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Teaching Algebra in the Middle Grades Using Mathmagic

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Teaching Algebra in the Middle Grades Using Mathmagic
Teaching Algebra in the Middle Grades Using Mathmagic
Author(s): HARI P. KOIRALA and PHILLIP M. GOODWIN
Source: Mathematics Teaching in the Middle School, Vol. 5, No. 9 (MAY 2000), pp. 562-566
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/41181758
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^HcflB
Math
LARGE NUMBER OF MATHEMATICS
EDUCATORS
and teachers argue for includingalgebra in
the middle school mathematicscurriculum
(Fouche 1997; Silver 1997). Recommended
algebraicconcepts to be taughtin the middlegrades include variable,expression,and equation (NCTM 1989),
and middle-gradestudentsshould be able to "applyalgebraic methods to solve a varietyof real-worldand
mathematicalproblems" (NCTM 1989,102). In spite of
this emphasis on teachingalgebra,a large numberof
and sixthmiddleschool students,especiallyat the fifthgrade levels,are nevertaughtalgebraicconcepts.
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Story
Manyteachersthinkthatalgebrais tooabstract
attheselevels.Evenwhenteacherstry
forstudents
to teach algebra,theirstudentsseldomsee the
ofx's andys
valueoflearningit.The introduction
contextdoes notmakesense
without
a meaningful
inmiddleschool.
ofstudents
tothemajority
in mind,we wantedto creWiththesethoughts
ate a contextthatwouldmakelearningthe algebraicconceptsofvariableandexpression
meaningThe contextthatwe used is called
fulforstudents.
in whichstudentsare invitedto play
mathmagic,
The MathmagicActivity
AT THE BEGINNINGOF THE CLASS,THE STUdentswere invitedto play witha simplemagic
suchas "thinkofa number,
add 7, add 3,
problem,
and subtractyouroriginalnumber."When they
completedthe computations,
theyall discovered
thattheyended up withthe same number,10.
intheclass,
Thissimplemagicsurprised
everyone
and theywantedto knowhow it worked.Aftera
briefdiscussion,the studentsrealizedthatthe
HARI P. KOIRALA
and PHILLIP
M. GOODWIN
Q
a.
$
<r
in
i
er
¡
Í
at
ë
cr
|
I
a.
ofa number,
add 4,
numbergames,suchas "think
itby2,"andso on.The ideaofusingmathmultiply
inlearning
mathematics
students
magictomotivate
educators.
has been recognizedby mathematics
that
Lovitt
andClarke(1988)reported
Forexample,
amountofexmathmagic
generatesa tremendous
To motivate
citement
andinterest
amongstudents.
ourstudents
andto makelearning
algebrafunand
witha combined
engaging,we triedmathmagic
and sixth-grade
class. Even thoughthe stufifthdentshad littleor no knowledgeof algebra,they
to learnit throughmathwere highlymotivated
becausetheycouldtrythemagic
magic,especially
members
athome.
withtheirpeersandwithfamily
teaches
mathHARIKOIRALA,
[email protected],
ematicsand mathematics
education
coursestopreservice
teachersat EasternConnecticut
State
and in-service
CT 06226. He is interested
in
Willimantic,
University,
students
to
learn
mathematics
conceptually.
motivating
PHILLIP
teaches
mathGOODWIN,
[email protected],
ematicsat LebanonMiddleSchool,Lebanon,CT 06249.
interested
in teaching
He isparticularly
concepts
algebraic
toyoung
children.
theirorigimagicworkedbecausetheysubtracted
forexample,
nal numberin theend. One student,
as shownin
clearlyexpressedherunderstanding,
figure1.
Indeed,themagicworksbecauseofthe"power
ofalgebra."
