Teaching Locus with a Conserved Property by Integrating Mathematical Tools and Dynamic Geometric Software

Teaching locus with a conserved property by integrating mathematical tools and dynamic geometric software

Moshe Stupel, Ruti Segal & Victor Oxman Shaanan College, Haifa, Israel stupel@bezeqint .net rutisegal@gmail .com victor .oxman@gmail .com

e present investigative tasks that concern loci, which integrate the use Wof dynamic geometry software (DGS) with mathematics for proving the obtained figures. Additional conditions were added to the loci: ellipse, parabola and circle, which result in the emergence of new loci, similar in form to the original loci. The mathematical relation between the parameters of the original and new loci was found by the learners. A mathematical explanation for the general case, using the ‘surprising’ results obtained in the investigative tasks, is presented. Integrating DGS in mathematics instruction fosters an improved teaching and learning process.

Introduction

A conserving property may be used to solve different problems in mathematics and the identification of such a property leads to a complete and deep understanding of mathematical concepts and the relationships between them. The locus, according to this definition, conserves a certain property and can therefore be used to prove a wide variety of problems and even serve in the

execution of construction problems using a straightedge and a compass, as Australian Senior Mathematics Journal vol. 30 no. 1 has been done in mathematics since antiquity. Loci can be divided into two groups: 1. Essential loci, such as: the straight line, parallel lines, the mid perpendicular of a segment, an angle bisector, the circle, the parabola, the ellipse and the hyperbola. 2. Loci which are a result of the essential loci or loci obtained as a result of satisfying several mathematical conditions. Analytical geometry is one prominent field in mathematics in which extensive use is made of loci. In most colleges for teacher training, prospective teachers of the high school (ages 12–17 years old) stream in mathematics are taught four sequential courses, which deal with loci.

