Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives

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1 Douglas A. Lapp Multiple Represetatios for Patter Exploratio with the Graphig Calculator ad Maipulatives To teach mathematics as a coected system of cocepts, we must have a shift i emphasis from a curriculum domiated by memorizatio of isolated facts ad procedures ad by proficiecy with paper-ad-pecil skills to oe that emphasizes coceptual uderstadigs, multiple represetatios ad coectios, mathematical modelig, ad mathematical problem solvig, accordig to the NCTM s Curriculum ad Evaluatio Stadards for School Mathematics (989, 25). Too ofte, educators sped time teachig skills. As a result, little time remais to cocetrate o cocepts that are essetial to uderstadig mathematics. Studets should ot be forced to play with symbols o a piece of paper; rather, they should be allowed to play with ideas that lead to coceptual uderstadig. Maipulatives ad techology ecourage discovery. Later, some of the desirable skills ca be addressed i a remarkably shorter time frame tha traditioal istructio usually requires (Heid 992). May times, as a first itroductio to proof by iductio, studets are asked to prove i = ( + ). Why would studets feel compelled to prove this summatio? It likely holds little meaig for them. To motivate studets, a teacher could discuss the followig problem, used by Thompso (985): Te blocks are eeded to make a staircase of four steps [as show i fig. ]. How may blocks are eeded to make te steps? How may blocks are eeded to make fifty steps? Pólya (957) suggests that a first step i solvig a problem is to be certai that the problem is uderstood. Maipulatives make a great startig Fig. Four-step staircase poit from which studets ca examie the process used i figurig the umber of blocks eeded to build a staircase of a certai size. The act of placig blocks or color tiles to build the figure ca be of great beefit i uderstadig a process by which the umber of blocks ca be determied. For example, i buildig a staircase of height 3, the studet obtais a figure cotaiig six blocks (see fig. 2a). To the cotiue ad build a staircase of height 4, all that he or she eeds is a row of four blocks to place o the bottom (see fig. 2b). The act of placig the blocks helps the studet uderstad the process. Oce the patter of as a expressio for the umber of blocks i a staircase of height is recogized, the brute force process ca be applied to calculate the umber of blocks eeded for a staircase of height, say, 000. Of course, this approach is ot ecessarily a desirable method for fidig a solutio. Douglas Lapp, teaches mathematics ad mathematics educatio at Cetral Michiga Uiversity, Mout Pleasat, MI His research iterests iclude studets views of techological authority ad its impact o the implemetatio of techology i classroom istructio. Why would studets feel compelled to prove this summatio? Vol. 92, No. 2 February Copyright 999 The Natioal Coucil of Teachers of Mathematics, Ic. All rights reserved. This material may ot be copied or distributed electroically or i ay other format without writte permissio from NCTM.

2 The importat aspect of the ivestigatio is mergig represetatios At this poit, techology, such as the TI-83 graphig calculator, ca be used to experimet with differet limits of the summatio (see fig. 3a e). However, the machie registers a error for a fairly small upper idex of 000 (see fig. 3e). Eve a powerful machie like the TI-92 takes quite log to evaluate Fig. 2 Extedig the three-step staircase to four steps This slowess gives the teacher the opportuity to pose the problem of how to fid a ice formula for the sum of the first positive itegers, thus elimiatig the log wait for the aswer. better acceptace i the mid of the studet. Maipulatives ca help studets discover the process that leads to a aswer. However, i lookig for mathematical relatioships, studets ofte fid a graphical approach useful. From a more graphical approach, the curve-fittig capabilities of graphig calculators such as the TI-83, the TI-82, or eve the TI-80 ca supply aother mode of exploratio. For example, cosider the list of data i figure 4a, where L represets the height of the staircase ad L2 represets the umber of blocks eeded. The typical graphig calculator cotais several regressio optios from which to choose for curve fittig (see fig. 4b). Data table Fig. 4 Regressio optios For example, the quadratic regressio commad yields the expressio 0.5x x for the geeral staircase of height x. The graph (fig. 5a) ad the algebraic expressio (fig. 5b) of this fuctio are give as displayed o the TI-83. Further ivestigatio with a cubic (fig. 5c) ad a quartic (fig. 5d) regressio yields algebraic expressios for the fuctios. Note that the cubic ad quartic expressios cotai x 4, x 3, ad costat coefficiets that are either zero or essetially zero. If these terms are igored, the machie gives the same expressio for (c) (d) (e) Fig. 3 Usig the TI-83 to fid the umber of blocks i a staircase of height 000 Graph of 0.5x x Quadratic algebraic expressio Teachers ca motivate studets to discover may differet represetatioal forms for solvig problems. Lapp (995) suggests that cofirmig a aswer across several represetatioal forms yields (c) Cubic algebraic expressio (d) Quartic algebraic expressio Fig. 5 Regressio optios for staircase of height x 0 THE MATHEMATICS TEACHER

