Crossing the Bridge to

420 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

MATHEMATICS EDUCATORS AGREE THATratio and proportion are important middleschool mathematics topics. In fact, re-searchers have stated that proportional rea-soning involves “watershed concepts” thatare at the “cornerstone of higher mathe-

matics” (Lamon 1994; Lesh, Post, and Behr 1988).Yet assisting students in developing robust under-

standing of the many concepts and procedures thatare related to using ratios, rates, and proportions isnot straightforward. For example, being able to rea-son proportionally and being able to represent thatreasoning symbolically do not always go hand-in-hand. As with many complex topics, students’ un-derstanding grows with time and experience.

Langrall and Swafford (2000) have classified stu-dents’ proportional reasoning solution strategies intofour levels. Students at level 0 do not use any propor-tional reasoning strategies. They tend to use additivestrategies and randomly selected methods and ar-rive at the correct solution only by luck. At level 1,students use pictures, models, or manipulative mate-rials to solve proportion problems. At level 2, stu-dents might continue to use materials to make senseof a problem, but they then use numeric calculations,particularly multiplication and division, to arrive at asolution. At level 3, students’ solution strategies areformalized; students set up and solve proportionsusing cross-multiplication or equal ratios.

In our work with middle school students, we havewrestled with how to further students’ understand-ing of ratio and proportion. Vergnaud (1994) points

SUZANNE CHAPIN, schapin@bu.edu, teaches at BostonUniversity, Boston, MA 02215. She is interested in the roleof discourse in learning mathematics. NANCY ANDERSON,ncanavan@bu.edu, was the lead mathematics teacher forProject Challenge, a program for grades 4–7 in the Chelseapublic schools, Chelsea, Massachusetts, when this articlewas written. She is interested in the education of gifted andtalented students.

Project Challenge (1998–2002) is funded by the Jacob K.Javits Gifted and Talented Students Education Program of the U.S. Department of Education (grant #R206A980001). The opinions expressed in this article do not nec-essarily reflect the position, policy, or endorsement of theDepartment of Education.

Crossing the Bridge to

Copyright © 2003 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.

Formal Proportional Reasoning

VOL. 8, NO. 8 . APRIL 2003 421

out that for students to conceptualize concepts prop-erly, they have to apply them to new domains of ex-perience. Thus, creating classroom situations thatextend students’ knowledge and help them recog-nize the same mathematical “structure” in differentcontexts can support transfer and generalization.When teachers assist students in classifying and for-malizing these structures and explicitly linking themto existing knowledge, learning is both strengthenedand deepened. One topic that is often new to stu-dents in the middle grades and involves proportionalreasoning is similarity. We have used this topic as avenue for discussing how level 2 solution strategieslink to the formal solution strategies of level 3. Whenexamining the symbolic representation of problems,we have also facilitated discussions about the multi-plicative relationships within and between ratios toassist students in expanding their solution methods.

In this article, we share some of our experiencesworking with a group of seventh-grade students whowere part of Project Challenge, a federally fundedcollaboration between a university and a publicschool system. We first provide some general back-ground information about solving proportions and

about the concept of similarity. We then discuss howstudents made the transition from informal concep-tual methods of solving proportional reasoning prob-lems to a more formalized method of solving propor-tions, using a similarity problem from the Stretchingand Shrinking unit in the Connected Mathematicsseries. (Anderson is the lead teacher of the seventhgraders, and Chapin is the project director.)

Relationships within and between Ratios

ALL RATIOS HAVE MULTIPLICATIVE RELATION-ships between the measures. For example, in theratio 2:10, if we multiply the first value (2) by 5, weobtain the second value (10); if we multiply the sec-ond value (10) by 1/5, we obtain the first value (2).In some ratios, these multiplicative relationships areeasy to identify using mental mathematics, but inother ratios, the relationships that exist between thevalues are not as obvious and must be determinedthrough multiplication or division. For example, inthe ratio 8:36, if we divide 8 by 36, we find that 8 is2/9 the size of 36. If we divide 36 by 8, we find that 36is 9/2, or 4.5 times, the size of 8.

Formal Proportional ReasoningS U Z A N N E H. C H A P I N A N D N A N C Y C A N A V A N A N D E R S O N

422 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

When two ratios are equal, the multiplicative re-lationships “within” each of the individual ratios arethe same, or invariant. As the example belowshows, multiplying the first number in both ratiosby 5 results in the second number in each ratio.