25 26 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman (MKT), asknowledge supporting theideaslinkedbyteacher and This knowledge includes the knowledge of what is acceptable to teach as a This knowledgeincludestheofwhat isacceptabletoteachasa In thissubjectthelociareabasisforadditionaltaskswhichincludemore Pedagogical content knowledge (PCK) has become one of the popular Pedagogical contentknowledge(PCK)hasbecome oneofthepopular line andendswiththecircle.Thethirdcourseisintegrationofcomputer required todiscoverwhatessentiallocusisobtained. requirements, andallowonetoobtainnewlocibasedonthegivenloci. repertoire of modes of representation, the most effective forms of representation, repertoire ofmodesrepresentation,themosteffective formsofrepresentation, the mathematicaltoolboxofsolvingdifferentproblemswhereidentification technology intheteachingofmathematicswherestudentsacquire the students. the sameproblemaccentuatedinterconnectednessofmathematicsfor andevendifferentialcalculus.Findingdifferentsolutionsfor trigonometry that thestudentsmustknowwhatisseenoncomputerscreennot Development ofpedagogicalcontentknowledge maximum useoftheknowledgeacquiredinthreepreviouscourses,and mathematical tools,suchas:Euclideangeometry, analyticalgeometry, mathematical proof,butonlyabasisforspeculation. mathematical proofsforthelocusshapeobtained.Itshouldbeemphasised obtaining lociwhicharesimilarinshapetotheoriginallocus. makes learning easy or difficult, knowledge (research and experimentation) makes learningeasyordifficult,knowledge(research andexperimentation) of conserved properties allows one to obtain a mathematical generalisation. of conserved whichbeginswiththestraight course istheprinciples of analyticalgeometry constructs studied by many scholars since it was defined by Shulman (1986). constructs studiedbymanyscholarssinceitwas defined byShulman(1986). coping and effective methods of reorganisation of knowledge. coping andeffectivemethods ofreorganisationknowledge. on theperceptionsandmisconceptionsofconcepts, teachingstrategiesfor intended to demonstrate the importance of thelocus as an importanttoolin in particular, theuseofcomputertechnology. is Advancedanalyticgeometry is advancedanalyticalgeometry, whichbeginswiththeparabola,ellipse powerful analogies, metaphors, explanations and demonstrations, which analogies,metaphors,explanationsanddemonstrations, which powerful formation ofalocuswhoseshapeisthesameasthatoriginallocus, Shulman anddefinedthe conceptofmathematicalknowledgeforteaching and thehyperbolaendswithuniquesubjects,dealsloci,making and othertimesadifferentlocusisformed.Thepresentpaperdealswith skills ofusingdifferentcomputerisedtechnologicaltools.Thefourthcourse The firstcourseistheelementsofEuclideangeometry. Thesecond In some tasks the students found several proofs as a result of usingdifferent In somecasesmathematicalconditionsaregivenandthestudentis Ball andBass(2003),relied onvarioustypesofknowledgeidentifiedby Naturally inallcasesthestudentswererequiredtoprepareformal Sometimes an introduction of additional requirements brings about Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 27 Stein, Smith and Henningsen (2000) described mathematical task classroom Stein, Smith and Henningsen (2000) Powerful tasks are meaningful rather than routine tasks that lead to Pedagogical knowledge of teachers includes selection, construction and selection, includes teachers of knowledge Pedagogical Pedagogical knowledge of teachers has a significant role in leading and of teachers has a significant Pedagogical knowledge Tasks that call for complex learning processes, called powerful that call for complex learning processes, tasks, are Tasks students showed greater understanding of mathematical concepts and were students showed greater understanding of mathematical concepts and pedagogical links. They are challenging, and encourage independent and pedagogical links. They are activity, which focuses student’s attention on a particular mathematical idea. attention on a particular which focuses student’s activity, a given problem can discuss, explain, connect and reflect on their learning can discuss, explain, connect and a given problem The process of tasks in a classroom environment. adaptation of mathematical feedback during the solution of problems (Alakoc, 2003; Martinovic & feedback during the solution of problems (Alakoc, 2003; Martinovic process (NCTM, 1991, 2000). process (NCTM, implementing reforms in mathematics education. Teachers ask for the use ask for the use Teachers in mathematics education. implementing reforms in teaching. Technological tools allow one to construct and represent and construct to one allow tools Technological teaching. in participants. Teachers engaged in solving tasks are also invited to reflect on invited to reflect also are tasks solving in engaged Teachers participants. of innovative technology to bring together students, working with tasks and to bring together students, of innovative technology collaboration and social interaction, and reveal mathematical thinking collaboration and social interaction, existing knowledge and their interaction with the task (Zaslavsky, 2007). with the task (Zaslavsky, existing knowledge and their interaction meaningful activities, creating a community in which students coping with creating a community in meaningful activities, through teaching and learning develops construction and integration of tasks mathematics and pedagogy for the existing knowledge, and incorporates emphasising integrating, the ability to plan and evaluate mathematical tasks, mathematical tasks, and evaluate the ability to plan integrating, emphasising mathematical objects in a dynamic manner while providing the user with mathematical objects in a dynamic manner while providing the models. In addition, it encourages different representations and connections better describe mathematical concepts and relations when compared to concepts and mathematical better describe between graphical descriptions while referring to mathematical concepts between graphical descriptions while referring to mathematical teaching that does not include a technological tool. In one study, for example, teaching that does not include a technological tool. In one study, the students to discover mathematical phenomena, as well as mathematical the students to discover mathematical phenomena, as well as mathematical understanding (Zaslavsky, Chapman & Leikin 2003). understanding (Zaslavsky, thought, and belief in processes of understanding of mathematical concepts. thought, and belief in processes necessary teachers for the development of knowledge and the teaching of together with the mathematical knowledge required to integrate and manage and manage required to integrate knowledge with the mathematical together these tasks. learning, and it is therefore also focused on the integration of technology learning, and it is therefore also focused on the integration of Use of computer technology Research in education seeks ways to improve the quality of teaching and Research in education seeks ways to improve the quality of teaching Manizade, 2013). Learning that includes use of dynamic software allows Manizade, 2013). Learning that includes use of dynamic software Classroom activity involves problems related to the same mathematical idea Classroom activity involves problems One model of powerful tasks is combining technological tools and conceptual (Kranier, 1993). (Kranier, (Wiest, 2001). Learning by using a technological tool allows students to (Wiest, 2001). Learning by using a technological tool allows 28 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman 1. & Hadas,1996;Hanna,2000;Laborde,2000). The computer’s abilitytogeneratemultiplediversifiedexamplesrapidly, to The studentswererequiredtoconstructthreeappletsforlociusingdynamic Geometry. Theaimofthistaskistoteachstudentsunderstanddeeplyand Lavicza, 2008). with information on the mathematical concept that constitutes the basis for the use of dynamic geometry software,whichinthepresentstudywasGeogebra. the useofdynamicgeometry to investigatethembothfromtheregularmathematicalaspectandthrough to differentgroupsofstudentsaspartthecourseAdvancedAnalytical the generalisationsandhypothesesthatrequireproof(Chazan,1993;Dreyfus to realisethechangeimmediately. there ismoreprocess-basedunderstanding(Lampert,1993). by learningfromexamples.Thestudentsdrawconclusionsexamples eventually teachthesubjectofloci.Alsoitmodelsintegrationtechnology mathematical world.Oncethestudentfindsahypothesis,orphenomenon, concerning theessentialmoves,trackcriticalpropertiesofconcept, in teachingthesubject.Thestudentsweregivenseveraltasksandasked internalise it andimplement it subsequently in thesolution of newproblems. inherently to the subject taught during the lesson in order to givestudentsthe during thediscussions.Theyalsoneedtobeattentivespeculationsthatrelate in the data and obtain thelocus by dragging. It also enhanced students’ ability feeling thattheirwayofthinkingisavitalcontributiontobroaderviewthe given accesstohigh-levelmathematicalideas(Hohenwarter, & Hoherwarter given byparticularnumericvalues. Stage Aoftheinvestigativetask store andretrievemovestoprovidequalityfeedback,providesthelearner software (Geogebra),which are basedon essential lociwhoseparameters are The useofthedynamicsoftwareallowsstudentstosolveproblems The investigative task presented in the present paper has been given The useofcomputertechnologyallowedstudentstomakedynamicchanges Teachers whointegratetechnologythroughteachingmustbeflexible Construct acanonicellipsewhichintersectsthex the triangleCAB,onsegmentOCsothatbymovingpoint C the curve. AtriangleCABisobtained.Connectthepoint C the curve. by constructionthepointM on whichM point M point point and B . SelectsomepointC O M , theorigin.ThesegmentOCisamedianintriangle.Place movesonacanonicellipse asshowninFigure1. shallmovecorrespondingly. Thequestionis:whatisthelocus moves?Correctconstruction oftheappletshowsthat ontheellipse,sothatitcanbedraggedalong , thepointofintersectionmediansin -axis andthepointsA withthe the Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 29 of each , the point the , on the parabola so parabola the on C -axis which intersects -axis which intersects shall move accordingly. The shall move accordingly. moves? Correct construction moves? M moves on a parabola whose vertex moves on a parabola whose vertex , the origin, and the segment OC is , the origin, and . Select any point any Select . B , the point M Figure 1 Figure Figure 2 Figure and A on the circle, and construct from it at least 6–8 … The question is: what is the locus on which point 3 M , 2 -axis, as shown in Figure 2. M , 1 is connected with point O is connected M C -axis at the points the at -axis x moves? so that by moving the point C so that by moving as shown in Figure 3. arbitraryof the midpoint M chords, followed by a construction M point the by construction Place triangle. the in median a Select a fixed point A question is: what is the locus on which locus the what is is: question point of the applet shows that the point M of the applet shows that the point of intersection of the medians in the triangle CAB, on the segment OC, of intersection of the medians in chord: the that it can be dragged along the curve.that it can be dragged along the A triangle CAB is obtained. The lies on the y Construct the circle based on the coordinates of its centre and its radius. Construct the circle based on the coordinates of its centre and its Construct a parabola whose vertex lies on the y Construct a parabola Correct construction of the applet shows that point M moves on a circle, Correct construction of the applet shows that point M moves on a M 2. 3. 30 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman (middle ofthechord). M Available onlineareGeogebraappletsinwhichtheparametersoflociand Applets fortasks The studentswereaskedtoinvestigatetasks1–2forthecaseinwhichpoint Upon completionofthecorrectconstructionappletstudentsfound Following aremathematical proofsforthecasesofanellipseandaparabola: First surprisingresult the divisionratioofsegmentscanbechanged: that pointM parameters ofthelocusanddivisionratio. Stage Boftheadvancedinvestigativetask and task3withpointM a locusthatissimilartotheoriginallocus. an interestingphenomenon:foreachratioofsegmentdivisionitbecameclear • • • dividesthesegmentOCinaratiootherthan1:2(intersectionofmedians), The resultingconclusionisthatineachofthecasespointM The studentswererequired topresentproofsforeachoftheabovecases. In eachoftheapplets there are rulersthatallowonetochangethe Locus inacircle:http://tube.geogebra.org/student/m149061 Locus inaparabola:http://tube.geogebra.org/student/m149060 Locus inanellipse:https://www.geogebratube.org/student/m149059 continuestomoveonalocusthatissimilartheoriginallocus. dividingthechordsinaratiodifferentthan1:1 Figure 3 moveson Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 31 , 0) and (a moves so that is the origin) is a median -axis at the points A . When we substitute these + 1) than the lengths of the 1 n = y + 2 m + 1)Y ⎞ ⎠ k m 1 1 2 ) and connect it to the points A = = + y b = Y , = (k k 2 n 2 m Y y y ⎛ ⎝ , b mark the points of intersection of the mark the points (x , ), as shown in Figure 4. = n + + Figure 4 Figure x ) moves on the ellipse and point M ) moves on the ellipse C 2 + 2 2 , Y M y m ⎞ ⎠ x a , M m O 1 2 that intersects the x + 1)X and B + 2 = a X (x in accordance with the subdivision of a segment in accordance with the subdivision k X ⎛ ⎝ are (X moves on the perimeter of a canonic ellipse, the – a = (k 2 = x -axis. Point C , 0). (–a In other words, point M Hence we obtain that: x We select on the parabola a point C We The coordinates of point M , so that the triangle ABC is obtained. CO (where O as shown in Figure 1, points A as shown in Figure and B axes of the original ellipse. inside a parabola inside an ellipse inside in a given ratio are: . We mark on the median a point M in the triangle ABC. We ellipse with the x , so that in any case the following ratio is on the segment OC, so that in any case correspondingly maintained between the segments: maintained between Mathematical proof for the formation of a parabola Mathematical proofMathematical the formation for of an ellipse lengths of whose axes is smaller by a factor of (k values in the equation for the ellipse, we obtain:

Given is the parabola y Given the canonic ellipse Given the canonic

The coordinates of point M B 32 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman

During thisstagethestudentswereaskedwhether thepointfromwhich they didnotknow. obtained locusandtheparametersoforiginal locusasafunctionofthe of x division ratio. parabola isnarrower, butitsvertexisalsolocatedonthey inside thelocus,suchthat pointM point M Stage Coftheinvestigativetask segments connecting to the points on the original locus can be anywhere The proofsdemonstrateaconnectionbetween theparametersof Thus, pointM This parabolaintersectsthex We movepointC Hence weobtainthatx Most ofthestudentsthought thattheanswerwasnoandsomesaid Based onthecoordinatesofsubdivisionsegment,weobtain: in theequationofparabolaweobtainanew move? movesonalocuswhichisparabola(Figure5).Theformed ontheparabola,andaskwhichgeometricalshapewill ⎝ ⎛ =(k k a X + Y 1 = +1)X = , -axis atthepoints 0 O m ( M k m ⎠ ⎞ M willstillmoveonasimilar locus. + C + x Figure 5 a n 1 n = , ) , d X y Y m n =(k 2 ⎝ ⎛ = = − − m k k k +1)Y a m a + + + 2 y 1 1 n , 0 ⎠ ⎞ . Bysubstitutingthevalues -axis. Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 33