Atfirst,
thestudents
didnotunderstand
howalgebracouldbe used to perform
themagic,
buttheywantedto learnmoreaboutitin thehope
thattheywouldbe abletodo themagicthemselves.
mathmagic
Figure2 (p. 564) showsa moreformal
after
thisinitial
discussion.
presented
activity
in
Whenthe studentscompletedthedirections
figure2, we toldthemthatmanyof themwere
ofa grayelephantfromDenmark.Many
thinking
students
weresurprised
thatthatwas indeedwhat
works
Themagic
because
timeyoustartoutwitha numevery
berandat theendyousubtract
so itreally
thatsamenumber
doesn't
matter
whatisinbetween
justas longas yousubtract
number
at
the
end.
your
Fig. 1 A student's explanation of how the magic works
VOL. 5, NO. 9 • MAY 2000 563
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Cover Story
and recordhow
Followthe directions,
Mathmagic
Activity:
changeas a resultofthenumberoperations.
yournumbers
a. Thinkofa number.
b.Add5.
c. Multiply
by3.
d.Subtract
3.
e. Divideby3.
number.
f. Subtract
youroriginal
thatis,1 = A,2 = B,
g. Mapthedigittoa letterinthealphabet;
=
3 C, andso on.
inEuropethatbeginswiththat
h.Pickthenameofa country
letter,
nameandthinkof
i. Take thesecondletterin thecountry's
an animalthatbeginswiththatletter,
j. Thinkofthecolorofthatanimal.
Fig. 2 A mathmagicactivity
BiHffJHM
Numbers
Change
Directions
a. Thinkofa number. 2 1 7 1 15
7
12
20
b. Add5.
21
36
60
c. Multiply
by3.
33
57
18
d. Subtract
3.
11
19
6
e. Divideby3.
4
4
4
f. Subtract
your
number.
original
Expressions
n
n+ 5
3» + 15
3m+ 12
a +4
4
We explainedthattheactivity
theywerethinking.
becauseitworksbased on imis calledmathmagic
concepts,such as variable
portantmathematical
whichare commonly
studiedin alandexpression,
gebra.We thenhelpedtheclassanalyzethemathematicalconceptsbehindthismagic.
We explainedthatthe magicworkedbecause
no matterwhat the students'originalnumbers
were,theyalwaysendedup with4 in stepf.When
theymapped4 to a letterin theEnglishalphabet,
namethatbeginswithD
theygotD. The country's
is Denmark.The second letterin thatcountry's
ofanimal
nameis E. The mostcommonly
thought
anditscoloris gray.
withE is elephant,
beginning
Thatis whya "grayelephantfromDenmark"is a
when
good guess forwhatpeople are thinking
The
stuthis
mathmagic
complete
activity.
they
dentsunderstood
thatthemagicworkedbecause
had thesamenumberat theend ofthe
everybody
Table 1 helpsstudentsunderstand
computation.
whyeveryonein the class ends up with4 in this
magic.
564
The middlecolumninthetableshowsthreeexamplesofstudents'numbers.Althoughthe original numberswere2, 7, and 15,thelastnumberis
columnis mosteffective
always4. The right-hand
because it shows thatthe algebraicexpressions
neverchange,regardlessoftheoriginalnumbers
didnothave
chose.The students
thatthestudents
anytroublestatingthat» is a variablethatrepresentsall theoriginalnumberschosenintheclass.
to teach
Creatingthetablewas a goodopportunity
the conceptsofvariableand expressionand simWe foundthata flowdiagramwas
ple operations.
usefulto help studentsmakeconnecparticularly
tionsamongthese concepts.Aftersome discussionabouthowtheflowdiagramsworked,thestudentsbegan to make theirown diagramsof the
magic.Some studentseagerlysharedtheirdiagramswiththeclass (see fig.3).
Inthenextpartoftheactivity,
we askedstudents
whethertheycouldcreatetheirownmagicto try
membersat
withtheirpeersin theclass or family
home.All the studentswereeagerto createtheir
own magic.We asked themto completea worksheet (shownin fig. 4) to showthe mathmagic
workandtoexplainwhattheyhadlearnedaboutalgebraandwhythemagicworked.
ofStudentLearning
Observations
THE STUDENTSIN THE CLASSLEARNEDBASIC
the
algebraconceptsin theprocessofcompleting
thesentence
worksheet.
Theyneededto translate
"Thinkofa number"to a variable,forexample,n,
thenextendit to such expressionsas n + 2 and
2(n + 2). This activity
helpedthestudentsunderoflike termsand
standadditionand subtraction
ofmultiplication
overadthedistributive
property
dition.Figure 5 showspartofa completedactivitysheet.
cc
i
er.