3 all istaces of curve fittig, amely, 0.5x x, that is, x(x + ). Oce a apparet patter emerges, the studets ca be asked to prove that it will always hold. Not all studets will be ready to give a formal proof by mathematical iductio or some other meas. At the early stages, a covicig geometric argumet aloe may suffice. Lettig the studets revisit the maipulatives is ofte helpful i gettig them to costruct a reasoable argumet. Cosider a specific case i which the staircase has height 4. By costructig two such staircases (fig. 6a) ad fittig them together to form a rectagle (fig. 6b), we ca argue that the dimesios i the th case will always be o oe side ad + o the other side, which gives the total umber of blocks i the rectagle as ( + ). But sice oly half of this rectagle is desired, we get ( + ) Fig. 6 Geometric argumet + Aother explaatio frequetly give by studets is the triagle approach. I lookig at a staircase as a triagle with jagged edges (fig. 7), we ca argue that the area of the staircase, which correspods to the umber of blocks, ca be determied by the area of the triagle, = 2, 2 2 alog with half the blocks alog the diagoal that were cut off to form the triagle. Sice blocks are alog the diagoal, the umber of blocks eeded to costruct a staircase of height would be give by 2 + = ( + ). 2 2 The importat aspect of this ivestigatio is the mergig of represetatios. The graphical represetatio afforded by techology aids i the search for patters. However, the use of other represetatios, such as physical or icoic models, ca play a importat role i the costructio of a logical argumet that explais the observed patter. 2 3 Fig. 7 Triagle approach It is importat to cosider the other represetatioal forms for solvig this particular problem permitted by such ew techologies as the TI-92. A represetatio for extedig this ivestigatio to the sum of squares uses matrices. Sice the sum of the first positive itegers yields a quadratic closed-form formula, might the sum of squares yield a cubic closed-form formula? To ivestigate, we ca sum to several upper idices of i 2 ad geerate data poits of (0, 385), (5, 240), (20, 2870), ad (25, 5525). If we assume that the formula is of the form ax 3 + bx 2 + cx + d, we ca costruct a liear system i which the ew variables are the coefficiets of the geeral cubic. a b 00 + c 0 + d = 385 a b c 5 + d = 240 a b c 20 + d = 2870 a b c 25 + d = 5525 Usig a matrix represetatio for the system, we get a b 240 = c d 5525 We ext use the matrix capabilities of the TI-92 to solve the system. Storig the coefficiet matrix i A ad the colum matrix o the right-had side of the equatio i B, we get a solutio by evaluat- Vol. 92, No. 2 February 999

4 ig A B (fig. 8). The result suggests that the cubic we are seekig is give by x 3 + x 2 + x Usig the factor commad (fig. 9), we get the form more commoly expressed as Fig. 8 Usig the matrix capabilities of the TI-92 Fig. 9 Usig the factor commad o the TI = /3( + )( + /2) + /2 Fig. 0 Proof without words (Siu 984) + x(x + )(2x + ). ``````6`````` The beauty of the matrix treatmet of this problem is that it allows the studet to exted polyomial curve fittig beyod the ormal choices offered by most graphig calculators. The TI-92 ad TI-83 calculators have caed polyomial curve-fittig capabilities up through a geeral quartic equatio. The use of matrices allows studets to try higher degree polyomials provided that the umber of data poits is sufficiet to produce a osigular coefficiet matrix. Eve though other calculators, such as the TI-83, also allow the use of matrices, the advatage of the TI-92 is that it keeps solutios i ratioal form rather tha as the decimal approximatios used by less powerful machies. Discoverig a formula modeled through a ivestigative method gives the studet a compellig reaso to seek a proof cofirmig the formula. As with the previous example of i, we wish to motivate studets to produce a proof by maipulatives of our discovered formula, i 2 = ( + )(2 + ). ``````6`````` Oe method, proposed by Siu (984), is give i figure 0. This approach is more complicated tha the physical proof offered previously for As a result, studets may have a more difficult time discoverig it o their ow. Some promptig by the teacher may be ecessary. However, the proof give by Siu does use strategies foud i both the triagle approach ad the two-staircase approach preseted previously for Studets do ot ofte have a opportuity to experiece what may mathematicias do regularly, that is, look for cocepts o the basis of patters, ad this approach will allow them to see the mathematical process firsthad. This ivestigatio itroduces the studets to the process of cojecture 2 THE MATHEMATICS TEACHER

5 followed by proof, ad the combiatio of maipulatives ad techology allows this process to be approached earlier tha traditioally thought appropriate. REFERENCES Heid, M. Kathlee. Fial Report: Computer-Itesive Curriculum for Secodary School Algebra. Report submitted to the Natioal Sciece Foudatio, NSF Project Number MDR , 992. Lapp, Douglas A. Studet Perceptio of the Authority of the Computer/Calculator i the Curve Fittig of Data. Ph.D. diss., Ohio State Uiversity, 995. Natioal Coucil of Teachers of Mathematics (NCTM). Curriculum ad Evaluatio Stadards for School Mathematics. Resto, Va.: NCTM, 989. Pólya, George. How to Solve It. 2d ed. Priceto, N.J.: Priceto Uiversity Press, 957. Siu, Ma-Keug. Proof without Words: Sum of Squares. Mathematics Magazie 57 (March 984): 92. Thompso, Alba G. O Patters, Cojectures, ad Proof: Developig Studets Mathematical Thikig. Arithmetic Teacher 33 (September 985): Vol. 92, No. 2 February 999 3

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