Using Magic Squares to Teach Linear Algebra

Using magic squares to teach linear algebra

Ricardo Teixeira University of Houston-Victoria, USA teixeirar@uhv .edu

Introduction

his article summarises activities that happened during the first three Tweeks of a fictitious high-school-level linear algebra section that used magic squares as a teaching tool to inspire students to further investigate the topics. The author has been working with students from high school and college levels for years, and although this situation can be considered very realistic, it was not based on a single classroom, it is a product of imagination based on actual experience. This study focuses in the development of activities related to linear algebra with use of technology. An active learning method was used as the main strategy to motivate discovery. It aligns with the Australian Curriculum not only for the encouragement of using computers and calculators but also for helping the student understand the concepts and techniques in matrices and apply reasoning skills to solve problems with matrices, which are learning outcomes from Unit 2 Specialist Mathematics.

Active learning

Educators have embraced innovation to better serve a constantly changing student population. Methods that once seemed to be the only rule, now are being challenged. With interactive technology available at virtually any daily task, students seem to not respond well to the outdated passive learning method anymore. Robert Zemsky [16] states that the current undergraduate structure is outdated and inadequately services the present and changing student demographic. Prince (2004) defines active learning as “any instructional method that engages students in the learning process.” This definition, in theory, includes many traditional classroom activities, such as instructor-centred lectures. However, in general, modern active learning processes commonly involve Australian Senior Mathematics Journal vol. 32 no. 2

