MAKING CONNECTIONS: A CASE FOR PROPORTIONALITY
Author(s): Kathleen Cramer and Thomas Post
Source: The Arithmetic Teacher, Vol. 40, No. 6, FOCUS ISSUE: EMPOWERING STUDENTS
THROUGH CONNECTIONS (FEBRUARY 1993), pp. 342-346
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/41195594 .
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
ACASEFOR
MAKING
CONNECTIONS:
PROPORTIONALITY
Kathleen Cramer and Thomas Post
is oneform
reasoning
ofmathematical
Many
reasoning.
Proportional
aspectsof ourworldoperateacrules.Inthescience
toproportional
cording
surfaceswhen
classroom,proportionality
beams
when
balance
is
density explored,
rates
areused,andwhenanytwoequivalent
classare compared.In the mathematics
whenpropsurfaces
room,proportionality
ertiesof similartrianglesare examined,
whenscalingproblemsare investigated,
are defunctions
and whentrigonometric
reaofproportional
fined.The importance
and
in
the
Curriculum
is
stressed
soning
EvaluationStandardsfor School Mathematics(NCTM 1989,82).
The abilityto reasonproportionally
developsin
students
through
grades5-8. Itisofsuchgreatimpormustbe
timeandeffort
whatever
tancethatitmerits
Students
expendedtoassureitscarefuldevelopment.
need to see manyproblemsituationsthatcan be
reamodeledand thensolvedthrough
proportional
soning.
Proportional
is more
reasoning
the
thanlearning
cross-product
algorithm.
whitepaintisneededif9 partsoftheblue
paintis used?(2/3= 9/v;Ix = 27; л =
13.5)
Post,Behr,and Lesh (1988) believed
tomissing-value
thatusingsolutions
probof proportional
lemsas thesole indicator
since
reasoningis muchtoo restrictive,
themselves
to
lend
answers
purely
algorithDefiningProportionality
Thefactors
micandpossiblyrotesolutions.
The abilityto solve missing-value
probreasonproportional
thata student involvedin defining
lemshasbeenusedtoindicate
are
more
complex.
reasoner
is a proportional
(Karplus,Pulas, ing
One way to documentknowledgein
andStage 1983;Noelting1980).Missingis to describethebehaviors
mathematics
arethetypicaltasksfound
valueproblems
We understand
that
depictunderstanding.
textbooks
in middleschool mathematics
tobe a proporwhat
it
means
about
whereinthreeor fourvalues in two rate enough
itinvolvesthe
that
thinker
to
realize
tional
istobe found.
pairsaregivenandthefourth
The followingis a missing-value
story following:
to
The
standard
algorithm
taught
problem.
solvethistypeofprobleminvolvessetting
andusingthecross-product
upa proportion
algorithm.
formixinga certainshade
The formula
ofbluepaintis 2 partsbluepaintand 3
partswhitepaint.Atthisrate,howmuch
KathleenCramerteachesat theUniversity
of Wisconsin,RiverFalls, WI 54022. She has studiedthe
teachingand learningof rationalnumbersfor a
number
ofyears.ThomasPostteachesat theUniversityof Minnesota,Minneapolis,MN 55455. He is
thecodirector
oftheNationalScienceFoundationsponsoredRationalNumberProject.
• Knowingthemathematical
characteristicsof proportional
situations
• Beingabletodifferentiate
mathematical
of proportional
characteristics
thinking
fromnonproportional
contexts
• Understanding
realisticand mathematical examplesof proportional
situations
• Realizingthatmultiplemethodscan be
tasksandthat
usedtosolveproportional
thesemethodsarerelatedto eachother
• Knowinghowto solvequantitative
and
tasks
proportional-reasoning
qualitative
• Beingunaffected
by thecontextof the
in thetask
numbers
the mathematical
charUnderstanding
acteristics
of proportional
situations
is the
mostimportant
partof thispicture.One
characteristic
ofprocriticalmathematical
portionalsituationsis the multiplicative
thatexistsamongthequantirelationship
Thismultithesituation.
tiesthatrepresent
can
be
explored
plicative relationship
and
tables,
algebraic
expressions,
through
mathcoordinate
the
By
graphs. examining
situematical
characteristics
ofproportional
ofmaking
ations,one sees theimportance
mathematical
connections
thatwill inevitofunction
intellitablyempowerstudents
gentlywhensolvingproblems.
