The Effectiveness of Using Scientific Calculator in Solving Non-Linear Equations by Newton- Raphson Method

Transcription

1 The Effectveness of Usng Scentfc Calculator n Solvng Non-Lnear Equatons by Newton- Raphson Method 1 Lm Kan Boon, 2 Grace Joy Yong, 3 Cheong Tau Han and 4 Tay Km Gak 1,2 [email protected], 3 [email protected], 4 [email protected] 1,2 Unverst Penddkan Sultan Idrs, 3 Unverst Teknolog Mara, 4 Unverst Tun Hussen Onn Malaysa Abstract: In ths paper, we wll report the result of the study on the effectveness of usng scentfc calculator n solvng non-lnear equaton by Newton- Raphson method from the aspect of students marks and duraton taken n solvng queston. A total of 38 mathematcs students who enrolled for numercal analyss course n the second semester sesson 2008/2009 n the Sultan Idrs Educaton Unversty are nvolved n ths research. The data taken from the pre test and post test were analyzed usng the Statstcal Package for Socal Scence (SPSS).The fndngs show that usng scentfc calculator gves postve effect on students ablty to obtan the correct soluton as well as reduces tme to solve the problems. 1. Introducton Calculator s a tool not only to perform mathematcal computatons, but t can be a tool for learnng mathematcs. Despte the myths of harmful consequences resultng from ther use, calculator s a pedagogcal tool of great value. Studes about the role of calculators had been conducted and Kssane [1] stated that students usng calculators have drect access to representatons of mathematcal deas whch are conceptually powerful and vtal for learnng. Students can now focus n understandng the mathematcal concepts and not become dstressed about computng the wrong answers wth the ad of calculators. Indeed calculators are crucal to help students to understand the mathematcal concepts but not to be feared due to ts rote computatons. CALC button n Caso fx-570es and fx-570ms scentfc calculator provdes the ablty to nput the functon by usng the varables and calculate for the values of the functon. Tay [2], Guerrero- García and Santos-Palomo, Á. [3] developed a collecton of non-trval keystroke sequence n Caso fx-570ms scentfc calculator to solve several numercal methods ncludng Newton-Raphson Method [4]. 2. Problem Statement Despte of all ther benefts and capabltes, scentfc calculators wll never replace the human mnd when t comes to readng and understandng a problem stuaton, wrtng an approprate equaton for the problem, choosng whch operaton to use n order to solve the problem,

2 nterpretng the soluton correctly that s dsplayed on the calculator, and determnng the approprateness of the answer. Scentfc calculators are only as effectve as the nformaton students enter nto them. Scentfc calculators, n combnaton wth mental, paper-and-pencl, and estmate sklls when approprate, comprse the tools to help students work through the computatons and manpulatons necessary for solvng problems. The Natonal Councl of Teacher Mathematcs [5] stated n the Technology Prncple that calculators as well as computers are vtal tools for teachng, learnng, and dong mathematcs. Electronc technologes provde vsual mages of mathematcal deas. It benefts the students by organzng and analyzng data. Hence, the students are able to compute effcently and accurately. When technologcal tools are avalable, students can focus on makng decson, reflectng, reasonng, and problem solvng. Technology enrches the range and qualty of nvestgatons by provdng a means of vewng mathematcal deas from multple perspectves. Scentfc calculator technology paved a way for the students to gan a hgher level of mathematcal understandng, rather than gvng up after beng frustrated or bored by these tedous manpulatons. Ths research wll study the effects of usng scentfc calculators n fndng the roots of non lnear equatons by the Newton- Raphson method. 3. Research Questons Ths study amed to determne the effectveness of usng the Caso fx-570es scentfc calculator n fndng the roots of non-lnear equatons by Newton- Raphson method by answerng the followng research questons:. Can students achevement mprove wth the help of the Caso fx-570es scentfc calculator?. Can the Caso fx-570es scentfc calculator reduce the duraton taken to answer queston? 4. Research Hypothess. By usng the Caso fx-570es scentfc calculator, students can mprove ther achevement n solvng non-lnear equatons by Newton- Raphson method.. By usng the Caso fx-570es scentfc calculator, students can reduce the duraton taken to answer the queston. 5. Methods A total of 38 mathematcs students who enrolled for numercal analyss course n the second semester sesson 2008/2009 n the Sultan Idrs Educaton Unversty were selected by smple random samplng to be the sample for ths research. A workshop was conducted for the partcpants. They were frst taught how to solve non-lnear equaton usng Newton- Raphson method wthout teachng them how to use a calculator. A pre test was conducted after they had completed the exercses and were able to solve non-lnear equaton usng the Newton- Raphson method. After that, the partcpants were taught how to use the calculator to solve non-lnear equaton usng the Newton-Raphson method as descrbed n Secton 7. The total marks for both

