Algebraic Explorations of the Error Author(s): RAFFAELLA BORASISource: The Mathematics Teacher, Vol. 79, No. 4 (APRIL 1986), pp. 246-248Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964883 .

Accessed: 09/07/2014 06:44

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 213.118.134.157 on Wed, 9 Jul 2014 06:44:22 AMAll use subject to JSTOR Terms and Conditions

Algebraic Explorations of the Error

1?T 1

By RAFFAELLA BORASI, University of Rochester, Rochester, NY 14627

Errors

can serve as a starting point and as a source of motivation for in

teresting mathematical explorations. When an obviously incorrect procedure yields a

correct result, we may feel puzzled and curi ous to know how and why this could have

happened. Trying to answer these questions can involve us not only in problem solving but in problem-posing activities as well. This experience can provide the op

portunity for creativity even in elementary mathematics.

The object of our investigation will be the following well-known "simplification":

10 _ 1

04 ~

4

What three-digit numbers can be simplified in this way?

Why does such an absurd simplification produce the correct result? Is this example the only case for which this kind of sim

plification works? We can attempt to answer both

questions at one time, by stating the more

general problem: For what values of the

digits a, ?, and c is

10a + b _af}

106 + c ~

c '

Or, equivalently, what are the integral solutions between 1 and 9 of the following equation?

(1) (10a 4- b)c - a(10? + c) = 0

The values (a, 6, c) = (1, 6, 4) satisfy this

equation, which explains why the result of the simplification turned out to be correct in the specific case presented.

Do other solutions exist? How can we search for them ? We do not have a straight forward algorithm that can be applied to solve equations of this kind, but we can try several approaches.

Given the limited range of our variables

(they must be digits), we can create a com

puter program to check all the possible combinations of a, 6, and c for those that

satisfy equation (1). The following BASIC

program, for example, will print all the pos sible solutions in a few seconds :

10 FOR A = 1 TO 9 20 FOR = 1 TO 9 30 FOR C = 1 TO 9 40 IF (10 * A + B) * C = A * (10 * + C)

THEN PRINT A, B, C 50 NEXT C 60 NEXT 70 NEXT A

However, we may not have access to a

computer. More important, this "

compu

tational' ' approach does not lend any in

sight into the nature of the solutions. We are left wondering why some combinations of a, 6, and c "work" and why most others do not. It seems worthwhile, therefore, to

try to approach the problem in a more alge braic way.

Just by looking at the example 10/04, we

realize that the simplification will work

"trivially" whenever a = b = c. We cannot

reasonably hope, however, to find other ex

amples by blind trials.

246 Mathematics Teacher

This content downloaded from 213.118.134.157 on Wed, 9 Jul 2014 06:44:22 AMAll use subject to JSTOR Terms and Conditions

How should we proceed? Frequently, ex

pressing an equation in different but equiv alent forms has logical as well as psycho logical advantages.

We can, for example, try to rewrite

equation (1) in different ways to see if any thing may be revealed. For example :

(2) (3) (4)

10a(6 -

c) = c(b -

a) IOa? = c(9a + ?) 9ac = ?(10a

- c)

Equation (2) may present some advantages, as all a, | b ? c |, c, and | 6

? a | must be less than 10. We can then observe that since 5 divides the first side and 5 is a prime number, either c = 5 or | b ? a | = 5. In our

example, we had, in fact, 6 ? a = 6 ? 1 = 5. We can now see if c = 5 in some solutions. With this extra condition, equation (2) be comes

10a(6 -

5) = 5(0 -

a)

or

(5) ? = 9a

2a - 1

Computing from (5) the values of b corre

sponding to a = 1, 2, ..., 9, we do find two new solutions besides a trivial one :

(a, 6, c) = (1, 9, 5) ^

= g

(a, 6, c) = (2, 6, 5) ^

= ?

We have thus found all the possible solutions with c = 5. If other solutions

exist, they must derive from | b ? a \ = 5, that is, when either ? = a + 5ora = ? + 5. At first sight checking this case may seem more complicated than checking c = 5, but it is actually less so. For 6 = a + 5, equa tion (2) becomes

10a(a + 5 ? c) = 5c

or

(6) c = 2a2 + 10a

1 + 2a

And this time we have only to check for a = 1, 2, 3, 4 in (6), as it must be that b = a + 5 < 10. We thus find two nontrivial

solutions, one of which is our original one :

(a, b, c) = (1, 6, 4) = \ 04 4

49 4 (a, 6, c) = (4, 8, 9)

^ = -

In the case of a = b + 5, equation (2) be comes

10(6 + 5)(? - c) = -5c or

(7) c = 2b2 + 106

9 + 26 '

Checking for 6 = 1, 2, 3, 4 (again it must be that a = 6 + 5 < 10) in (7), we find no other solution.

In conclusion, with only seventeen "checks" (instead of the 9 9 9 = 729 we

might have supposed necessary at the be

ginning), we have been able to find all the cases in which the "outrageous" sim

plification would work.