Thisarticlefirst
exploresthemathematiof proportional
situacal characteristics
tionsandthenexplainsthatthisknowledge
willenablestudents
tosolvea proportionalreasoningproblemin severalways.The
articleconcludesby analyzingthemathematicalconnectionsthatare made as
mathematical
characteristics
are explored
and applied.
QKJ
Mathematical
Characteristicsof
ProportionalSituations
ofall
Asjuststated,
thecritical
component
is
situations
the
multiplicative
proportional
thatexistsbetweenthequantirelationship
TEACHER
ARITHMETIC
342
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In all proportional
the nusituations,
mericalrelationship
between
can
quantities
be expressedbya ruleintheformу = mx9
Scolino problem
wherem is one of the constantfactors
thetwoquantities.
1 cm
2 cm
3 cm
4 cm
relating
5 cm
Map distance
The graphof theruleу = 5л:,withу =
Actualdistance
5 km
10 km
15 km
20 km
25 km
actualdistanceandx = mapdistanceshows
another
mathematical
characteristic
ofprosituations
The
(see fig.1).
portional
graph
ofу = 5x is a straight
line,climbsfromleft
to right(has positiveslope), and passes
theorigin.The graphofу = 1/5*
through
has
similar
characteristics.
In all proporThe graph offу s 5x is a proportionalsituation*
tionalsituations,
thepointsofthegraphlie
on a straight
line. In real-world
settings
theselinesalwayshavepositiveslope.The
on the
pointsfromtable1 arehighlighted
25
with
two
graphalong
points,(1.5,7.5) and
~T/r'5',2?'
on the
(2.5, 12.5),whicharenotrecorded
table.Notethatthesepointsfallontheline.
All ratepairsforthe givenproportional
situation
fallon theline.
20
linearequationsis a topicdeGraphing
/ 4'| |Ь
in
the
middlegrades.Students
veloped
variations
of
thegenerallinearequagraph
tionу= mx+ b,exploredifferent
definitions
8
,5
of
describe
the
characteristics
ofline
15)-/-(3,
slope,
and
relate
the
characteristics
ofline
graphs,
to
the
is
graphs
generalequation(m slope
and b is j-intercept).By applyingour
ofstraight-line
tographs
<
knowledge
graphs
10
HM2, 10)of proportional
we
see
thatfor
situations,
theruley= mx,mistheslopeoftheline.The
situations
slopeofthelineofproportional
is alwaystheconstantfactorrelating
the
twoquantities.
For ourmap examplethe
constantfactor5, whichcan be used to
definetherelationship
betweenmap distanceand actualdistanceas v = 5л' is the
slopeofthelineforthegraphofу= 5л'We
also
knowthatу = mxcrossestheorigin
123456789
10
becausein thisequation"ft"is 0.
Map distance (cm)
Another
characteristic
ofprointeresting
situations
can
be
shown
portional
by rethe
different
rate
cording
pairs(actualdisfound
in table 1 as
tiesthatrepresent
distance)
thesituation.
Consider twoways.Ifwe multiply
tance/map
mapdistanceby
fractions:
5/1,
10/2,
15/3,20/4,and 25/5.
thisproportion
5 km/cm,
we findthecorresponding
number
problem:
All
these
fractions
have
a valueof5. Again,
ofkilometers
fortheactualdistance.If we
The scale on a map suggeststhat 1
we
see
the
of
the
constant
factor
5.
presence
theactualdistanceby 1/5cm/km,
centimeter
an actualdistance multiply
represents
The
rate
reciprocal pairs,1/5,2/10,3/15,
we findthecorresponding
number
ofcenof 5 kilometers.The map distance
timeters
forthemapdistance.