50 Using magic squares to teach linear algebra Australian Senior Mathematics Journal vol. 32 no. 2 51 semester grade is dependent on how well they are prepared for the time semester grade is dependent on assignments, or reflection upon a topic; assignments, or reflection upon a gaining first exposure prior to class: with YouTube videos, pre-lesson class: with YouTube gaining first exposure prior to in classroom; performing cognitive learning: in-class activities that focus in higher-level once students are well-prepared, class time is used to promote deeper promote to used is time class well-prepared, are students once organise knowledge in ways that facilitate retrieval and application. organise knowledge in ways that receiving incentive to come well-prepared to class: part of students’ receiving incentive to come well-prepared understand facts and ideas in the context of a conceptual framework; understand facts and ideas in the learning and to increase their skills at using new material. learning and to increase their skills Demographic data was not available, but students were in majority white Demographic data was not available, but students were in majority have factual knowledge; Bonwell and Eison (1991) conclude that active learning leads to better (1991) conclude that active Bonwell and Eison Our method of instruction follows the standard practices in most recent instruction follows the standard Our method of Terenzini et al. (2001) shows that students taught in a way that incorporates Terenzini (Bransford, 2004) • • • • • • small-group learning procedures generally achieve better knowledge small-group learning procedures generally achieve better science major, and 2 preferring a Biology program. and 2 preferring a Biology science major, students’ attitudes and improvements in students thinking and writing. More and improvements in students students’ attitudes students need to: and Hispanic. a spring semester. There were 26 students registered, all of them willing to There were 26 students a spring semester. anecdotal stories seem to support the claim that students learn best when they seem to support the claim that students anecdotal stories research suggests that to develop competence in a topic, articles. In order Students in our linear algebra class will achieve this by the following elements: will achieve this by the following Students in our linear algebra class going to class, where with peer and faculty support a higher level of cognitive with peer and faculty support going to class, where four to five members. day of classes, students were assigned into six different groups, ranging from day of classes, students were assigned into six different groups, ranging peer-assisted and problem-based learning approaches. Michael (Michael, Michael (Michael, approaches. learning and problem-based peer-assisted classroom method. This approach requires students to gain knowledge before knowledge before students to gain requires method. This approach classroom each time. It mocked a college-level class structure and lasted 15 weeks during weeks 15 lasted and structure class college-level a mocked It time. each engage with course material and actively participate in their learning. material and actively participate engage with course retention than students working by themselves. Hence, during the veryretention than students working by themselves. Hence, during the first work can be performed. have a college degree: 8 saying they will pursue math major, 16 computer have a college degree: 8 saying they will pursue math major, 2006) highlights the success for active learning, especially the flipped- learning, especially for active the success 2006) highlights About the course Class met for three hours a week, divided into two meetings of 75 minutes Class met for three hours a week, divided into two meetings of This special and optional linear algebra section was aimed at senior students. This special and optional linear algebra section was aimed at senior 52 Australian Senior Mathematics Journal vol. 32 no. 2 Teixeira (specifically, rankandinverse). The goalofstartingthecoursewith‘magicmodule’isawayintroducing About the‘magicmodule’ Magic squareshavefascinatedmathematicians for centuries.Theearliest Products, LinearTransformations, andEigenvaluesEigenvectors. Fuh-Hi describedthe“Loh-Shu”,or“scrollof river Loh”.Sincethen,many For thefirstweeks,magicsquareswereusedasathemetomotivatedeeper Magic squares known magicsquarewasa3byonerecorded around2800BCinChina. with: dominoes(Springfield &Goddard,2009),vectorspaces(Ward, 1980), where heteacheshow to createfourbymagicsquaresbasedon a would dedicatemoretimetoproofs,andmathematicalprecisionin would developsolidideasandintuitionsregardingthematerial.Bynomeans volunteer’s birthday (Benjamin,2006).Otherstudiesrelatemagic squares Mathematical concepts learning activities. required byacourseatthislevel. the differentstudentgroups,weuseletters:A,B,C,D,E,andF. Andtothe the modulemaysubstitute the formalandpreciseapproachtolinearalgebra manipulating matrices,andinhowtocomputerank,determinants,more. of linearequations,andsomeapplicationskeycharacteristicsamatrix involving magicsquares.Forinstance,Benjamin andYasuda (1999)had people inmanynationshaveenjoyed,studied,and recordedmagicsquares. inquiries regarding some linear algebra concepts, such as solutions of systems and MatrixAlgebra.Laterchaptersinthecoursewere:Vector Spaces,Inner ofthefirsttwochaptersonly:SystemsLinearEquations assist thedelivery students wereencouragednottouseit.Themagicsquarethemewasused students asnumberswithinthegroup:A1,A2,andsoon. squares. Benjaminalsopublishedanarticleon amagazineformagicians studied interestingpropertiesforsquaringnumbers formedonthemagic several linear algebraconcepts through an effective approachsostudents Although the class requires a textbook, during the first weeks of classes, After the completion of the three-week introductory module,students After thecompletionofthree-weekintroductory The followingmaterialwasonlypresentedtothestudentsafterouractive Recently, severalresearch papershavebeenpublishedinquestions For privacy, namesshallbeomitted.Inourreport,whenreferringto Using magic squares to teach linear algebra Australian Senior Mathematics Journal vol. 32 no. 2 53 . . 5 2 1 = 3 × + 1)n ) 2 1 2 + 2 = (n 3 ( = t numbers in this magic 2 2 . Let’s use a small trick to . Let’s + 1). So, 2S 2 2 = 65. , what is the expected sum of that solve the following three solve the following that + n 2 2 5, t n 2 ) 1 n – 1) and z ) + 2 y + 1) + (n 1 2 2 , 2 + x n 2 ( n = – 1) + … + 2 + 1 ( 2 S n = = 1 + 2 + … + n = S t columns, there are n columns, + (n rows to have the same sum, then the number 2 n 1 + 2 + … + (n square and answer some key questions. + 1) + … + (n = 34. And for a 5 × 2 = × n 3 magic square, the sum will be will sum the square, magic 3 rows and n × + 1) + (n 2 S = = (n 4 magic square t

Let us think of an n If we expect each of the n A magic square is a display of numbers in a square form having nice having nice in a square form of numbers square is a display A magic square. squares may have other collections of numbers whose sum adds up to the other collections of numbers squares may have may a variation of the problem, a volunteer in Or, same specific number. areas of the magic that the rows/columns/diagonals/special specify a number to repeat numbers within to (in this case, we may have square will be equal any column or in any diagonal is always the same number. One other ‘nice’ number. any diagonal is always the same any column or in System of equations Since there are n determine a nice formula for it: property that some magic squares may show is having all numbers different, magic squares may show is having property that some magic 1. Some starting at integers consecutive display will even fact, some in example, how we would find values for how we would find example, equations simultaneously: each row/column/diagonal? must be mathematical properties. Normally, the sum of all numbers in any row or in of all numbers in any row or the sum Normally, mathematical properties. the square). weights and centre of mass (Behforooz, 2012), and even the US election even the US and 2012), (Behforooz, and centre of mass weights For a 4 × How many numbers will it contain? Hence: 2S If we display the consecutive numbers 1, 2, … n If we display the consecutive numbers