: is located at (1) R (2) 1 + 1 (3) k ) = 2R move along the perimeter of the move along the = ) = 2R r + n k r ) = 2r using the ruler. = o (m + n + l n m Figure 6 Figure R (t )(m ) + l m m + m + l … on the original locus, and subsequently the and subsequently original locus, … on the + n n 3 A = (m , t … are marked with a constant ratio on the formed a constant ratio on the formed … are marked with r 2 3 A M , , 1 2 is selected inside the locus, from which straight lines are lines are from which straight inside the locus, is selected M , 1 is the centre of the circle. and change the division ratio k There is a mathematical relation between the radius of the original circle There is a mathematical relation An applet for the formation of a circle inside a circle is available at a circle a circle inside of formation An applet for the An example of this case is shown in Figure 6, where point P case is shown in Figure 6, where An example of this some location inside the circle and points M some location inside segments. It turns out that these points are located on a locus with the same out that these points are located segments. It turns locus. shape as the original in the radius of the obtained circle as a function of the division ratio k in the radius of the obtained circle point P division points M internal circle. drawn to the points A drawn to http://tube.geogebra.org/student/m149062. In the applet one can move http://tube.geogebra.org/student/m149062. In the applet one Point O By dividing expression (3) by expression (2), we obtain:

A very result surprising An arbitrary point P 34 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman The general case: An arbitrary locus The generalcase:Anarbitrary This transformationiscalledahomothetywithcentreO

In otherwords,

Hence itisclearthatif:

Point homothety transfers some curve (shape)G homothety transferssomecurve which transferseachpointC then thepointsC to G b) c) connect itbystraightlineswiththepointsC connecting thepointO is a curve G is acurve divide eachsegmentinthesameratio a realnumberk a) We considerthegeneralcase.ApointO On what curve dothepointsM On whatcurve On thesegmentsOC In otherwordsthelocusofallpointsC . C we markpointsC point we select some arbitrary pointO we selectsomearbitrary divide eachsegmentinthesameratio points C moves on a given curve. We movesonagivencurve. selectsomepointO ' thatissimilartoG C movesonagivencurve, 1 , C

≠ 0isgiven.Letusconsiderthetransformationofplane 1 2 , , C C 2 3 , … onthecurve, C C withthepointsC O ' M O 1 1 3 , , ' C … move on a curve thatissimilartothegivencurve. … moveonacurve 1 M 1 O C OC C O C ' 1 1 C 1 ' 1 C 2 toapointC , = = 2 ' , C = (Figure7). C M O OC O ' k 3 ' C … onthesegmentsOC M 2 2 C 1 C ⎝ ⎛ O 3 ' , M O 2 2 Figure 7 … wemarkthepointsM o 2 2 C C r = = ' 2 C O m n , M connectedbystraightlines,withthe C = M O O C , i.e.: ' k C ', sothat ' M that lie on the curve, suchthat thatlieonthecurve, C 3 3 C ' 3 into a curve G intoacurve C ' … move? 3 = isgivenintheplane.Inaddition 3 3 1 3 , C m n k = = . k − … … 2 , C 1 = ⎠ ⎞ ' thatlieonthesegments = 3 m n m … onthecurve. n outside the curve and outsidethecurve 1 ', sothatG , andafactork OC 1 , 2 , M OC 2 , M 3 ' issimilar … which 3 … that . The Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 35 . , which ', which = 1. is a positive real number. is a positive real number. ' on the straight line OX, and therefore ' are located on the same side of the centre ' are located on the same side of the is a given point and k on the geometric location is a point A and X with a point X is called the similarity factor. OA when O ×

OA' = k O is negative, the two points are located on both sides of the centre ⋅ OX, where k is positive, the points X k Different transformations are known. One transformation preservesOne transformation known. are transformations Different Making the transformation strengthens the notion that a geometrical a that notion the strengthens transformation the Making In all examples cited in the article transformation was applied on all the In all examples cited in the article transformation was applied on In the studies of plane geometryIn the studies of 12–17) use is made of in high school (ages If We explain the phenomena mentioned throughout this article using explain the phenomena mentioned throughout this article We The potential of combining mathematical tools and computerised The potential of combining mathematical tools and computerised The special approach to the solution of tasks having to do with loci is loci with do to having tasks of solution the to approach special The The solution of even simple problems using transformations develops The solution of even simple problems The image of each point A The use of different transformations, whether basic or complex, allows one transformations, whether basic The use of different , and if k shape is not a ‘frozen’ object, but rather something that can be moved. shape is not a ‘frozen’ object, but accompanied by visual representation of the different parameters affecting accompanied by visual representation of the different parameters auxiliary constructions, but almost and even simple alternative constructions geometric insight, provides alternative tools and illustrates how transformation geometric insight, provides alternative geometry. features that explain the phenomena discovered in the investigation: distances: it is called an isometric transformation. When compared to such distances: it is called an isometric intended to provide and impart additional tools allowing one to deal with intended to provide and impart commonly used methods. in many cases to obtain a simpler solution of problems in plane or in analytic obtain a simpler solution of problems in many cases to is “a science that investigates the properties of shapes that do not change investigates the properties of is “a science that matches each point X mathematical challenges, sometimes in a simple manner when compared to mathematical challenges, sometimes or a point, rotation or homothety. out. of a geometrical shape can be carried maintains transformations, homothety is a transformation relative to a centre O transformations, homothety is a the shapes of the loci. technological tools is veryis tools technological powerful learning dynamic allows it since not preserve a constant distance between the points unless k no use is made of transformations of shift, reflection relative to a straight line transformations of shift, reflection no use is made of under transformations of the plane (space)” (Klein, 2015). of the plane (space)” (Klein, under transformations locus points. homothety. Homothety is also a basic transformation but it is not isometric because it does Homothety is also a basic transformation but it is not isometric because It seems that this transformation, which is called homothety, contains two It seems that this transformation, which is called homothety, According to the definition of the great mathematician Felix Klein, geometrymathematician of the great to the definition According OX' = k O The importance transformation of the 36 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman (Figure 10). 1. If The centreofthecircleisagivensize.ThinkwhoseP

