Procedia - Social and Behavioral Sciences 128 ( 2014 ) EPC-TKS 2013
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1 Available olie at ScieceDirect Procedia - Social ad Behavioral Scieces 18 ( 014 ) EPC-TKS 013 Iductive derivatio of formulae by a computer Sava Grozdev a *, Veseli Nekov b a Professor, DSi, Istitute of Mathematics ad Iformatics, Acad. G. Bochev Street 8, Sofia 1113, Bulgaria b Associated Professor, PhD, Techical Colledge Lovech, Sajko Saev Street 31, Lovech 5500, Bulgaria Abstract The aim of the paper is to preset visual derivatio of classic arithmetic formulae by a computer. The GEOMETER S SKETCHPAD software is used for the purpose. The approach has bee experieced with 6-grade studets, who are already familiar with powers. The results are ecouragig, sice the studets show successful applicatio of the formulae that have bee taught. Several applicatios are proposed. 014 The Authors. Published by by Elsevier Ltd. Ltd. This is a ope access article uder the CC BY-NC-ND licese ( Selectio ad peer-review uder resposibility of EPC KTS ad Guest Editors Dr Cristia Vasile, Dr Mihaela Siger ad Dr Selectio Emil Sta. ad peer-review uder resposibility of Petroleum-Gas Uiversity of Ploiesti, Educatio Scieces Departmet. Keywords: positive iteger; sum; power; formula; computer; circle; uit square; triagle; geometric figure; area 1. Itroductio The possibility to cosider umbers as geometric figures is of iterest ot oly from iterdiscipliary poit of view. Such a possibility is of cosiderable importace i Mathematics educatio, because it helps studets to perceive ew otios ad facts by additioal orgas of sese, eyesight icluded. Sometimes, the approach is called proofs without words (Nelse, 1993). Sice each formula has two sides, the mai well-kow combiatorial tool for demostratio is to represet them i two differet ways. I our case this meas to itroduce two geometric objects or otios from Geometry. The GEOMETER S SKETCHPAD (GSP) software is used for the purpose. Applicatio of other software like GeoNext, GeoGebra, etc., is possible too. The approach has bee experieced with 6-grade studets. The choice of the age is depedet o the curriculum. Kowledge of powers is ecessary. The results are ecouragig, sice the studets show successful applicatio of the formulae that have bee taught. * Tel.: ; fax: address: sava.grozdev@gmail.com The Authors. Published by Elsevier Ltd. This is a ope access article uder the CC BY-NC-ND licese ( Selectio ad peer-review uder resposibility of Petroleum-Gas Uiversity of Ploiesti, Educatio Scieces Departmet. doi: /j.sbspro
2 400 Sava Grozdev ad Veseli Nekov / Procedia - Social ad Behavioral Scieces 18 ( 014 ) The authors believe that measuremets for the level of perceptio are ot eeded i this case sice correct uses are eough.. Methodology Usually, mathematical iductio is icluded i the last grades curriculum of secodary school. However, prelimiary steps i its mai idea learig are possible to be exercised earlier by meas of a computer. It is ot meat a full iductio but small iductive steps that lead studets to kowledge. The authors propose a approach, which is similar to the so called Socratic style gettig the truth via iquiry. The origial Socratic method is ot advocated here, because it is ot used a questioig to dismatle or discard preexistig ideas. It is emphasized oly o the ultimate goal of such a style to icrease uderstadig through iductive steps. Iductio is a hidde reality of sciece ad the computer is a coveiet tool to reveal a part of it or at least to give a possibility of ivestigatig to a level, which supports its full study. The iductio method could be realized as a experimetal approach to a give mathematical problem. This approximates the problem to oe from the domai of the experimetal sciece, by which a mathematical relatio is discovered. Well-kow arithmetic formulae are derived i the sequel ad some applicatios are proposed. The followig formulae for arbitrary positive itegers are attacked by mathematical iductio, usually: , 135 1, 1 3, However, it is ot clear where these formulae come from. A certai clarificatio will be doe usig geometric represetatios by meas of GSP. 3. Sum of the first positive itegers Usually, the Gauss method is applied to fid the sum S 13. We have: S A geometric variety of this idea has bee used i Aciet Greece. Here, the positive itegers are idetified with correspodig umbers of circles (Fig. 1). Cosider the cases: S1 1, S 1, S3 1 3 ad S I each case arrage the circles i a right triagle, copy the triagle ad reflect the copy symmetrically, as show. Fially, stick the symmetric image to the iitial triagle. Rectagles are obtaied of the kid: 1, 3, Fig. 1
3 Sava Grozdev ad Veseli Nekov / Procedia - Social ad Behavioral Scieces 18 ( 014 ) ad Assumig that the area of each circle is equal to 1, the the sums are expressed by the areas of the correspodig rectagles: 1.1., 1..3, ad The coclusio is that i the geeral case of S a 1 rectagle 1 is obtaied. Thus, S. Note, that because of the shapes, by which the represetatios have bee realized, the umbers 1, 3, 6, 10, are called triagular umbers. The triagular 1 umbers are described by the formula. 4. Sum of the first odd positive itegers I order to derive a formula for the sum P 135 1, cosider the cases P 1 1, P 1 3, P ad P Use agai a circle for each added. I each case arrage agai the circles i a right triagle, which is with a toothed hypoteuse ow (Fig.). Rearrage the obtaied triagles i squares step by step, as show. The computer aimates the procedure. Iductively we coclude that P. The shapes are the reasos to call the sums square umbers. 5. Sum of the squares of the first positive itegers The formula for the sum Q 1 3 could be discovered usig the two forms of the square umbers. Firstly, represet each square of the itegers from 1 to 8 by the toothed triagles from the previous chapter (Fig. 3). Costruct a Eiffel tower usig full horizotal rows of circles: begi with the logest row, cotaiig 1 circles (i our case 15 circles); the put all rows with 3 circles (i our case two rows with 13 circles each); cotiue with the ext three rows i a similar way ad s. o, as show. The Eiffel tower, thus obtaied, is symmetric ad cotais as may circles as the value of Q. Because of the symmetry it is possible to stick two upside dow copies of the tower to both sides of the iitial oe. The obtaied figure is a rectagle with sides 1 ad 13, cotaiig three times the sum Q (Fig. 4). Cosequetly, 3Q 1 13, i.e. Q Fig.