Hence for a 3 a for Hence Thus This expression gives the sum of all numbers. The first topic the class covered was solving systems of linear equations. For The first topic the class covered was solving systems of linear equations. S The total sum of all numbers is Problem 1 Problem Problem 2 Problem (Behforooz, 2009). (Behforooz, Answer 1 Answer 2 54 Australian Senior Mathematics Journal vol. 32 no. 2 Teixeira A setofvectorsislinearlyindependentifnovectorintheis: Definition 1 The processiscompletedbyrewritingtheproblemwithmatrices. In ordertounderstandtheconceptofrankamatrix,weneedsetsome

Rank ofamatrix Finally, thematrixonright-handsidecanbesimplycalled linear combinationoftheothers. language: rows. Inpractice,therankofann vector, andwenameitvectorB of theothers,theninpracticeitmeansthatapiece ofinformationisbeing commutative, thatis,orderisimportant,thematrixproductE in the coefficient matrix, perform aleft-multiplication onbothsides(matrixmultiplicationisnot perform from theproductF called thevariablevector(aissimplyamatrixhavingonlyonecolumn). Specifically, theleft-handed sideofeachequationwillprovideadifferentrow are linearlyindependent.Thatis,ifthereisno row thatcanbewrittenasa somehow repeated. • • The rankofamatrixisthemaximumnumberlinearlyindependent The firstmatrixA In asystemofequations,eachequationcanbeseen asapieceofinformation. One strategytosolveisfindtheinversematrixA equal tothesumofscalarmultiplesothervectors. a linearcombinationofothervectorsintheset,thatis,ifonevectoris a scalarmultipleofanothervectorintheset;or ·E iscalledthematrixofcoefficients.ThesecondX ): A ⎝ ⎜ ⎜ ⎜ ⎛ 3 2 1 A –1 . If a row can be written as a linear combination ∙A 1 2 5 ⎩ ⎪ ⎨ ⎪ ⎧ x . Sotherepresentationofthisproblemis: 3 2 ∙X x x − − + 4 3 A 1 + +

2 =A × ∙X 5 y y ⎠ ⎟ ⎟ ⎟ ⎞

y m − + ⎝ ⎜ ⎜ ⎜ ⎛ − matrixwillben z 4 –1 =B 3 x z z y = ∙B z = − ⎠ ⎟ ⎟ ⎟ ⎞ = 2 8 . SotheresultisX = − 4 ⎝ ⎜ ⎜ ⎜ ⎛

− − 8 4 2 ⎠ ⎟ ⎟ ⎟ ⎞ ifandonlyallrows –1 (ifitexists)and =A ·F –1 maydiffer (System 1) ∙B . is Using magic squares to teach linear algebra Australian Senior Mathematics Journal vol. 32 no. 2 55 = 2: + 8y (System 2) will have a have will B = be a vector with with vector a be X will have inverse. B ·

4 − 8 2 = − . = z 2 = z 3 4 z = − + − y y ) = n identity if and only if all entries are zero, y y 8 5