When students understand that the multiplicativerelationships are the same within each ratio in a pro-portion, they can use this knowledge to find a missingvalue, a solution method often referred to as thescalar method. For example, to determine the value ofx below, we identify the multiplicative relationshipwithin the first ratio, which is multiplying by 5, thenmultiply the given value in the second ratio, 6.8, bythe same factor (6.8 × 5 = 34). In this ratio, x equals 34.

Multiplicative relationships “between” the two ra-tios in a proportion also exist. In the first exampleabove, if we multiply both numbers in the ratio 2:10by 1.5, we obtain the values in the second ratio, 3:15.We can use this method, often referred to as thefunctional method, to find x in the proportion below.

The number 6.8 is 3.4 times as large as 2. Thus, wemultiply 10 by the same factor, 3.4, to determinethat the value of x is 34.

Students can use relationships within ratios andbetween two ratios to solve missing-value proportionproblems. Frequently, however, students do not an-alyze the relationships among symbols, do not rec-ognize patterns, and thus, do not use the relation-ships within and between ratios. Instead, they tend

to use the cross-product method. The cross-productmethod is most appropriate when multiplicative rela-tionships within or between ratios are not obvious,but it should be one of a number of methods thatstudents use, not the only one. We believe that in-struction should help students to identify the moststraightforward relationship that requires the leastamount of computation, then use that relationship tosolve the problem by finding the missing value. Thisapproach allows students to use their conceptual un-derstanding of proportional relationships to developprocedural fluency with proportions.

Why Similarity?

THE CONCEPT OF SIMILARITY IS AN IMPORTANTone that is encountered in many situations, such asenlargements, reductions, scale factors, projections,and indirect measurement. Understanding similarityhelps students not only to link geometry with numberbut also to develop an understanding of the geometryin their environment. Furthermore, because thelengths of the corresponding sides of similar figuresare proportional, the many relationships within andbetween ratios can be investigated and discussed.

For example, in figure 1, triangle ABC is similarto triangle DEF because the measures of their corre-sponding angles are equal and the lengths of theircorresponding sides increase by the same factor,called the scale factor; in other words, the lengthsare proportional. The scale factor from ▲▲ABC to▲▲DEF is 3; multiplying the length of each side of▲▲ABC by 3 results in the lengths of the correspond-ing sides of ▲▲DEF. Likewise, the scale factor from▲▲DEF to ▲▲ABC is 1/3 because multiplying thelength of each side of ▲▲DEF by 1/3 gives thelengths of the corresponding sides of ▲▲ABC.

One reason we introduce the topic of similar fig-ures to shift students into using level 3 solution meth-ods is that many equivalent ratios exist that relatesimilar triangles. For example, one ratio that can beconstructed compares the length of a side of one tri-angle to the length of the corresponding side of thesimilar triangle, that is, AB:DE. This ratio is equiva-lent to the comparison of the two hypotenuses of thetriangles, or AC:DF. One way to determine the lengthof DF is to set up a proportion, note that the “within”multiplicative relationship in the first ratio is the sameas the scale factor, which is 3, and realize that the re-lationships in the second ratio are invariant and, thus,will also be linked by the scale factor. Hence, we solvefor x by multiplying 2.5 by 3 to get 7.5.

= or × 3 = × 3 x = 7.5ABDE

ACDF

1.54.5

2.5x

2:10 = 6.8:x or = 210

6.8x

× 3.4

× 3.4

× 3.4

× 3.4

2:10 = 6.8:x or 5 × = × 5 ⇒ x = 34 210

6.8x

× 5 × 5

2:10 = 3:15 or 5 × = × 5 210

315

× 5 × 5

Fig. 1 Similar triangles ABC and DEF

A

C

D

x

E F

B

1.5

4.5

2.5

VOL. 8, NO. 8 . APRIL 2003 423

Another proportion that can be constructed to findthe length of DF is shown below. In this situation,the “between” multiplicative relationship from oneratio to the other ratio is the scale factor, 3, and theprocedure for finding the missing value (DF), is tomultiply 2.5 by the scale factor (2.5 × 3 = 7.5).

Classroom Implementation

HOW DO TEACHERS HELP STUDENTS MAKE THEtransition from using the informal conceptual solu-tion methods of levels 1 and 2 to the formal propor-tions of level 3? The next sections describe an activityand discussion from a Project Challenge class thathelped students cross this bridge. Before and duringthe five-week unit on similarity, Project Challengestudents had many opportunities to reason aboutproportional situations; to find unit rates; to constructequivalent ratios; and to link ratios, percents, andfractions. They were using level 2 solution strategies,even though some instruction had focused on settingup and solving proportions. Discussions of solutionstrategies emphasized students’ understanding ofthe multiplicative relationships inherent in the prob-lems. By the end of the similarity unit, studentsseemed ready to link informal and formal strategies.One particular problem emerged as being extremelyuseful in helping students link what they knew aboutproportions across contexts and was discussed fortwo hour-long mathematics periods.

