and volumes.

The formulas for the volume and total surface area can be used to solve more

complex problems.

Suppose that you own a food-packing com

pony that manufactures cans. You wish to

pack a certain volume of liquid into a can us

ing the least amount of metal as possible for the can.

In lower level applications, certain as

sumptions and simplifications need to be

618 Mathematics Teacher

This content downloaded from 110.146.133.181 on Sat, 13 Sep 2014 04:13:28 AMAll use subject to JSTOR Terms and Conditions

made before a solution can be attempted. First, you probably do not want to consider the metal wasted when the cans are cut or

punched from the metal sheets. One possi bility is seen in figure 1. You are consid

ering only the top, bottom, and sides of the can.

Fig. 1

We are ignoring the thickness of the metal sheet and the metal needed to weld seams around the top and bottom, and down the side. In this model we will also assume the completed can will be com

pletely filled, with no air space. The problem has been stated in such a

way that the volume of the can has been fixed. Therefore, the only variables to be considered are the radius r and the height A. Since is a constant in V = nr% it is

more convenient to write the formula in a form that can be solved for A or r. Let's choose A, and divide both sides of the equa tion by irr2. The result is

or in computer format,

H = V/(3.14*RA2). Since V and m are constant, the correct

height can be found once the radius is known. The computer can generate a table of A values for given values, and graphs of r vs. A can be constructed for a given V. The nonlinearity of the data will be obvi

ous. Each student should be assigned a dif ferent value for the volume to encourage original work.

This development still does not solve the

original problem, however. Now is the time to bring in the surface area formula.

S = Imrir + A) The computer program to solve for the sur face area for each input radius would use the following:

H = V/(3.14*RA2) S = 6.28*R*(R + H)

The student would need to input the de sired volume, input the radius, print the de sired results, and graph the surface areas vs. the radii. The graph of the final results should look like figure 2.

Fig.

Table 1 is a typical run of a student pro gram that used a FOR-NEXT loop to com

pute the surface area for cans of various radii with a fixed volume. A comparison of R and H at the minimum value for surface area, 354.232, reveals that H is approxi mately twice R. Needless to say, this run

TABLE 1

RUN THE VOLUME OF OUR CAN IS 512 CUBIC CENTIMETERS

THE FIRST R IS ? 4.28 THE STEP INCREMENT IS ? .02

R H S 4.28 8.90128 354.292 4.3 8.81868 354.257 4.32 8.73721 354.237 4.34 8.65687 354.232 4.36 8.57763 354.243 4.38 8.49948 354.268 4.4 8.42238 354.308 4.42 8.34634 354.363 4.44 8.27131 354.432 4.46 8.1973 354.516 4.48 8.12427 354.614

November 1981 619

This content downloaded from 110.146.133.181 on Sat, 13 Sep 2014 04:13:28 AMAll use subject to JSTOR Terms and Conditions

was not the first made by the student. A

good original estimate for the correct value for R reduces the number of preliminary runs that must be made. The incremental

change for R can be refined until you run into the accuracy limits of your computer, not to mention the accuracy of the tools to construct the can. As the student becomes more sophisticated, the assumptions can become more complex. Students can return to the model realizing that they have had success with a similar one in the past.

This application can accomplish several

important objectives.

Beginning Computer use: solving formulas; use of BASIC statements INPUT and

PRINT; use of arithmetic operators MULTIPLY (*), DIVIDE (/), and RAISE TO A POWER ( ).

Advanced uses: loops to automatically in crement the radius; changes in the STEP value for FOR-NEXT loops (after a

rough minimum has been obtained, nar row the values of r used by lowering the value of STEP); multiple graphs on one set of axes (using different values of V); class comparison of results in preparing a composite report.

Alterations to problem: What happens if

you are given the amount of metal to be used (surface area) and have to find the maximum volume possible?

Extension of the problem: Remove some of the original assumptions, allow for was

tage on the seams and rims, or add in the effects of the metal's thickness.

Noncomputer applications also abound. The economics of a can-manufacturing company could be investigated. Other

questions might include the following: Why aren't all cans shaped to use the least amount of metal possible (diameter

=

height)? Is there a shape that would use less metal than a cylinder? Why was a cylinder chosen for the shape of a can in the first

place? I'm sure that you can come up with more

ideas. The purpose of this article is to pro

vide a starting point. Keep in mind, how

ever, the following:

Make the application practical. Use real-life objects where possible. Choose a rich application for wide uses.

Make the application adaptable to vari ous competency levels.

Provide various levels for student differ ences.

Use related noncomputer uses in the classroom also.

Provide for meaningful class discussion of the applications and the results ob tained.

Encourage creativity. Make mathematics fun.

REFERENCES

Clyde, Donald. "The Tin Can Problem." Calculators/

Computers Magazine (November 1977): 57-64.

-. "Recycle the Tin Can." Calculators/Comput ers Magazine (February 1978): 35-44.

GUIDELINES FOR EVALUATING

COMPUTERIZED INSTRUCTIONAL MATERIALS

for users and creators of educational software with sample evaluation instruments

by an international organization dedicated to the improvement of mathematics instruction

1981 ISBN 0-87353-176-0 #122 $3.75

-1 NATIONAL COUNCIL OF dB TEACHERS OF MATHEMATICS QESJJ 1906 Association Drive

Reston, Virginia 22091

See NCTM Materials Order Form in "Professional Dates"

620 Mathematics Teacher

This content downloaded from 110.146.133.181 on Sat, 13 Sep 2014 04:13:28 AMAll use subject to JSTOR Terms and Conditions