2 + + + n

+ x x x × 6 2 3 x

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ be a vector with n variables, and and variables, n with vector a be X matrix, n

matrices, only the ones with ‘rank’ equal to n ×

Equation 4 = Equation 1 + Equation 2 + Equation 3 1 + Equation 2 + Equation Equation 4 = Equation n n × matrix is called the n n

be an be × A

Not all matrices have inverse, only the ones having non-zero determinant. Not all matrices have inverse, only the ones having non-zero determinant. It looks like that the new equation adds a new piece of information. the new equation adds a new It looks like that Only square matrices (the same number of rows and columns) may have Only square matrices (the same number of rows and columns) may The following result is valid. It is mentioned in the introductoryThe following result is valid. It is module, The process of finding the inverse of a matrix can be elaborated. For our the The process of finding n numbers. Then the system of equations represented by represented equations of system the Then numbers. So technically, this fourth equation is redundant. So technically, from all n initial goals, we may use computational tools that will find them. permutations) for now and we will concentrate instead on its use. stating that, inverse. An equivalent way of determining the ones with inverse is inverse of a matrix. Let’s start with some definitions (remember that the main (remember that definitions with some start Let’s matrix. of a inverse position is the same as the column position). diagonal is the one where the row other matrix will be equal to the given matrix, provided the multiplication the provided matrix, given the to equal be will matrix other can be done. except on the main diagonal, which displays only ones. but its proof is only presented later in the course. but its proof is only presented later three equations: A unique solution if and only if rank( Inverse of a matrix Let However, it does not, the fourth equation is simply the addition of all initial fourth equation is simply the it does not, the However,

Our approach to solve a linear system of equations relies on finding the Our approach to solve a linear We will avoid the formal definition of the determinant (that implies notions of will avoid the formal definition of the determinant (that implies We n This name is consistent with its main property: an identity matrix times any This name is consistent with its main property: an identity matrix x equation 6 and add a fourth the previous system for instance, Take, Definition 2 A Theorem 1 Theorem 56 Australian Senior Mathematics Journal vol. 32 no. 2 Teixeira “rank{{1, 2,−1},{2,1,4},{3,5,−3}}”inWolframAlpha, forexample,wecertify A Theorem 2 A waytoovercomethis issue would be giving a valuefor any ofthe variables as An example There areseveralcalculators,software,apps,websites, thatwouldworkwith n Technology z Consider thesystem:

Let having therankofA the result is that the rank is 3, so we may find unique solutions for the variables. We know represented byA remaining twoequationsareindependent. Hence, the‘rank’istwo.Thatis, the resultofmanipulationothersbyanequationthatsimplygivesavalueto there areonlytwopiecesofinformation. that thethirdequationisnota‘new’pieceofinformation,since be consideredredundant,soitmaydeleted.Ifwerewritethesystemas computed therank),and B matrices. Forinstance,WolframAlpha (www.wolframalpha.com) maybeuseful. may beseenasamathematicalpieceofinformation),quickanalysisreveals can betheresultofaddingfirstandsecondequations.Thisequation it wouldbex inverse matrixofA is three,sothisproblemnowhasuniquesolution. a variable.Forinstance,ifwesaythatx a thirdpieceofinformation.Thatis,wewouldsubstituteanequationthatis =1. · variables,andB Although itlooksliketherearethreepiecesofinformation(eachequation The rankof Now goingbacktothefirstsystemofequations of thischapter, bytyping Since, we are trying tosolveforthreevariables,accordingtheorem1 Since, wearetrying X A = beasquarematrixn B , we conclude that the ‘rank’ of the matrix X +0y = A ·X +0z –1 isavectorwithn exists,representedbyA

· equalto2wouldnotbeenoughgetauniquesolution. =B B =4,andthethirdrowofmatrixA , whereA hasauniquesolutiongivenbyX istheright-handside.The solutionisx