4 40 Sava Grozdev ad Veseli Nekov / Procedia - Social ad Behavioral Scieces 18 ( 014 ) Fig. 3 Fig. 4
5 Sava Grozdev ad Veseli Nekov / Procedia - Social ad Behavioral Scieces 18 ( 014 ) Fig Sum of the cubes of the first positive itegers Differet geometric ideas exist for the determiatio of the sum R 1 3. We will cosider the oe of Warre Lushbaugh (Garder, 1988). Cosider a square. Sice it cotais 4 uit squares, its area is (Fig. 5). Pack this square with squares of the same size. Eight squares are eeded. The area of the ew figure is 4... Pack it with 3 3 squares. We eed such squares ad the total ew area is Cotiue the packig util a pack of 4 squares with side legth is reached. The ew surface is 4.. (Fig. 5). The figure, thus obtaied, is a cosistet square (without holes ad overlappig). Its side legth is equal to, while the area is O the other had, from the way of costructig, it follows, that the area is equal to I such a way we obtai the formula 7. Exercises ad applicatios , i.e. R 1. Problem 1. Fid the sum Solutio: The studets were asked to use the idea for the determiatio of the sum S i the case 10. All of them costructed isosceles right triagles with pecil like the oe i Fig. 6. After that they were asked to take a copy of the triagle ad by drawig to stick it with the iitial triagle i such a way, that their hypoteuses coicide exactly. The obtaied figure was a square like the oe i Fig. 6. The studets couted the circles i the square: The teacher gave the geeral result
6 404 Sava Grozdev ad Veseli Nekov / Procedia - Social ad Behavioral Scieces 18 ( 014 ) Fig. 6 Fig. 7 Problem. Fid the umber of all squares, which ca be foud i a chess-board. Solutio: The classic chess-board is a 8 8 square, but we cosider the geeral situatio of a board. Take the cases 1,, 3 ad 4 (Fig. 7). I each case the first step is to cout the iitial square. At the secod step cosider all possible equal squares with side legth, which is smaller by oe uit square. At the third step cosider all possible equal squares with side legth, which is smaller by two uit squares, ad s. o. Note that at each step the equal squares are grouped i a square ad it is eough to cosider the dimesios of this square. Thus, the coutig of the cases uder cosideratio gives: 1, 1, 1 3 ad The umber of the squares i a board is equal to Q 1 3. I the case of a classic chess-board the umber is Q The last problem was solved by the teacher, while the ext oe was give to the studets for homework. Istructios were give to follow the idea of the solutio of problem. Problem 3. Fid the umber of all rectagles, which ca be foud i a chess-board. 8. Coclusio The formulae ad the problems, which are discussed i the preset paper, are of arithmetic character. Very ofte such kid of material is borig. Usually, studets feel themselves more comfortably whe umbers, terms or otios are visualized. Geometry suggests may possibilities i this directio. Geometric objects ad iformatio techologies stimulate cogitive activity. They have the power to trasform the profoud idifferece from the begiig to desirable actios. The teachig ad learig become successful ad the proof is i the listed examples.
7 Sava Grozdev ad Veseli Nekov / Procedia - Social ad Behavioral Scieces 18 ( 014 ) Refereces Nelse, R. (1993). Proofs without words. The Mathematical Associatio of America, ISBN Garder, M. (1988). Hexaflexagos ad Other Mathematical Diversios: The First Scietific America Book of Puzzles ad Games. Uiversity of Chicago Press, ISBN
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