3 pre test and post test s 10 and the maxmum duraton to answer both tests s 10 mnutes. The scores for both tests and duraton taken to answer both tests were analyzed usng the Statstcal Package for the Socal Scence (SPSS) to test whether there are mprovement n the scores and reducton n the duraton taken to answer queston. 6. The Newton- Raphson Method In numercal analyss, the Newton- Raphson method s one of the best known methods to approxmate the roots of non-lnear equatons. Newton's method can often converge remarkably quckly; especally f the teraton begns "suffcently near" the desred root. Gven a functon ƒ(x) and ts dervatve ƒ '(x), we begn wth a frst guess x 0 and terate usng the followng formula: Repeat the teratons untl f ( x ) 0, f ( x ) or x 1 x. (1) 7. Algorthm of the Caso fx570-es Scentfc Calculator for Newton-Raphson Method Example: Fnd the root of y Iterate untl x 1 x cos( x) 3x 7 wth x 0 = 2 by usng Newton-Raphson Method. Soluton: Knowng that the teraton formula for Newton- Raphson Method s gven by; x f ( x ), f ( x ) 1 x 0,1, 2... Step 1: Change the setup nto radan mode and fx mode nto 3 decmal places. Press qw4 for radan mode snce the functon s n trgonometrc functon. Press qw6for fx mode. Press qw3 for 3 decmal places. Step 2: Input the teraton formula ( ) nto the calculator whch s as follows:

4 Press Q)-a Shft the cursor up to the numerator box and key n the functon nto t. Press kq))-3q)+7 Shft the cursor down to the denomnator box and key n the functon nto t. Press RzjQ)) -3 Step 3: Start by calculatng the frst teraton value by substtutng the ntal value, x 2. 0 Press r and you wll be asked to specfy the value of x nto the functon. n order to be substtuted Press2p Step 4: Contnue to calculate the next value by pressng r M p buttons. Note that the last answer calculated s stored n answer memory. The answer memory can be recalled by pressng M button. Press r M p Step 5: Repeat Step 4 untl x 1 x. SOLUTION: Table 1 Iteraton Table x x - x Snce, hence s the root. The root for s.

5 8. Result and Analyss Students scores The students mean marks n the pre test and post test are shown n table 2. The mean value for the pre-test s , whle the mean value for the post-test s Thus, there s a dfference of where the mean marks ncreases n the post-test. Obvously, the students performances n solvng the non-lnear equatons ncrease after treatment was gven to them. Table 2 Mean and standard devaton for students marks Mean N Standard devaton Par 1 Pre test Post test We compared the pre test and post test marks by usng the pared sample t-test analyss n order to analyze whether there s any mprovement on the students performances. The maxmum mark for both tests s 10. The result of ths analyss s shown n Table 3: Table 3 Pared sample t-test for students marks Pared Dfferences Mean Standard Devaton Standard Error Mean t df Sg. (2-taled) Pre test - Post test Based on the result n Table 3, the test statstc value obtaned s and the p-value obtaned for ths test s where we dvde by 2 gven that ths s a one taled test. The p-value obtaned s less than the sgnfcance value, Thus, we have enough evdence to support the hypothess that the students marks ncreases after they were taught how to use the calculator to solve the non-lnear equaton by Newton- Raphson Method. Duraton taken As shown n Table 4, the mean value duraton for the pre-test s mnutes, whle the mean value duraton for the post-test s mnutes. The mean duraton taken n solvng the nonlnear equatons decreases as much as mnutes after the students were taught the technque to use the calculator. Hence, we can state that the duraton taken by students n solvng non-lnear equatons by Newton-Raphson method was reduced when the respondents solved the mathematcal problems by usng the scentfc calculator. Table 4 Mean and standard devaton for duraton taken Mean N Standard devaton Par 1 duraton_pre_test duraton_post_test