2. If then the curve obtainedbydragginggathersallthedotsthatareequidistant then thecurve the radius obtained throughtransformation. from and thus explain the similarity between the original curve and the curve andthecurve and thusexplainthesimilaritybetweenoriginalcurve A circleisthelocusofallpointswithequaldistancesfromagivenpoint. Another layerofexplanationispossibleifwerecallthedefinitionsloci. These featuresoftransformationcharacterisephotographymagnification A P This attribute stems from imaginary trianglesDOA'B This attributestemsfromimaginary transformation canalsobefoundononeline.Thisattributestems then a'+b=180°(Figure9). expansion) ofABwithrespecttok from the similarity of triangles: Because ' constantandequaltok 'B A A ' = ' andB , R B . If P k andC

× is the centre of the circle, obtained through transformation, AB. In other words, the section ' arerespectivelyimagesofpointsA areononeline,thenthepicturesobtainedthrough

× R Figure 10 Figure 9 Figure 8 , sowegotanewcirclewithradiusk R . k a = a' and A 'B andB ' is the shrinking (or b = ' ~DOAB(Figure 8). b', if , thenthereis a + b = 180° and

× R

Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 37 , . We . We 2 MQ = on the ×

– 9x 3 ' images of the of images ' MP + k ) = x Q ' image obtained ' image × (x

' and and ' ' = k P Q 'Q Q' M ' + M 'P M' k = A P B B Figure 11 Figure Figure 12 Figure k P' (MP + MQ). along the graph of the function and tracing point B P point on the original ellipse and M the original ellipse point on outside the graph and connect it with some point B outside the graph and connect it ' is a fixed size, k Similarly, an ellipse is the locus of all the points where the sum of their where the sum of all the points is the locus an ellipse Similarly, Similarly, we can show that in the case of the parabola, the curve we can show that in the case of obtained Similarly, By dragging point B In fact, homothety is a transformation preservingIn fact, homothety is a transformation of any graph similarity In other words, all the points on the curveIn other words, obtained by using the drag ' and Q (MP + MQ). and any geometric shape, as we can see in the following examples: and any geometric shape, as we can graph of the function. We divide the segment AB by some ratio divide the graph of the function. We focal ellipse. If the M focal ellipse. will result that M from using transformation Examples of homothety using dynamic softwareExamples of homothety pick some point A distances from two given points is fixed. They will be will They fixed. is points given two from distances engage the defining feature of the ellipse: the sum of distances of the points feature of the ellipse: the sum engage the defining by dragging reserves of the place (Figure 11). the geometric features we obtain a curve that is similar to the original curve, as seen in Figure 12. Example A: The graph of an algebraic function Example A: The graph of an algebraic

k We take the graph of an algebraic function, for example f take the graph of an algebraic We P 38 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman We connectthispointbystraightlinestopointsonthesidesoftriangle. Given are two parabolas with a common vertex at the origin On eachoftheobtainedsegmentsweconstructacorrespondingpointM triangleABC.WeGiven isanarbitrary selectsomepointP By draggingthepointsD http://tube.geogebra.org/student/m149069. Example B: Demonstrating the property on an arbitrary triangle onanarbitrary Example B:Demonstratingtheproperty with which dividesthesegmentbyratiok tracing thepointsM the locationofverticestriangleandpointP not dependonc the originaltriangleABCareformed. triangle, asseeninFigure13. be changed.Thereisalsoarulerthatallowsthedivisionratioto method arelocatedonthesidesofatrianglethatissimilartooriginal of theparameterfunctionanddivisionfactor. other geometricalshapesaswell. passes throughtheorigin intersectstheparabolasatpointsAandB. Similar parabolasandconstantsegment ratio algebraic function.Theappletcontainsrulersthatallowchangingthevalue available athttp://tube.geogebra.org/student/m149066. This applet demonstrates the method of obtaining a similartrianglewhen The appletfortheformationofatriangleinsideisavailableat This demonstrates the creation of the similar locus on the graph of an The creationofthesimilarlocusongraphanalgebraicfunctionis The studentswererequiredtochecktheexistence ofthesepropertiesin Prove that the ratio of the lengths of the segments a , b > 0 and . a > 1 , b M , asshowninFigure14.Thestraightliney 2 , , M E , 3 , thesidesofinnertrianglethatissimilarto F alongthesidesAC, Figure 13 . ThepointsM AB, A B O A is constant and does obtainedusingthis BC respectivelyand insidethetriangle. y = ax 2 and = cx that y