n x x ⎝ ⎜ ⎜ ⎜ ⎛ isthematrixofcoefficients (thisonewejust +3y whoserankisexactlyequalton +2y +y 1 1 2 numbers,thenthesystemofequations +3z +4z 2 0 1 –z =4,theninsteadofthethirdequation, =–2 − 4 0 =6 =8 1 −1 ⎠ ⎟ ⎟ ⎟ ⎞ . Furthermore,ifX A is, then, notthree.The wouldbe[100]. =A −1 ·B isavectorof =3;y . , thenthe =–2; Using magic squares to teach linear algebra Australian Senior Mathematics Journal vol. 32 no. 2 57 and (first row) (third row) (second row) (first column) (third column) (main diagonal) (second column) second diagonal) row 9. i f c = 15 = 15 = 15 = 15 = 15 = 15 = 15 = 15 ( e b h + i + c + f + g + i + i + h + g + e + e + f + b + h + e + d + e g d a = 5, which is a reasonable number since it is the = 5, which is a reasonable number row 5 + row 7 + row 8 = row 1 + row 3 + 3 × 3 magic square 3 magic Let us replace it by giving the value 1 to a cell. Do we want this low value Let us replace it by giving the value 1 to a cell. Do we want this low That is, from the 9 rows, we can eliminate two that are results of others. That is, from the 9 rows, we can eliminate two that are results There are only eight equations for nine variables. We may want to create nine variables. We There are only eight equations for Again, without many details at this point, we notice that adding rows 5, 7 Again, without many details at this point, we notice that adding × = 1 (that is the sixth equation now). and 8 we end up with: {1, 1, 1, 0, 3, 0, 1, 1, 1} which is the same as adding rows 1 and 8 we end up with: {1, 1, 1, 0, 3, 0, 1, 1, 1} which is the same as adding and 3, and three times row 9. So: a ninth equation. We can accomplish this by giving value to a variable. For can accomplish a ninth equation. We get rank = 7. that all rows, all columns and both diagonals have sum equal to 15: columns and both diagonals have that all rows, all do, we point out that adding the first three rows gets the same result as adding do, we point out that adding the first three rows gets the same result instance, let us say that e on one of the corners, or in the middle? Let us try on the middle, for instance compute the rank (through an app, calculator or available websites), and we and websites), available or calculator app, an (through rank the compute middle value and the middle cell. After adding this piece of information, we middle value and the middle cell. After adding this piece of information, represents giving value to a new variable (since we already gave value to b represents giving value to a new variable (since we already gave value rows 4, 5, and 6. So, we can eliminate one of these rows. Eliminate sixth row rows 4, 5, and 6. So, we can eliminate one of these rows. Eliminate From the above discussion, we want to display numbers 1, 2… 9 in such way discussion, we want to display numbers From the above 3 a d g a b c a Hence, we can eliminate one of these rows and replace it with a row that Hence, we can eliminate one of these rows and replace it with

c b The equations are: And we add two new values to variables. Without going on details of how to And we add two new values to variables. Without going on details (basically, we noticed that row 1 + row 2 + row 3 – row 4 – row 5 = row 6). we noticed that (basically, The problem 58 Australian Senior Mathematics Journal vol. 32 no. 2 Teixeira 4 e

