ONE POINT OF VIEW: The Least Common Denominator

ONE POINT OF VIEW: The Least Common DenominatorAuthor(s): Warren W. EstySource: The Arithmetic Teacher, Vol. 39, No. 4 (DECEMBER 1991), pp. 6-7Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194997 .

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ONE POINT OF VIEW

The Least Cemmen Denominator Warren W. Esty

people of all ages have

difficulty adding and subtracting fractions. I think that the usual emphasis on finding the least common denominator is at fault. Three basic steps are followed

to add fractions whose denominators are not the same:

1. Find a common denominator. 2. Express each fraction in terms of that

common denominator. 3. Add (or subtract) the numerators and

put the sum over that common denominator.

An additional step devoted to finding a least common denominator greatly complicates the whole procedure, adds nothing to the mathematical correctness, and leaves many students incapable of adding fractions. Furthermore, it is bad training for algebra, as example 2 will demonstrate. The product of the two denominators serves equally well and is conceptually simpler; its use is a better preparation for learning to add quotients expressed in algebraic notation.

Example 1

Find the sum: 1 1

i

4 6

Students using the least-common- denominator approach have to search for

Warren Esty teaches at Montana State University, Bozeman, MT5971 7. He does research in probability and statistics as well as mathematics education.

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the least common denominator: 12. This process can be complex. Then they must find the equivalent fractions in terms of twelfths. To find the appropriate "equivalent of 1," they must find the factor by which 4 must be multiplied to get 12, that is, 3, so that 1/4 can be rewritten as ( 1/4X3/3) = 3/12. Then they must find the factor by which 6 must be multiplied to get 12, that is, 2, to rewrite 1/6 as 2/12. Note that neither the 3 nor the 2 is evident in the original problem. Then they obtain, of course, 3/12 + 2/12 = 5/12.

However, the product of the two denominators is always a common denominator. For example, in 4 X 6 = 24, the 6 is used to convert fourths into twenty-fourths and the 4 is used to convert sixths into twenty-fourths. The conversion occurs using numbers that are already exhibited in the original problem. The students get 6/24 + 4/24 = 10/24. A simple conceptual aid works well with this approach. Divide a piece of paper into fourths with vertical lines and into sixths with horizontal lines. Now the twenty-fourths are evident, and the equivalent of a fourth or a sixth in terms of twenty-fourths is easily seen. I know of no such simple conceptual aid that illustrates the least-common- denominator process.

The least-common-denominator method more frequently yields an answer that is already in "lowest terms" (but not always, e.g., 1/3 + 1/6 = 3/6). Is this virtue impor- tant? Why do we care whether 2/7 or 4/14 is used? The real reason is historical: Reducing fractions makes the evaluation of the decimal equivalent by long division easier. Do we evaluate fractions by long division any more? I don't.

Some calculators can even find the

reduced form of fractions. Calculators have changed the role of computational skills, but the fundamental computation concepts remain. For example, the "product of the denominators" approach is essential in algebra.

Example 2

Find the difference:

J_ 1_ x x + 1

Many students who have successfully completed two years of high school -algebra cannot correctly solve this problem. Why do students find so difficult a process using letters that is precisely parallel to the process using integers? It is trivial to find a common denominator, but those who have been discouraged (from fourth grade on up) from using the product of the denominators are often uneasy with the obvious possibility.

J_ 1_ _ 1(дг + 1) 1(jc) x x + 1 x(x + 1) x(x + 1)

x + 1 - x 1

x(x + 1) x{x + 1)

The concept and process of the product-of-the-denominators method is concisely represented algebraically by the following:

_ ad be ad + be Td +

~bl =

Ы At no stage does a question arise about

what to do next or how to do it. This

ARITHMETIC TEACHER



EDITORIAL

Manuscripts Needed

you have a great classroom-tested lesson that you would like -to share

with other teachers? Have you written for one of our affiliate journals, and would you like to share your ideas with a larger audience? Would you consider working with a coauthor who can help you trans- late your classroom practices into journal- istic prose? If the answer to any of these questions is "yes," the Editorial Panel of the Arithmetic Teacher: Mathematics Education through the Middle Grades encourages you to take the next step for-

ward and submit material for publication. See the masthead page for details on the submission of manuscripts.

We welcome manuscripts on any aspect of K-8 mathematics. However,the following topics are currently underrepre- sented in the journal, and manuscripts on those topics will receive priority treatment when they reach us at the Headquarters Office in Reston. Pick one, or combine several, of the following topics and write about successes you have had with your students: • Making connections

•Involving primary-level students in meaningful mathematics

• Meeting the mathematics needs of special students

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• Addressing the concerns of underrepre- sented groups in mathematics

• Integrating technology into the teaching of mathematics

•Facilitating students' communication (talking, reading, and writing) about mathematics

We look forward to reading your manuscripts.

Editorial Panel Arithmetic Teacher

sequence of equations summarizes the whole process, both arithmetically and algebraically. Conversely, it is not possible to express the least-common- denominator method of addition in a simple algebraic sequence, which serves to con- firm that the least-common-denominator process is overly complex. Mathemati- cally, the least-common-denominator approach adds nothing conceptual, makes the process more confusing, and is incon- sistent with the usual approach of algebra.

Unfortunately, some instructors still insist that students use the least-common- denominator approach. Students should be encouraged to use the product of the denominators, not discouraged by taking off points or by such remarks as, "You should have used twelfths, not twenty- fourths."

We learn some subtle lessons when we consider what we are teaching our students by emphasizing lowest terms. Although adding fractions requires regarding reduced fractions as equal to unreduced fractions (in example 1, 1/4 is converted

DECEMBER 1991

to 3/12, or 6/24), students learn that 1/4 and 3/12 are not the same; the former is "better." And as an answer to some prob- lems, "3/12" may even be marked "wrong." Thus the whole process of adding fractions becomes somewhat suspect, since "good" numbers like 1/4 are changed into numbers like 3/12, which sends up a flag: Danger! I might get a point off! It's not reduced (and is, therefore, wrong) ! Mathe- matically, 1/4 and 3/12 are equal, but we convey to students a preference for the reduced fraction.

The process of reducing fractions to lowest terms can and should be conceptually separated from the process of adding and subtracting fractions. Of course, students still need to know how to express equivalent fractions (to reduce as well as "unreduce" them). But if using the multistaged least-common-denominator approach and obtaining fractional answers in reduced form are ever deemed necessary, they should not be required until after the three-step conceptual process outlined here has been mastered. W

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The Math Department seeks applications for tenure track positions at the assist, or assoc. professor rank for Aug. 1992. Applicants should have a Ph.D. or Ed.D. by Aug. 1992 in Math Education or other comparable degree; the ability to teach a wide range of math ed courses including content and methods courses; an active research program; a commitment to teacher education and to quality teaching at both the undergraduate and graduate levels; and some K-12 teaching experience. To apply send vita, official transcripts and three letters of recommendation by Feb. 1 to: Dr. Jane O. Swafford, Math Dept., Illinois State, Normal, IL 61761. AA/EEO.

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ONE POINT OF VIEW: The Least Common Denominator