= can bx 2 Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 39 ), and both 2 = bx > 0) is similar to (y (a 2 2 2 c + = bx = ax . What is surprising is that 1 , and multiplying it by a real ⋅ 2 ⎞ ⎟ ⎠ c a 2 is maintained. The calculation The maintained. is b c = A O , B A = ax 1 1 c b O ⎛ ⎜ ⎝ − + A B , b b a a 2 , = = c ⎞ ⎟ ⎠ + 2 A A Figure 14 Figure O O a c 1 B B A A , remains constant and does not depend on remains constant and does not ⋅ c a A O ⎛ ⎜ ⎝ 1 a B A A c − 1 b = > 0), and the two parabolas maintain a constant ratio A B (b 2 = bx 0, we obtain a similar parabola which is narrower or wider or ≠ 0, we obtain a similar parabola which is narrower or wider or

> 0) and the other one has a maximum y k (a 2 By equating the equation of the straight line and the equations of the By equating the equation of the It should be noted that the action of creating the similar parabola is natural, It should be noted that the action of creating the similar parabola In other words, the ratio In the same manner, in the case that one of the parabolas has a minimum in the case that one In the same manner, = ax share a common vertex, as shown in Figure 15, the property of the constant share a common vertex, as shown in Figure 15, the property of the since when taking a parabola of the type y and therefore inverted, depending on the value and the sign of k parabolas, we obtain the points of intersection parabolas, we obtain the points of conserved property. of the lengths of the segments gives the ratio origin. between the lengths of the chords formed by straight lines issuing from the between the lengths of the chords formed by straight lines issuing the ratio of the segments between two similar parabolas is constant and is a the ratio of the segments between two similar parabolas is constant the parabola y the slope of the straight line. Thus, any parabola y the slope of the straight line. Thus, number ratio between the lengths of the segments the of lengths the between ratio

