5.2 What Can We Do With a Common Numerator

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528 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 14, No. 9, May 2009

area models and investigating severalcontexts for fraction addition. Thisopening task raised the idea that the

conceptual work we had been doingin class would be directly useful intheir future teaching. I had also hopedto expose a bias towardtelling that iscommon in this population, expect-ing to hear, “Tell Sally that she needsto have common denominators, andthat she should not add the denomi-nators.” Although my students didmake this comment, one student alsosuggested, “Or you could tell Sally touse common numerators.”

I had not expected this answer, andit was a new response for me—a teach-er with almost twenty years of math-ematics experience teaching studentsand teachers. At the board, I thought

briey that the student was thinkingof the common numerator strategy forcomparing fractions, but we wereadding

fractions, in which case we do indeedneed common denominators.Back in my ofce, though, I won-

dered if I might have spoken too soon.Might it be possible to add fractionsby common numerators?

What follows is an account ofmy experience developing a novelalgorithm for adding fractions. Thealgorithm originated from a task givento elementary licensure students at thestart of a mathematics content course. The structure of the article follows theformat of a metacognitive journal, asoutlined by Roberts and Tayeh (2006).

THE STATEMENTOF THE PROBLEMI learned late in life that there arealternatives to the standard algorithmfor comparing fractions, in which we nd a common denominator andcompare numerators. In this standard

Sally claims that it is easier to addtwo fractions if she adds the numera-tors and then adds the denominators.

How can you help her?

This article was inspired by a stu-dent’s wrong answer to the precedingopening question from my class. Thestudent is in a mathematics contentcourse that is required for prospectiveelementary school teachers, most of whom are rst-year and second-yearstudents. We spend a substantialamount of time dealing with rationalnumbers. Although this is a contentcourse and not a pedagogy course,I take opportunities to tie the work we are doing in class to the teachers’future roles in the classroom.

In class, we had been drawing

Christopher Danielson

Christopher Danielson, [email protected], teaches math-ematics and mathematics methods courses at Minnesota State University,Mankato. He is interested in student thinking in all areas of mathematics,especially number.

Comparing fractions with a common numerator may not be news,but exploring the use of common numerators for adding fractionswas an adventure into unchartered territory for aspiring teachers.

What Can We

Common N

Copyright © 2009 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.



Vol. 14, No. 9, May 2009 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 529

Do with a

umerator



530 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 14, No. 9, May 2009

method, we can see that 2/3< 4/5 byusing the common denominator of 15:10/15 < 12/15. One powerful alterna-tive is to use common numerators. Inthis case, we can see that 2/3< 4/5 by

using the common numerator of 4:4/6 < 4/5. With these new fractions, we immediately see that we have 4pieces in each case, but the pieces aredifferent sizes. With common de-nominators, we standardize the size ofthe pieces and compare their number. With common numerators, we stan-dardize the number of the pieces andcompare the size of each piece.

I learned this way of thinking frommy seventh-grade students as they worked with fraction strips and in con-texts that supported their independentreasoning about fractions. Now that Iteach mathematics content courses tomostly aspiring elementary teachers,I make sure to suggest that they usecommon numerators as a strategy forcomparing fractions; I also make surethat my students know that this reason-ing came from children.

The problem I sought to solve

follows from this idea: Can we usecommon numerators in place of com-mon denominators in other situations,as well? If we can compare fractions with common numerators instead ofthe traditional common denomina-tors, can we use common numerators wherever we would normally usecommon denominators? Can we usecommon numerators to add fractions?Specically, how can we add fractionsby using common numerators?

A SOLUTION STRATEGYI began with what seemed to be aneasy example:

When we add fractions using commondenominators, the sum has the samedenominator as the addends. By anal-ogy, I chose to investigate the situation

in which the sum would have the samenumerator as the addends. In the caseof this problem, that numerator is 1. Therefore, I had this equation:

From what I know about addition,I could see that the question mark doesnot represent an integer. Consider thefollowing: One addend is 1/2, whichis the largest unit fraction less than 1.(A unit fraction is dened as a fraction with 1 as the numerator and a positiveinteger as the denominator. By thisdenition, 1/1 = 1 is a unit fraction.) The other addend is 1/4, which is lessthan 1/2, so my sum would be less than1 but greater than 1/2. So the answer isbetween the two largest unit fractions, yet the numerator would be 1.

This means that the denomina-tor of the result is between 1 and 2,specically, a compound fraction. Myrst temptation was to try 1 1/2 asthe denominator. Although averagingdenominators does not usually givethe expected results (e.g., 1/4 is not

halfway between 1/3 and 1/5; rather,it is closer to 1/5). I thought somemore and again asked myself,

When I add 1/2 and 1/4, I get one piece, but what is the size of the piece?