Exploring a Problem

THE FOLLOWING PROBLEM FROM STRETCHINGand Shrinking was used to explore proportionality:

Mr. Anwar’s class is using the shadow method toestimate the height of their school building.They have made the following measurementsand sketch [see fig. 2]:

Length of the meterstick = 1 mLength of the meterstick shadow = 0.2 mLength of the building shadow = 7 m

Use what you know about similar triangles to findthe building’s height from the given measurements.Explain your work. (Lappan et al. 1997, p. 60)

The lesson began with a short discussion about

whether the triangles were similar. Once the studentshad determined that the angles in the two triangleswere congruent and, thus, that the triangles weresimilar, they set to work using what they knew aboutsimilar shapes to find the missing height. Working inpairs, most students decided to use the scale factor tofind the missing height. Many students used thesenumber sentences to solve the problem:

0.2 × ? = 7 7 ÷ 0.2 = 35 1 × 35 = 35

The majority of students were using level 2 solu-tion strategies; they recognized the relationships inthe problem as multiplicative and used division andmultiplication to find the scale factor and missingheight. They were not at level 3, because they hadnot set up a proportion and solved for the unknown.A few pairs of students, however, did use formalproportions to find the height of the building, whichenabled us to introduce more students to formalproportional solution methods through discussion.Our goal was for students to link their informal level2 approaches that used proportional reasoning tothe more formal symbolism and process of settingtwo equivalent ratios equal to each other.

Discussion of Strategies and Solutions

DURING THE FIRST PART OF THE CLASS DISCUS-sion, many students explained how they used multi-plication to solve the problem. One student said, “Ifigured out that 0.2 × 35 is 7, so I multiplied 1 × 35 toget the height of the building, which is 35.” Anotherclassmate elaborated—

I knew that I had to find the scale factor to find the height. Ifigured out that 0.2 × 35 was 7 since 7 ÷ 0.2 was 35. Thistold me that the scale factor for the two triangles was 35. Sothen I could use the scale factor times 1 to get the height ofthe building since it corresponds with the meterstick.

= or = x = 7.5

× 3

× 3

ABAC

DEDF

1.52.5

4.5x

Fig. 2 Similarity problem (adapted from Lappan et al. 1997, p. 60)

?

7 m 0.2 m

1 m

424 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

A third student drew the picture shown in figure 3on the board, linking the legs of the triangles witharrows and the values with expressions.

These responses, as well as others made in pairsand whole-class discussions, indicated that the stu-dents had a strong understanding of the componentsof proportional reasoning. All students knew that thechange between corresponding side lengths was mul-tiplicative, not additive in nature. No student tried tofind an additive relationship between the side lengths.Instead, all students, through a variety of strategies,tried to identify the factor that described the multi-plicative relationship between 0.2 and 7. Further, theyknew that the multiplicative change between corre-sponding sides was invariant. Once students foundthe multiplicative relationship between the shadows,they immediately reasoned that if the change be-tween shadows was “times 35,” then the change be-tween actual heights must also be “times 35” becausethe scale factor is constant in similar figures.

Having demonstrated such a strong understand-ing of proportional reasoning while using informalstrategies, the students seemed ready to exploremore formal level 3 strategies. The teacher calledon one pair of students who had solved the problemusing a proportion to explain their solution:

We used a proportion to solve the problem. We wrote 1over 0.2 equals N over 7. You can solve this problem byfinding what times 0.2 equals 7. To get this [number], youdivide 7 by 0.2, and that’s 35. So then, up top, 1 × 35equals 35; N is 35.

As this student spoke, the teacher recorded hersteps on the board (see fig. 4).

Once this relationship between ratios was shownwith arrows, many other students contributed to thediscussion by articulating their understanding oftheir classmates’ methods. Students could explainhow to set up the proportion by making sure thatcorresponding sides were compared accurately. Forexample, many students stated that one of the mostimportant aspects of setting up this proportion cor-rectly was that “the height has to go with the heightand the length has to go with the length.” Whenasked to compare the solution strategy that usedmultiplication and division number sentences (level2) with the proportion method (level 3), studentsmade the following observations:

• “The methods look different since the first oneuses number sentences, but the other uses ra-tios. They are similar because, in each, you aremultiplying and dividing the same numbers.”