Theconstant and so on, all have a value of 1/5.This
betweentwo townsis 8 centimeters.
of equivalentrate
factors
5 or 1/5canbe usedtoexpresseither special characteristic
Whatis theactualdistance?
situations
formofthenumerical
relationship
algebra- pairsthatexistinallproportional
A tablecan helphighlight
enablesone to use thecross-product
thisrelation- ically:
algoship(see table1).
actualdistance= 5 km/cm
x mapdistance rithm.
The numericalrelationship
thatexists
Thefollowing
listsummarizes
themathor
between
thetwoquantities
ofmapdistance
ematical
characteristics
of
situproportional
= - cm/km
x actualdistance.
mapdistance
and actualdistancecan be expressedin
ations:
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FEBRUARY
1993
343
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All use subject to JSTOR Terms and Conditions
1. A constantmultiplicativerelationship
exists between two quantitiesand can be
expressedin two ways.
2. All ratepairs describinga given proportionalsituationareequivalent.The same
statementis trueof the reciprocalof these
ratepairs. These two constantsof proportionalitydefinethemultiplicativerelationship.
3. The rulethatexpressesthemultiplicativerelationshipis alwaysy = mx,wherem
is one of the constantsof proportionality.
4. Graphically,all points for a proportionalsituationfall on a straightline passing throughthe origin. In real-worldsettingstheselinesalwayshave positiveslope.
All ratepairsfortheparticularsituationfall
on the line.
5. The slope of the line is the m in the
equationy = mx and is one of the constant
factorsrelatingthetwoquanmultiplicative
titiesy and x.
can
Proportionality
be explored
tablesand
through
algebraic
sentences.
11111Ж1
Running-lapsproblem
Julie's laps
3
4
5
6
7
8
Sue's laps
9
10
11
12
13
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The importanceof understandingthe
ofproportional
characteristics
mathematical
situationscan be highlighted
bythefollowtwo
problems:
ing
1. If you travelto a foreigncountry,you
exchange dollars for the currencyused
there.In Englandyou could exchange$3
for2 pounds. How many pounds could
you exchange for$21?
2. Sue and Juliewere runningequally fast
around a track.Sue startedfirst.When
she had runninelaps, Juliehad runthree
laps. When Juliehad completedfifteen
laps, how manylaps had Sue run?
thetwoproblemslookalike.
Superficially
Each containsthreepieces of information
withone unknown.Theylook likemissingvalue problemsfoundina sixth-or seventh-
1
0 I I I I 1 I I I I I I If I I I I I I I I I I I
1
234
56789
Julie'slaps
out of
grade mathematicstext.Thirty-two
students
solved
both
thirty-three
preservice
a
problems by settingup proportionand
using thecross-productalgorithmto reach
an answer.The cross-productalgorithmis
an appropriate strategyfor the moneyexchange problem because that context
representsa proportionalsituation.It is not
an appropriatestrategyfortherunning-laps
problem because that problem is not a
proportionalsituationand depictsan additive, and not a multiplicative,situation.
An understandingof why procedures
10
work and under what conditions procedurescan be applied are objectivesthatare
oftenlacking in mathematicsinstruction.
When one has a superficialunderstanding
of a concept,it is easy to applymemorized
rules in the wrongsituation.
If we look closely at the running-laps
problem, checking it against our list of
characteristicsof proportionalsituations,
we can see the importanceof having a
deeper understandingof this context.Being able to look beyond the superficial
characteristicsof proportionalsituations
ARITHMETIC
TEACHER
344
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All use subject to JSTOR Terms and Conditions
^^^^^^^н
Problem:Completethetable.Are the
data relatedproportionally?
Definethe
ruleforthedatapredicting
у whenx is
given.