Expressing an equation in different forms has logical as well as psychological advantages.

Although we have now solved our orig inal questions, we might still be puzzled by the fact that in all the nontrivial solutions

found, 6 is a multiple of 3. A look at equa tion (4) can provide some justification for this unexpected result. As 9 divides the first side of the equation, we can deduce that either (10a

? c) is a multiple of 9 or b is a

multiple of 3. This does not mean that "6 is a multiple of 3" is a necessary condition for a set of solutions of equation (4)?a coun

terexample is provided by the trival solu tion a = b = c = 1. However, we can easily prove, by using some divisibility consider

ations, that the only cases in which

(10a ?

c) is a multiple of 9 is when a = c.

Since 10a ? c = 9a + (a ?

c) and 9 divides

9a, 9 will divide (10a ? c) if and only if it divides (a

? c). With the given restrictions

on the variables, this means a ? c = 0.

Therefore, all nontrivial solutions must have 6 as a multiple of 3.

April 1986 247

This content downloaded from 213.118.134.157 on Wed, 9 Jul 2014 06:44:22 AMAll use subject to JSTOR Terms and Conditions

Let us now list briefly some mathemat ical and educational considerations about this situation that may be of interest to al

gebra teachers.

The curiosity for finding out when the

simplification would work provided a moti vation for stating and solving an equation.

We dealt with an equation with more

than one unknown and with limitations on

the range of the variables. This situation can easily occur in applications, but it does

not generally receive enough attention in

school. As no "sure" algorithm was avail

able, finding all solutions, or even just some of them, involved some creative prob lem solving.

The procedure used to solve the orginal equation clearly pointed out that although

logically equivalent, the different ways in which an equation can be written may have

specific roles in the search for, and analysis of, solutions. The argument that allowed us

to limit considerably the values to check

could, in fact, be based only on equation (2).

Equation (4), however, helped yield further

explanations of the results obtained. Problem posing is essential in the analy

sis of this error and can take various forms. In the beginning we had to state the prob lem mathematically?by formulating the

first equation. The absence of an algorithm made it necessary to ask several "unusual"

questions, such as, How can we eliminate some values to check? What values are

more likely to give solutions? Even when the original problem was solved, we felt the

urge to pose a new question: Why does b turn out to be a multiple of 3?

This situation can become a rich source of new problems once we challenge the way we have previously stated the problem (equation 1) or modify some of its elements. For example, we have implicitly assumed that the numbers were written in the usual decimal notation. What if the base of nu

meration were not ten but another natural number k? The problem would then be to

find the integral solutions between 1 and

(k ?

1) of the equation

c(ka + 6) - a(kb + c) = 0.

It may be interesting to discuss the values

of k to which we can still apply the argu ment used in this paper (with proper modifi

cations). In this article, we have also limited our

consideration to two- and one-digit num

bers. Can we come up with analogous "sim

plifications" using more digits? For exam

ple, what about 504/207 = 54/27? Finding all "three-digit fractions" that can correct

ly be simplified in this way will now involve a lot more cases. Even if we use a computer,

we will face the real challenge in writing an efficient program and eliminating a

priori as many trivial solutions as possible

(you can expect hundreds of solutions in

this case!). What are other possible sim

plifications that can occur with "three-digit fractions"? What is the percentage of "cor

rect" versus "wrong" results of each sim

plification? Does any pattern occur in the

solutions? This problem can provide concrete ma

terial and the stimulus for a discussion

about the difference between necessary and

sufficient conditions for solutions and

about the values and limitations of heuris

tic procedures versus algorithms in solving equations. It can also provide further re

flection on the use of computers in math

ematics, in comparison to more "classical" mathematical activities.

BIBLIOGRAPHY

Brown, Stephen I., and Marion I. Walter. The Art of Problem Posing. Philadelphia: Franklin Institute

Press, 1983.

Carman, Robert A. "Mathematical Misteaks." Math

ematics Teacher 54 (February 1971): 109-15.

Meyerson, L. . "Mathematical Mistakes." Mathemat

ics Teaching No. 76 (September 1976):36-40.

IS THE TEACHING OF MATHEMATICS A TEACHER DEPENDENT SUBJECT? If your answer is YES, then you should

adopt the WEEKS & ADKINS MATH SERIES.

A FIRST COURSE IN ALGEBRA A COURSE IN PLANE & SOLID GEOMETRY

A SECOND COURSE IN ALGEBRA The above books come with review tests, answer keys, teacher & solution manuals.

These are no-nonsense books that are pedagogically correct, and have been widely used for over 25 years for college bound students. Write for brochure to: BATES PUBLISHING CO., 277 Nashoba Road, Concord, MA 01742.

248-Mathematics Teacher

This content downloaded from 213.118.134.157 on Wed, 9 Jul 2014 06:44:22 AMAll use subject to JSTOR Terms and Conditions