Hence we obtain

y 40 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman The methodicalaspect The paperpresentsinvestigativeactivitythatdealswithfindingloci Work withdynamicsoftwareallowedthestudentstogainabetterandmore when the division ratio of the relevant segment changes and the location of roles ofdynamicsoftware,theychoosetoaccentuate theroleofdynamic the technologyexemplifies. the recurringpedagogicprocess:reflexivepedagogy inaction.Ofthedifferent the originalpointthroughwhichpenciloflines passeschanges.Martinovic the useofdynamicsoftwareallowedstudents toidentifysurprising to allowtrackingthemovementofapointwhiledraggingdifferentpoint). the pointthatislocatedatoriginallocus.(The‘tracing’operationone through whichthestudentscouldseeactualcreationoflocusina the studentsdealwithtaskshavingtodofindinglociwithoutconsidering case usersrelyontheirown mathematicalknowledgeinordertojustifywhat comprehension ofthelocus.Insecondstage, asinthesubsequentstages, of itsconstruction,theprocessthatstudentsandpupilsclass of thepossibilitiesprovidedforbydynamicsoftware(Geogebra)inorder theshapeoforiginallocus,whereusuallyinclassenvironment conserve profound understanding of the concept ‘locus’, and especially of the dynamics dynamic manner, while draggingapointontheoriginallocusandtracing presented. Duringthefirststageusewasmadeofdynamicgeometricsoftware possible generalisations.Severalstagesofinvestigativeactivityhavebeen find hardtoseewhenthetaskisgivenstaticallyinatextbook.Thealgebraic generalisations that demonstrate the conservation oftheoriginallocus,even generalisations thatdemonstratetheconservation accelerates processesofunderstanding andmathematicaljustification.In this and Manizade(2013)describethecontributionof dynamicsoftwareaspartof software asapartnerinthe learningprocess.Theuseofthedynamicsoftware solution alsopresentsalgebraicprocessesthatdonotsufficientlyaiddynamic Figure 15 Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 41 Stupel & Ben-Chaim (2013) suggested the concept “semi-proof” in “semi-proof” the concept suggested Ben-Chaim (2013) Stupel & In addition, the investigative activity presented in the paper offers students In addition, the investigative activity The investigative task presented in the paper is a powerfulThe investigative task presented in task that exposes The selected loci examples demonstrated in this article, are to our in this article, are demonstrated loci examples The selected As mathematics teacher educators who are charged with the professional As mathematics teacher educators a collection of multiple special case examples and hypothesising (De Villiers, special case examples and a collection of multiple and pedagogic reasoning (Krainer, 1993). The processes of construction of 1993). The processes of construction and pedagogic reasoning (Krainer, activity that is relevant to the class environment. The way the students and activity that is relevant to the class environment. The way the students analytic geometry ratio). Exposing at a given of the segment division (locus, generalisation put an emphasis on the link between different subjects in put an emphasis on the link between different generalisation groups of practising teachers with an activity that is suitable for two teaching groups of practising teachers with an activity that is suitable for two properties and relations between them which remain constant while creating between them which remain properties and relations in the presentation of the importance and necessity of mathematical proof to of the importance and necessity in the presentation proof to the infinite range of special cases. importance of valid mathematical objects, identifying a conserving property, generalising, etc. The dragging etc. The dragging generalising, identifying a conservingobjects, property, objects; identification of a dynamic view of mathematical operation offers development of teachers, we have presented pre-service presented have we teachers, of development two and teachers on the methods of their teaching in the class environment and to interact of their teaching on the methods expanding pedagogic and mathematical knowledge (Segal, Stupel & Oxman, expanding pedagogic and mathematical knowledge (Segal, Stupel environments: the teaching environment of teachers (academic educational environments: the teaching environment of teachers (academic to improve college) and the teaching environment of pupils in class. In order collection of disjoint subjects (House & Coxford, 1995; NCTM, 2000). collection of disjoint subjects (House mathematics: Euclidean geometry and (segment ratios, triangle similarity) mathematics, which has essential value in mathematical education. mathematical and pedagogic aspects of the task provide the opportunity for opportunity the provide task the of aspects pedagogic and mathematical be similar to the tasks given in the classroom environment (Zaslavsky, 2007, be similar to the tasks given in the classroom environment (Zaslavsky, the process of generalisation. In other words: in making it clear to students In other words: in the process of generalisation. generalisations and phenomena of identification on based semi-proof a that the objects using the dynamic software and the processes of mathematical the objects using the dynamic software the students to a complex process of learning and to associating mathematical the students to a complex process the teachers dealt with the task and the discussions held with regards to the teachers dealt with the task and the discussions held with teaching, the students and the teachers were presented with an investigative teaching, the students and the teachers were presented with an the opportunity to performopportunity the mathematical phenomena, from of generalisation the students to relations of these types provides them with an opportunity the students to relations of these science and not as a discrete to consider mathematics as an interconnected working with a technological tool while dragging different points and points dragging different tool while with a technological working led the teachers to find more interest, to cooperate, to be surprised, to reflect led the teachers to find more interest, to cooperate, to be surprised, while working with that the ecological tool is not enough, and stressing the that the ecological tool is not while working with with the tasks. In order to achieve these goals the tasks for the teachers must with the tasks. In order to achieve these goals the tasks for the teachers which they can draw conclusions with regard to similar tasks. which they can draw conclusions knowledge original and are very since they represent the beauty of important 2008). 2015). Tasks that are relevant for integration in the class environment have environment class the in integration for relevant are that Tasks 2015). 1998). Teaching in a technological environment poses a pedagogic challenge poses a pedagogic challenge in a technological environment 1998). Teaching 42 Australian Senior Mathematics Journal vol. 30 no. 1 Stupel, Segal & Oxman The commentsofstudents The commentsofteachers Investigative activityhasbeenpresentedbefore45studentsaspartoftwo Evidence oftheprocessesspecifiedabovewereobtainedwhileconducting with relative ease corresponding applets that describe the similar curves with relativeeasecorrespondingappletsthatdescribethesimilarcurves were exposed.Theyexperiencedinpracticethestrengthoftechnological task incombinationwiththetechnologicaltool: tool as a significant partner in the investigative activity. The ability to construct the participantsweresurprisedbymathematicaldiscoveriestowhichthey teachers aspartofteachertraining.Inaddition,20mathematicsteacher- comments ofthestudentsfollow: experienced whilesolvingtheactivityintechnologicalenvironment.Some educators werepresentedwiththisactivityaspartofnationalconferences.All computerised tool: populations, whichreflectedtheirawarenessofthecontribution Participants’ responses in theirownwords responses Participants’ furthered the awareness on several levels of all those e solving the investigative a reflective discussion with the students concerning the processes they study groups in a course of advanced analytic geometry, and to 20 practicing • • • • • • • • “Working withdynamic softwareallowedustodiscoversurprisingproperties “Working withthecomputer allowedustochangethevaluesof “The computerisedtoolpresentsinavisualandclearmannertheproperty “Now (withintegrationofthedynamicsoftware)Iseeandunderstandwhat “I willdomybesttointegrate thetechnologicaltoolinmylessonsbecause The