3 6 9 12 15
j
~x 2 4 6 8 10 16 19
Thegraphof y s (3/2)x
Possiblesolutions:
Ifthenumber
ofpairs
are relatedproportionally,
thenthe rate
fracpairsshouldforma setofequivalent
tions.The fractions
3/2,6/4,9/6,12/8,and
15/10 allhavea valueof3/2,so thedataare
related.This factvalidates
proportionally
theuse ofthecross-product-algorithm
solutionstrategy
to findthecorresponding
y-valuesforx = 16-3/2 = y/16,2y= 48,
andу = 24; and forx = 19-3/2 = y/19,
2y = 57, and v = 57/2.Since thisis a
the constantrate,
situation,
proportional
24
21
6
Manyaspectsof
ourworldoperate
accordingto
rules.
proportional
-Jh
1 2 3 4 5 6 7 8 9 10 111213141516171819
20
v
will enablestudents
to makeappropriate
can
decisionsas towhenandifa procedure
be applied.
the
a tablehelpstoidentify
Constructing
betweenthe two
numericalrelationship
(see table2).
quantities
betweenthe
The numerical
relationship
number
of Julie'slaps andthenumberof
Sue's lapscan be expressedalgebraically:
Sues laps = Julieslaps + 6. Thenumerical
is a constant
sum,nota conrelationship
stantfactor.
line
The graphin figure2 is a straight
withpositiveslope,butthelinecrossesthe
y-axisat (0, 6) andnotat theorigin.Ifthe
thenthegraph
situation
wereproportional,
theorigin.If we
wouldhave intersected
- 9/3,
expresstheratepairsas fractions
1
we
see
that
10/4, 1/5,12/6
theydo not
situahavethesamevalue.In proportional
are
all
rate
tions,
equivalent.The
pairs
based
on equivalent
proportion
algorithm
soluis nottheappropriate
rates,therefore,
forthisexample.
tionstrategy
of the
Havinga deeperunderstanding
characteristics
ofproportional
mathematical
to solve probenablesstudents
situations
As
lemsin multiple
Behr,and
Post,
ways.
with
students
Lesh( 1988) state,
"Equipping
a variety
ofperspectives
andsolutionstratnotonlybetter
understanding
egiesfosters
but also a more confidentand flexible
approachto problemsolving."
ApplyingKnowledge
of the Mathematical
Characteristicsof
ProportionalSituations
A student
of
can use hisor herknowledge
themathematical
characteristics
ofproportionalsituations
tosolvethefollowing
task
in severalways.
3/2, is the constantfactorforthe rule
relatingthedata in thetable;у = (3/2)uc.
of 3/2,is the
Notethat2/3,thereciprocal
otherconstant
factorrelating
x andy. The
=
x
would
be
usedifone
equation (2/3)y
was givena y-valueandx was unknown.
A solutionstrategy
usingthegraphing
is possible.The graphin
characteristics
3 connectsthedatapointsfromthe
figure
table by a straight
line. The graphis a
line
with
straight
positiveslope,andwhen
itis extendeditpassesthrough
theorigin.
One can concludethatthedataareproportionallyrelated.
Sinceall datapointswillfallontheline,
thelinecan be extendedupwardand the
corresponding
y-value forx = 16 can be
foundas showninfigure
4. Thegraphisless
for
the
helpful
finding
corresponding
y-valueforx = 19,sinceitcrossesbetweeen
the
values.Therulefordescribing
integral
datawouldbe helpful.
To findtherulefirst
findtheslopeofthe
which
can
be
found
fromthegraph.It
line,
is theratioofthevertical
distancebetween
FEBRUARY
1993
345
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All use subject to JSTOR Terms and Conditions
twopointson thegraphto thehorizontal
4
Theright
seeninfigure
distance.
triangle
showstheslopetobe 3/2.Notethatforany
twopointsselectedtheslope will have a
valueequalto 3/2.The slopeofthelineis
them in theequation,so theruleis y valueforx = 19 can
Thecorresponding
(3/2)дг.
be foundusingtheformula
у = (3/2)(19).