• “Both methods use the same lengths. Both usemultiplication and division. It’s just that in one,you use a proportion.”

• “They are similar because, in the end, you endup dividing 7 by 0.2 to get the scale factor.”

• “The proportion is more organized. The height isover the length, and you have an arrow goingover to the other height and length. In the othermethod, there’s a picture and number sentencesall over the place.”

The discussion took another turn when one stu-dent noted that the relationship between ratios wasthe scale factor; that is, if you multiply the values inone ratio by 35, the result is the second ratio. An-other insightful student asked, “What if you made adifferent proportion of shadow over shadow equalsheight over height? Do you still multiply one ratio bythe scale factor to get the other ratio?” The class de-cided to investigate the proportion that followed fromthis question, as shown below:

One student said, “You don’t multiply by thescale factor. Since 0.2 × 5 = 1, you multiply by 5. So

0 2

7 N

1.=

Fig. 3 A student’s informal solution method

Fig. 4 A student’s formal solution method

VOL. 8, NO. 8 . APRIL 2003 425

7 × 5 equals 35. The answer is thesame no matter which way you set upthe ratios.” Next, the teacher asked,“Where is the scale factor?” Anotherstudent responded, “In the ratio ofshadow over shadow, you ask your-self what times 0.2 equals 7, and thatis 35! So you times 1 by 35 to find N.The scale factor is going up anddown—between the numbers in theratio, not across.”

The teacher then asked students tocompare the two proportions. Studentsconcluded that when they constructedratios so that corresponding parts ofeach of the triangles were in differentratios (see fig. 4), multiplying by thescale factor connected the ratios. Yetwhen they constructed ratios that com-pared corresponding parts of each tri-angle in the same ratio (see the exam-ple on page 424, bottom of page),multiplying the first number in eachratio by the scale factor produced thesecond number in the same ratio.

Closing Comments

STUDENTS’ COMMENTS SUPPORTEDthe decision to push them beyondlevel 2 strategies into level 3 strate-gies. The students could readily iden-tify many relationships between theinformal strategy of using multiplica-tion and division and the formal use ofproportion. For example, they sawthat both methods used the samenumbers and the operations of multi-plication and division to find the scalefactor, relationships between sidelengths, and ultimately, the missingheight. They understood that depend-ing on how they constructed the ra-tios, the multiplicative relationshipthey discovered by dividing (namely,the scale factor) appeared within ra-tios or between ratios. Students sawhow a proportion can be used to orga-nize the multiplicative relationshipsthey identified in the problem and tosolve problems. As a result of linkingthe informal and formal strategies, in-stead of presenting them as unrelatedsolutions or focusing only on one, thestudents could see the benefits ofusing formal proportions when solv-

ing similarity and other types of mea-surement problems.

Many research studies indicatethat students are more likely to useand remember mathematical strate-gies if they understand them. Stu-dents should be encouraged to set upand solve formal proportions oncethey can recognize a situation as hav-ing invariant multiplicative relation-ships within and between measures.Specifically linking informal and for-mal procedures also appears to helpstudents move toward symbolic repre-sentation of proportionality.

References

Lamon, Susan. “Ratio and Proportion: Cogni-tive Foundations in Unitizing and Norm-ing.” In The Development of MultiplicativeReasoning in the Learning of Mathemat-ics, edited by Guershon Harel and JereConfrey, pp. 89–120. Albany, N.Y.: StateUniversity of New York Press, 1994.

Langrall, Cynthia W., and Jane Swafford.“Three Balloons for Two Dollars: De-veloping Proportional Reasoning.”Mathematics Teaching in the MiddleSchool 6 (December 2000): 254–61.

Lappan, Glenda, James T. Fey, William M.Fitzgerald, Susan N. Friel, and Eliza-beth Difanis Phillips. Stretching andShrinking. Palo Alto, Calif.: Dale Sey-mour Publications, 1997.

Lesh, Richard, Thomas R. Post, and Mer-lyn Behr. “Proportional Reasoning.” InNumber Concepts and Operations in theMiddle Grades , edited by JamesHiebert and Merlyn Behr, pp. 93–118.Reston, Va.: National Council of Teach-ers of Mathematics, 1988.

Vergnaud, Gerard. “Multiplicative Con-ceptual Field: What and Why?” In TheDevelopment of Multiplicative Reason-ing in the Learning of Mathematics,edited by Guershon Harel and JereConfrey, pp. 41–60. Albany, N.Y.:State University of New York Press,1994. �