levelofteachers’ teachers: theimportance of exposingteachers’ The teachers’level:theimportanceofusing the technologicaltoolfor The students’level:thetechnologicaltoolasanassistantinsignificantand Following aresomeexamplesofcommentsgivenbythedifferent that canbegeneralised.” then subsequently try tounderstandwhythisisalways true.” then subsequentlytry the task,whichcontribute totheunderstandingofstudymaterial.” technological toolforthepurposeofinstructingfutureteachers. teachers toinvestigativeactivitythatcombinesthecomputerised of the new locus which is similar to the original locus, which is very of thenewlocuswhichissimilartooriginal locus,whichisvery of theimmediatevisualimpact, thesurprisingnatureandaestheticsof parameters andtoseetheirinfluenceonthelocus immediately., We could integrating investigativeactivityinteachingtheclassroomenvironment. deeper understandingofcomplexanddynamicmathematicalconcepts a locusmeans,howisformed.Itwasdifficulttoimaginebefore.” and phenomena. surprising.” Teaching locus with a conserved property Australian Senior Mathematics Journal vol. 30 no. 1 43 students to acquire the skills for using the tool. At most I will use it as a I will use it as a tool. At most skills for using the to acquire the students Proceedings of the 2002 annual meeting of the for teaching In B. Davis & E. Simmt (Eds), Proceedings accessible in each class and offer the students the option to use them freely; class and offer the students the accessible in each pre-service improves the teachers in mathematics education, because it significant learning.” quality of teaching and brings about itself in many diverse areas, and this is the right time to integrate it in the in integrate it time to right the is this areas, and many diverse in itself presentation to demonstrate to the students without them experiencing it to the students without presentation to demonstrate independently.” educational system as a powerful tool.” to the workload in the program of studies, I do not have the time for the the time for the I do not have of studies, in the program to the workload thus we will bring them closer to the digital age even in the classroom.” them closer to the digital age even thus we will bring hard to recognise, but are important for improving the quality of teaching.” hard to recognise, but are important Mathematics teachers require a broad PCT and MKT training which includes Mathematics teachers require a broad PCT and MKT training which (pp. 3–14). Edmonton, AB: CMESG/GCEDM. Canadian Mathematics Education Study Group Turkish Online Journal of Educational Technology, 2(1), 1303–6521. Online Journal of Educational Technology, Turkish “The task allowed us to encounter generalisation processes which are usually “The task allowed us to encounter “The use of technology tools must be included in most courses taught to taught in most courses be included tools must of technology “The use “The effectiveness and the importance of the computerised tool proved tool computerised the of importance the and effectiveness “The “Despite all the contribution of the technological computerised tool, due tool, due computerised of the technological all the contribution “Despite “The school must make sure that the computerised technological tools are make sure that the computerised “The school must “The task gave me a new view of the teaching of loci.” “The task gave me • • • • • • and mathematical proofs. The tasks presented in this paper are powerful tasks, Summary proofs were given that corroborate the results of the surprising graphical proofs were given that corroborate the results of the surprising presentation, and point to the importance of the computer tool. or a geometric shape. original locus and there is a connection between the parameters. The activity original locus and there is a connection of data that relate to the original locus. In each task, when DGS software was of data that relate to the original References used, a surprising discovered feature was that the new locus is similar to the used, a surprising discovered feature the use of different representations of mathematical objects, dynamic use of the use of different representations of mathematical objects, dynamic processes technology suitable for integrating tasks that combine exploration resulted in conserving properties that can be generalised. Full mathematical Ball, D. L. & Bass, H. (2003). Toward a practice-based theory knowledge of mathematical Ball, D. L. & Bass, H. (2003). Toward which demand the discovery conserved of surprising of the curve properties We presented investigative tasks of creating a new locus through the addition presented investigative tasks of We Alakoç, Z. (2003). Technological modern teaching approaches in mathematics teaching. The modern teaching Alakoç, Z. (2003). Technological The words of teacher educators of teacher The words Chazan, D. (1993). Students microcomputer-aided explanation in geometry. In S. I. Brown & M. I. Walter (Eds), Problem posing: Reflection and application (pp. 289–299). Hillsdale, NJ: Lawrence Erlbaum. De Villiers, M. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer & D. Chazan (Eds), Designing learning environment for developing understanding of geometry and space (pp. 369–393). Mahwah, NJ: Lawrence Erlbaum. Dreyfus, T. & Hadas, N. (1996). Proof as answer to the question why. 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Stupel, Segal & Oxman der Mathematik, 28(1), 1–5. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1), 5–23. Hohenwarter, J., Hohenwarter, M. & Lavicza, Z. (2008). Introducing dynamic mathematics software to secondary school teachers: The case of Geogebra. Journal of Computers in Mathematics and Science Teaching, 28(2), 135–146. House, P. A. & Coxford, A. F. (Eds). (1995). Connecting mathematics across the curriculum. Reston, VA: NCTM. Klein, F. C. (2015). Royal Netherlands Academy of Arts and Sciences. Retrieved 22 July 2015. Krainer, K. (1993). Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction. Educational Studies in Mathematics, 24(1), 65–93. Laborde, C. (2000). Dynamic geometry environment as a source of rich learning contexts for the complex activity of providing. Educational Studies in Mathematics, 44(1), 151–161. Lampert, M. (1993). Teachers’ thinking about students’ thinking about geometry: The effects of new teaching tools. In J. L. Scwartz, M. Yerushalmy, & B. Wilson (Eds), The geometric supposer: What is it a case of? (pp. 143–177). Hillsdale, NJ: Lawrence Erlbaum. Martinovic, D. & Manizade, A. G. (2013). Technology as a partner in geometry classrooms. The Electronic Journal of Mathematics and Technology, 8(2). Retrieved from https://php. radford.edu/~ejmt National Council of Teachers of Mathematics (NCTM). (1991). Standards for evaluation of the teaching of mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author. Segal, R., Stupel, M. & Oxman, V. (2016). Dynamic investigation of loci with surprising outcomes and their mathematical explanations. International Journal of Mathematical Education in Science and Technology, 47(3), 1–20. DOI: 10.1080/0020739X.2015.1075613. http://www.tandfonline.com/eprint/WPMwrksiAtkgw7JqdYwV/full Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 5(2), 4–14. Stein M. K., Grover B. W. & Henningsen M. A. (1996). Building student capacity for mathematical thinking and reasoning; analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, 455–488). Stupel, M. & Ben-Chaim, D. (2013). One problem, multiple solutions: How multiple proofs can connect several areas of mathematics. Far East Journal of Mathematical Education, 11(2), 129–161. Wiest, L. R. (2001). The role of computers in mathematics teaching and learning. Computers in the Schools, 17(1–2), 41–55. Zaslavsky, O., Chapman, O. & Leikin, R. (2003). Professional development in mathematics education: Trends and tasks. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds), Second international handbook of mathematics education (pp. 877–917). Dordrecht, The Netherlands: Kluwer Academic Publishers. Zaslavsky, O. (2007). Mathematics-related tasks, teacher education, and teacher educators: The dynamics associated with tasks in mathematics teacher education. Journal for Mathematics Teacher Education, 10(4), 433–440. Zaslavsky, O. (2008). Meeting the challenges of mathematics teacher education through design and use of tasks that facilitate teacher learning. In B. Jaworski & T. Wood (Eds), The mathematics teacher educator as a developing professional (pp. 93–114). Rotterdam, The Netherlands: Sense. Australian Senior Mathematics Journal vol. 30 no. 1