Multiplesolutionstrategiesgive stuthat
dentsthepowerto choosea strategy
bestfitsthedata;theresourcestheyhave
available,such as a graphingcalculator;
Ingeneralit
andtheir
personal
preferences.
is alwaysa good idea to view a concept
frommultiple
perspectives.
щл^тЯ
Using the graph of у в (3/2)х to findmissing data and slope
-ШШШШШ
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24 ---------------4
Mathematical
Connections
i
i
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i
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i
1(16,24)
:шшрат
havebeenmadeas we
Whatconnections
characteristics
the
mathematical
explored
ofproportional
situations?
First,we repre12
-ýh
contextsin a tableto
sentedproportional
We thentransexaminenumberpatterns.
latedthe numericalrelationship
relating
intoalgebraicsentences.
thetwoquantities
thealgebraicrepresentation
We translated
to a graphicalrepresentation.
Examining
the relationships
repreamongdifferent
is important.
Different
sentations
represendifferent
tationshighlight
aspectsof the
andintereach
situation, fostering
insights
totheother.NCTM's curricuconnections
1 2 3 4 5 6 7 8 9 1011 12 131415 1617181920
lum standards
(1989) emphasizethe imof
connections
among
portance making
x
tables, algebraic generalizations,and
Proportionality
graphicalrepresentations.
a goodwaytomaketheseconnecpresents
tions.
ofthemath- todetermine
ourexploration
To continue
whether
thecontext
is propor- References
situ- tionalor
characteristics
ofproportional
ematical
stunonproportional.
Empowering
Karplus,Robert,StevenPulas,andElizabethStage.
ations,we used our knowledgeof line dentsmeansnot
them
with
onlyequipping
"Proportional
ReasoningandEarlyAdolescents."
to
make
adand
^-intercept
graphs,slope,
InAcquisition
to solve problemsbut helping
ofMathematics
Conceptsand Prostrategies
about graphsof
ditionalgeneralizations
cesses,editedbyRichardLeshandMarshaLandau.
themunderstand
so
underlying
concepts
New York:AcademicPress,1983.
Studentsneed to
situations.
proportional
ata
theycan applystrategies
appropriately
NationalCouncilofTeachersof Mathematics.
Curunderstand
thattheycanoftenuseoneform later
stage.
riculumand Evaluation Standardsfor School
of a mathematicalidea to help them
Mathematics.
In conclusion,by exploringthemathReston,Va.: The Council,1989.
anotheridea. This "persistent ematical
understand
characteristics
ofproportional
situ- Noelting,Gerald.TheDevelopmentofProportional
attention
to recognizing
anddrawingconReasoningand theRatio Concept,Part 1- the
ations,studentslearna varietyof very
nectionsamongtopicswill instillin stuDifferentiation
ofStages,EducationalStudiesin
mathematical
forsolvimportant
strategies
Mathematics
II, 217-53. Boston:ReidellPublishthatthe ideas they
dentsan expectation
ingproportional-reasoning
problems.
ingCo., 1980.
learnareusefulin solvingotherproblems
ofproblems
Different
serveas difPost,Thomas,MerlynBehr,andRichardLesh."Prorepresentations
andexploring
other
mathematical
concepts" ferentlenses throughwhichstudentsinterpret
and the Developmentof Prealgebra
the
portionality
In TheIdeas ofAlgebra,K-I2,
Ifstudents
andthesolutions.
aretobecome
(NCTM 1989,85).
Understandings."
problems
1988YearbookoftheNationalCouncilofTeachmustbeflexible
mathematically
they
powerful,
enough
byusing to
Understanding
proportionality
ers of Mathematics,
editedby Arthur
ina variety
ofwaysandrecogF. Coxford
approachsituations
to
severalrepresentations
enablesstudents
and AlbertP. Shulte,78-90. Reston,Va.: The
nizetherelationships
amongdifferent
pointsofview
and
evaluateproblemsituations
Council,1988.•
(NCTM 1989,84).
critically
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TEACHER
ARITHMETIC
346
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