4
9:
Î
i
X
ë
iñ
I
Fig. 3 A studentsharingher flowdiagramwiththe class
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
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Story
The workproducedby thisstudentis veryimthe
pressive.She was able not onlyto perform
name,
magicbutalsotogivethevariablea different
c, whichwas notdiscussedin class. She had no
thata variablecanbe repreproblem
understanding
sentedbyanyletters,
suchas c,n,x,ovy,as longas
theirvaluesvary.Anotherstudent,
who was in a
showeda remarkable
specialeducationprogram,
interestin this activityand producedthe work
showninfigure6 (p. 566).
As we can see in theirwork,thestudents
were
able to represent
numbersusinga variableand to
constructdifferent
algebraic expressionswith
thesevariables.The flowdiagramswereparticuforthestudents'
ofthe
larlyhelpful
understanding
algebraicconceptsof variableand expressionas
theycreatedtheirmagic.The studentscouldverbalizealgebraicexpressions,
suchas that12added
ton is n + 12,thatn + 12 multiplied
by3 is 3« + 36,
andso on.
We also observedthatthestudents
evaluatedalgebraicexpressionsas theycreatedtheirmagic.
a tableofvaluesfroma singleexTheygenerated
pressionwhentheychangedthevaluesofthevariableanddiscovered
thatan infinite
numberofsolutionscouldbe generatedthrough
For
mathmagic.
had no problemunderstandexample,thestudents
n + 12becomes
ingthatthevalueoftheexpression
14, 16, or 18 when the value of the variableis
changedfrom2 to4 to6.
The students
wereso excitedaboutmathmagic
thattheywerestillplaying
itseveraldaysafterthe
lesson.Theywereevenusingitwithotherteachers and withtheirfamily
membersat home.We
also observedseveralstudentsusingmathmagic
on theirfriendsin the fifth-grade
All
lunchroom.
wereable to do themagic,and severalwereeven
using the algebraicexpressionsto shape their
magic.
Fig. 4 A mathmagicworksheet
ClosingComments
WEOBSERVED
THESTUDENTS'
ENTHUSIASM
AND
excitementabout this activitywithsatisfaction.
We were particularly
pleased because students
werenotonlymotivated
to perform
themagicbut
also interestedin learningadditionalalgebraic
concepts.The studentsdevelopedconceptualunof the meaningsof variableand exderstanding
pression.Manystudentswereable to manipulate
and evaluatealgebraicexpressionsand did so eagerly.The resultsfromourclassroomexperience
indicatethatalgebracan be introducedsuccessand sixthgraders
fullyto a largenumberoffifth
if it is taughtusing an interesting
contextlike
mathmagic.
D^
Fig. 5 A student's explanationof how mathmagicis related to algebra
VOL. 5, NO. 9 • MAY 2000 565
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Cover Story
References
a. Think
ofa number. 2
4
6
h
K. "AlgebraforEveryone:Start
Fouche,Katheryn
Early."MathematicsTeachingin the Middle
School2 (February
1997):226-29.
b.Add11
14
16
18
*+12
c.Multiply
by3.
U
48
54
3* * 36
d.Subtract
3.
39
45
51
3** 33
Bank,vols.1 and2.
Development
Package.Activity
Canberra,Australia:Curriculum
Development
e.Pivideby3.
13
15
17
*+ll
1988.
Centre,
I learned
I alsolearned
a lotaboutalgebraic
aboutflow
expression
I
will
and
an
at
the
bottom.
diagram,
put example
n^n+12^3n
+36^>3n +33 ^ n +11^> (jj)
Fig. 6 A special education student's explanationof how mathmagicis related to
algebra
Lovitt,
Charles,and Doug Clarke.TheMathematics
Curriculum
and TeachingProgram:
Professional
National Council of Teachers of Mathematics
and EvaluationStandards
(NCTM). Curriculum
Va.:NCTM,1989.
forSchoolMathematics.
Reston,
Silver,EdwardA. "AlgebraforAll:IncreasingStudents'Access to AlgebraicIdeas,NotJustAlgebraCourses."Mathematics
intheMiddle
Teaching
School2 (February1997): 204-7. <S)
LjÄfllJlM^liiMill
566
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
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