Let ussaywewanttocreatea4× which is: will have the same sum, which we call S. According to previous discussion, let’s we eliminate8throwandleta would alsoresultinsimilarmagicsquares. whether theyneedtostart givingvaluestodifferentcells. the areasbelow. Thereare eight newareas:each‘quadrant’,anareacomposed totalling 10equations.Andsomeequationsmay even beredundant. the variablevector(X rotation oranyvertical/horizontalreflectioncombinationsofthesemoves rectangles. by thecornersa of equations.Also,they need tochecktherankofmatrixA done bytrial-and-errororlittleexperience.Alsonoticethatany90degree above square: could uselinearalgebraandsolveit.So,ourmagicsquareistheresultofA say • • , wecangivevaluetoanyothervariablethanh × Columns: 834 How did we guarantee that the numbers did not repeat?Well, thatcan be Rows: 816 Notice thefollowingniceproperties(Benjamin&Yasuda, 1999)ofthe Now, So, we may create some other ‘special areas’ that keep the same sum. See Students needtocheck thenumberofvariablesandtotal 4magicsquare S =34.Thereare16variables,butonly4rows,columns and2diagonals, A A hasrank9(computedwithatechnologyresource).Hence,we = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 2 +357 1 0 0 1 0 0 1 0 0 , 2 d +159 1 0 0 0 1 1 0 0 0 , 2 n +492 , 1 0 0 0 0 0 0 1 0 ) andtheright-handedvector(B q 2 , anareawithfour‘centre’numbersf +672 0 1 0 1 0 0 0 0 0 2 =618 0 1 0 0 1 0 1 1 1 2 =8.Thecoefficientmatrix(whichwecallA =438 0 1 0 0 0 0 0 0 0 4 3 8 4magicsquare.Eachrow, column, diagonal 2 +753 0 0 1 1 0 0 0 1 0 2 +951 0 0 1 0 1 0 0 0 0 1 9 5 2 +294 0 0 1 0 0 1 0 0 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ 2 6 2 7 +376 , X 2 = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 2 . d , sinceb h a g b c e i f ) become: ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ , B = +e , ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ g , +h 1 1 1 1 1 1 8 1 5 k . Andcheck 5 5 5 5 5 5 , l = 15).Say ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ , plustwo −1 B ), Using magic squares to teach linear algebra Australian Senior Mathematics Journal vol. 32 no. 2 59 q h d m l c g p . Each group f b k o = B · X j e a n , and how to read the · B −1 q h d m l c g p f b k o j e a n q h d m l c g p f b k o Once all groups show a relative knowledge on both methods, instructor Once all groups show a relative knowledge on both methods, instructor Instructor asks each student to download a ‘matrix app’ on their cell phones Instructor asks each student to download a ‘matrix app’ on their cell Two students are required to go to the tutoring centre to work on matrix students are required Two Then instructor shows WolframAlpha website, and taught students how Then instructor shows WolframAlpha j e a n solve problems via either ‘elimination process’ or ‘substitution’. are well-known, but some students show deficiency in matrix product. A are well-known, but some students also presents their own problems as matrix multiplication. and try a wrong app (it just computes to use them. A student downloads on how available. All others download good ones. A class discussion emerges Class experience Second day decides to present the systems as a matrix multiplication A instructor’s computer. All answers match. computer. instructor’s determinants), a student has no battery left, and another has no memory equations. Different groups solve a different system, and members choose to equations. Different groups solve a different system, and members multiplication. Both go. matrices, and the basic operations (sum, difference and product). All topics matrices, and the basic operations based on a birth date. the first few weeks would have a different structure: the active learning method. the first few weeks would have a different twice a week for 75 minutes each time, there were a total of only six meetings. twice a week for 75 minutes each to type matrices, how to solve the system by A results. Each group sends a representative to redo their own problems at the results. Each group sends a representative to redo their own problems watch TED talk by Arthur Benjamin where he shows a 4 by 4 magic square watch TED talk by Arthur Benjamin with 4 or 5 members. The instructor goes over the syllabus and explains that and explains syllabus over the instructor goes 5 members. The with 4 or First day homework on these operations is then assigned. Also, students are asked to to asked are students Also, assigned. then is operations these on homework He also informally assesses students’ level of familiarity with classification of classification with familiarity of level students’ assesses informally also He It starts with groups solving systems of equations with two variables and two and variables two with equations of systems solving with groups starts It On the very each first day of classes, the 26 students are divided into six groups, There was a three-week period reserved Since class meets for this experiment. That is how it worked: 60 Australian Senior Mathematics Journal vol. 32 no. 2 Teixeira assumes information thatwasnotgiven. assumes