I knew that fractions have amultiplicative structure. For example,multiplying the numerator and de-nominator of a fraction by the samenumber yields an equivalent fraction,but adding to the numerator and de-nominator does not. So my next guess was that the unknown denominatoris the product of the denominators.However, the product was an integer,in this case, 8, which had already beenruled out. Also, using the product would yield the product of these twofractions, not their sum.

I needed a fractional denominatorand decided to investigate incorporat-

ing the product of the denominatorssomehow. With little more reasonthan that, I considered the fraction

which looked like the following:

My new complex fraction is equiva-lent to 6/8 as well as 3/4, which I knewto be the correct sum of 1/2 + 1/4.

Was this a coincidence, or had Istumbled onto a correct algorithm?I tried my new algorithm on a dif-ferent problem, whose answer I didnot immediately know. I calculated2/3 + 3/5, as shown ingure 1a . Ifound the result to be 1 4/15, which

(a)

(b)

Fig. 1 Calculating the sum of twofractions by the standard commondenominator algorithm



Vol. 14, No. 9, May 2009 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 531

looked to be about right; the answer was greater than 1 but less than 1 1/2,and the denominator seemed correct.I conrmed the answer by using thecommon denominator algorithm. Itried one more, 1/3 + 1/6, and foundthe sum to be 1/2 (seeg. 1b). This

example came out nicely, as well. Although briey pleased andcontent, I know enough mathematicsto know that examplessuggest but donot prove . Next, I considered thingssymbolically. Here was my algorithm:

I sought to prove that the right-hand side was indeed equivalent to theleft. Assuming thatb and c are posi-tive integers, my attempt at a proof was successful, as shown ingure 2.Figure 3 shows the correspondingalgorithm for subtraction.

REFLECTIONS After solving the problem, I felt asthough this proof was one that veriesbut not one that illuminates. Symboli-cally, although knowing that the algo-

rithm works, I do not feel as thoughI know why it works. The algorithmfeels separate from, rather thanintegrated with, my other fractionknowledge. I continue to seek waysto think about the algorithm so thatit truly makes sense to me and havestarted using the algorithm whenevera fraction addition problem comes up.(I acknowledge, of course, that mystatus as a mathematics teacher meansthat I am prone to add fractions moreoften than the average citizen.)

I was unsure whether I had a newalgorithm on my hands. Clearly it was new to me, but was the algorithmnew to human knowledge? I havedone a lot of reading, writing, andteaching about rational numbers and

have never run across it. That fact,coupled with the relative complexityof the algorithm and the obscure wayI stumbled into it, led me to believethat it might be completely new.

It is not. An Internet search on“common numerator” and “addition”turned up many sites that discuss botha common numerator strategy forcomparing fractions and a commondenominator strategy for adding frac-tions. A few sites also discuss a com-mon numerator algorithm for addingfractions. One article available on theInternet refers to resistance in electri-cal circuits (Ellerman n.d.). Evidently,serial resistors add resistance in theusual way, and the author refers to theusual addition as serial addition anduses the plus sign (+) to symbolize it.Parallel resistors, however, add resis-tance differently. The author denesparallel addition as

using a colon (:) to symbolize it. Par-allel addition, then, is the product di-

vided by the (serial) sum. Recall thatthis is the denominator of the sum inmy common numerator algorithm, so we can write:

I was delighted to learn that myalgorithm, in fact, does have connec-tions to useful applications. How-ever, I have not found references to acommon numerator algorithm in anysources that a middle school teacher islikely to come across.

In the examples that were workedto explore this problem, I noticed thatsometimes the denominator works outto be an integer but is often a fraction.I am curious under what circum-stances the denominator is an integer.In symbolic terms, I am curious whena + b dividesab. Surely, this is a well-known result in number theory, but I

do not know it offhand.

CONCLUSION As with all good mathematical ques-tions, this simple, seemingly bizarrequestion has expanded my ownthinking and has led to new ques-tions I have yet to answer. I hope toremind teachers of mathematics thateven the mathematics we teach on adaily basis holds possibilities for newinsights.

REFERENCESEllerman, David. “Series-Parallel Duality

and Financial Mathematics.” www.ellerman.org/Davids-Stuff/Maths/Math.htm.

Roberts, Sally, and Carla Tayeh. “It’s the Thought That Counts: Reecting onProblem Solving.” Mathematics Teach-ing in the Middle School 12 (January2007): 232− 37. ●

and then

Fig. 2 Verication that the algorithmfor adding fractions by using a commonnumerator was mathematically sound

Fig. 3 The common numerator algorithmfor subtraction is easy to guess once wehave the addition algorithm.

5.2 What Can We Do With a Common Numerator

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