6 We compared the duraton taken by the partcpants n the pre test and post test usng the pared sample t-test analyss n order to analyze whether there s any mprovement on the students effcency on solvng the problem. The maxmum duraton taken for both test s 10 mnutes. The result of the analyss s gven n Table 5: Table 5 Pared sample t-test for duraton taken Pared Dfferences Mean Standard Devaton Standard Error Mean t df Sg. (2-taled) Pre test Post test The result shown n Table 5 confrms that the test statstc value equals to and the p-value s where we dvde by 2 gven that ths s a one taled test. The p-value obtaned s less than the sgnfcance value, Thus, we have enough evdence to support the hypothess that there s a reducton n the duraton taken by the students to answer the queston after they were taught how to use the calculator to solve the non-lnear equaton by Newton- Raphson Method as descrbed n Secton 7. Common Mstakes Table 6 shows the number of respondents that had performed the common mstakes. Table 6 Common mstakes done by the partcpants Type of common mstakes Pre Test Post Test 1. Wrong calculaton Incomplete soluton 3 0 Wrong calculaton and ncomplete soluton are the common mstakes done by the partcpants n the pre-test. Both common mstakes are greatly reduced n the post test after the technques of usng the calculator were taught as gven n Secton 7. Three partcpants dd not have the complete soluton n the pre test whch mght be caused by ther nablty to solve the problem effcently. However, there s no partcpant that faced ths problem n post test. Thus, ths shows that they can answer the queston effcently wth the algorthm n Secton Concluson The effectveness of the scentfc calculator n solvng non-lnear equatons by the Newton- Raphson method has been verfed n ths study. The usage of the scentfc calculator n solvng non-lnear equatons by the Newton-Raphson method mproves students marks. Students marks ncreased after the respondents were exposed to the method of solvng the non-lnear equatons by the Newton-Raphson method by usng scentfc calculator. Tme reduced tremendously when the

7 partcpants answered questons utlzng the Caso fx-570es scentfc calculator. Ths research also shows that the common mstakes made by the partcpants are reduced after they were taught the technque to solve the problem usng the calculator. Acknowledgements We express our thanks to the Sultan Idrs Educaton Unversty, Malaysa for awardng us the grant to carry out ths research (Grant Code: ). We would also lke to thank Nadah Yan for her valuable comments and nput. References [1] Kssane,B. (2006). Technology n secondary mathematcs educaton: The role of calculators. Second Thaland World Teachers Day Congress, Bangkok, Thaland, October 5-7, 2006 (Onlne proceedngs at [2] Tay, K. G. (2006). Numercal Methods wth Calculator Caso fx-570ms. Malaysa: Penerbt KUTTHO. [3] Guerrero-García, P. and Santos-Palomo, Á. (2008): Squeezng the Most Out of the Caso fx-570ms for Electrcal/Electroncs Engneers. Proceedngs of the Internatonal Conference on Computatonal and Mathematcal Methods n Scence and Engneerng (CMMSE). Retreved 17 December 2008 from the World Wde Web: nvestgacon/tr/ma08_01.pdf. [4] Chapra, S. C. (2008), Appled Numercal Methods wth MATLAB for Engneers and Scentsts (2 nd ed). Boston: McGraw Hll. [5] Natonal Councl for Teachers of Mathematcs (2000). Prncples and standards for school mathematcs. Reston, VA: NCTM. New York: Gulford Press.

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