information Third day Class startswithasystemthreevariablesandtwoequations.Studentsare would bevalid.Studentsdebateandconcludethatitmakessense,but variables. zero toavariable.Heasksaboutwhatwouldhappenifothervalueswere linear combinationoftheothers. until nextclass. to watchothermagicsquarevideos,includingtheoneonNumberphile two equationsandvariables.Theinstructordoesnotconfirmwhetherit that itwouldbe“impossible.”Astudent,let’s callhimstudentA1,proposesa two groupswereabletoalmostsimultaneouslysolve theproblem.Instructor to enterdata.Theyrealisethatdifferentprogramshaveformsof channel. Studentswhowereunabletoinstall an apparerequiredtodoso correct solution.Hedecidestomakex of whether he couldhaveconsidered that y each half. group,leavingsomestudentswithoutmuchparticipation. Afterawhile, every equations. Groupsdebatetoseeiftheycansolve.Studentleadersemergein encourages theclasstoseeifthatwouldbepossible.Anothersolutionwas entering rows. divides theclassintotwohalves,andarepresentative ofthesetwogroupslead infinitely-many solutions. Everyone seemstoacceptwithoutanyproof. infinitely-many solutions.Everyone different memberswouldassignthesamevalue,butforvariables. found. StudentC1answersthesystemoncez Several newsolutionsarecreated.StudentA1conjecturesthattherewouldbe asked to find “a” solution. Most students try tousetheirapps,andconclude asked tofind“a”solution.Moststudentstry aware, therewereonlyfourindependentequations, sincethelastonewasa assigned. Differentgroupshavedifferentvalues,andevenwithineachgroup, some variables,differentsolutionswouldbefound.Studentsareencouraged says that the solution exists, but it would not be unique. By attributing values to solvable systemswithlessequationsthenvariables.Onthehandout,itsimply system of five equations with seven variables. However, and students are not Another student from other group, student B1, raises the question Class isalmostover, wheninstructorassignsthehomework.Itissolving a Homework involvessomesystems.Italsoincludedatwo-pagehandoutfor Class endswithstudentssolvingsystemsofthreeequation Instructor, then, decides to write a problem with four variables and three Instructor asks why students were simply fixing on the possibility of assigning equaltozero,andsolvesasystemwith =0. and not x is zero. Instructor Using magic squares to teach linear algebra Australian Senior Mathematics Journal vol. 32 no. 2 61 into their apps, into their i f c e b h g d a the magic square, with discussion on rank. Do a list with ten systems of equations. Give a through solution for a system of equations that would summarise Give a through solution for a system of equations that would summarise Due to the increased number of variables and equations, it is hard to Due to the increased number of variables and equations, it is Finally, instructor introduces a generic 3 by 3 magic square and asks instructor introduces a generic 3 by 3 magic Finally, Group D gives values to three variables and solve the system. However, Group system. However, to three variables and solve the Group D gives values Instructor explains the relationship of determinant and existence of an of determinant explains the relationship Instructor ‘magic Instructor shows a generic 4 by 4 magic square, and by selecting the Class answers the problems relatively easily. Instructor assigns as homework Instructor easily. Class answers the problems relatively any input in the discussion. Students conjecture that the difference between difference the that conjecture Students discussion. the in input any and asks them to compute its ‘rank’. At least one person per group is able compute its ‘rank’. At least one and asks them to an YouTube about rank of matrices, and gives each student an option: about rank of matrices, and gives each student an YouTube and assigns homework problems and two linear algebra videos. areas’ they come up with the number of equations. Their apps help them help apps Their equations. of number the with up come they areas’ give value to four variables and finds an inconsistency. Instructor refrains from an inconsistency. and finds four variables give value to problems 1 and 2 (see previous sections): identify which equation is redundant. Instructor reviews linear independence identify which equation is redundant. Instructor reviews linear independence inverse. Then, with previous examples, he shows that the square matrices square the that shows he examples, previous with Then, inverse. identify the rank, or the number of different pieces of information. eight variables. equation since it does not add new piece of information. Students take few does not add new piece of information. equation since it explain their own solutions to students who had not chosen their options. explain their own solutions to students who had not chosen their minutes to come up with the solution. minutes to come the rank and the number of variables has anything to do with the amount of the rank and the number of variables new data to be included. Instructor gives another problem: six equations and new data to be included. Instructor to do so. The rank was 4. Instructor then shows that the fifth equation is a was 4. Instructor then shows to do so. The rank linear combination of the first four. He briefly talks about excluding the fifth He briefly talks about of the first four. linear combination whose rank were less than number of rows all had zero determinant. Fourth day Fifth day Most of the students were unable to solve the only problem on the homework. on the homework. the only problem unable to solve students were Most of the 2. C only gives values to two variable and struggles. Meanwhile, Group A tries to A tries Group Meanwhile, struggles. and variable two to values gives only C Instructor then asks them to type the matrix of coefficients A of coefficients to type the matrix then asks them Instructor That took longer than anticipated. About half of the class chose each option. Instructor decides to let students About half of the class chose each option. Instructor decides to 1. 62 Australian Senior Mathematics Journal vol. 32 no. 2 Teixeira With the perceived success of this strategy, to create a the next step is trying When otherconceptswerelaterexplained,instructordidhisbesttorelate Other days Instructor showstheentiredevelopmentofproblemusingconcepts whole semestercoursesolelybasedonactivelearning. uses. Forinstance,Teixeira (2017)teachesanicecardtrickwhilerelating to inspiringstudentsviamagic,thereareseveralexamplesthathighlightsgood to theproblemoftypingtoomanyequationswithvariables.Some to solvea5bymagicsquare.Sincetheproblemofidentifying‘magicareas’ more magictricksinthisandothercourses. mathematics (probability and statistics). We conclude that we should include magic square.Forinstance: probability concepts.Also,LesserandGlickman(2009)usemagictoteach preparation, since he tested each and every appthatwerebeingused. preparation, sincehetestedeachandevery Students seemedparticularlyexcitedwithallthesereferencestomagicsquares. for the material, also served asareviewoftheconcepts,andexamples for thematerial,alsoserved Sixth day Conclusions and future work Conclusions andfuture apps limittheamountofcharactersperequation.Instructorhadalong additional applications.Homeworkonwordproblemswasassigned. • • • • • • • Also, studentsseemtoenjoymotivationfromdifferentareas.Whenitcomes That class, which was the last one on the three-week period that was reserved That class,whichwasthelastoneonthree-weekperiodthatreserved hence thespaceofmagicsquaresisavectorspace; zero isavectorsubspace; the eigenvalueofamagicspaceisnumber; the transposeofamagicspaceisalsosquare; the spaceofallmagicsquaresacertainorderwhosenumberis the multipleofamagicsquareisalsosquare,sameorder; the sumoftwomagicsquaressameorderisalsoasquare; a vectorwithonlyonesisaneigenvector. Using magic squares to teach linear algebra Australian Senior Mathematics Journal vol. 32 no. 2 63 School of Education and Human Development. School of Education rectangles. Bulletin of the ICA, 101–108. content/probably-magic. Learning vs. lecture/discussion: Students’ reported learning gains. Journal reported learning gains. Learning vs. lecture/discussion: Students’ of Engineering 283–286. Higher Education Report No. 1). Washington DC: The George Washington University, University, DC: The George Washington No. 1). Washington Higher Education Report American Magicians), 68-69. American Magicians), Education, 90(1), 123–130. Education, 30(4), 159–167. Education, 93(3), 223–231. Recreational Mathematics, 1(351), 37–38. Recreational (4), 265–274. Model Assisted Statistics and Applications, 4 106(2), 152–156. Springfield, M. & Goddard, W. (2009). The existence of domino magic squares and (2009). The existence Springfield, M. & Goddard, W. References Lesser, L. & Glickman, M. (2009). Using magic in the teaching of probability and statistics. probability and of M. (2009). Using magic in the teaching L. & Glickman, Lesser, (ASHE-ERIC in the classroom excitement J. (1991). Active learning:Bonwell, C. & Eison, Creating Press. DC: National Academic How people learnBransford, J. (2004). . Washington, work? A review of the research. JournalPrince, M. (2004). Does active learning of Engineering Journal of Recreational Mathematics, 4(36), Mathematics, squares. Journal magic of Recreational H. (2012). Weighted Behforooz, years. Journalsquare with US election magic H. (2009). Behforooz-Franklin Behforooz, of of (Magazine for the Society Double birthday magic square. M-U-M Benjamin, A. (2006). Monthly, Magic “squares” indeed! The American Mathematical K. (1999). Benjamin, A. & Yasuda, Michael, J. (2006). Where’s the evidence that active learning works? Advances in Physiology the evidence that active learning Michael, J. (2006). Where’s Ward, J. (1980). Vector spaces of magic squares. Mathematics Magazine, 53(2), 108. spaces of magic squares. J. (1980). Vector Ward, Terenzini, P., Cabrera, A., Colbeck, C., Parente, J. & Bjorklund, S. (2001). Collaborative Cabrera, A., P., Terenzini, Probably magic! Retrieved 12 February R. (2017). Probably https://plus.maths.org/ 2018 from Teixeira,