Available nline at www.sciencedirect.cm Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 INTERNATIONAL EDUCATIONAL TECHNOLOGY CONFERENCE IETC01 Apply Discvery Teaching Mdel t Instruct Engineering Drawing Curse: Sketch a Regular Pentagn Chin-Hsiang Chang * Natinal Sua Marine & Fisheries Vcatinal High Schl, N. 31, Su-Gng Rd. Su-A, I-Lan, ( 70), Taiwan Abstract The drawing methd f a regular plygn has been explained in a variety f different methds in the relevant literatures. Mst f the students against the pltting described prcedure t cmplete the drawing step by step, but ften withut knwing the principle. It s imprtant t develp a reasnable drawing scheme with simple gemetry applied in the engineering drawing technlgy. A new mdel fr regular pentagn (regular plygn, n=) drawing will be develped with Discvery teaching methd (as experimental grup). The perfrmance is better than Demnstrating methd (as reference grup) that students cntrast the drawing prcedure t finish a regular pentagn step by step. We chse the students f vcatinal high schl prvided with engineering backgrund as a sample space. The pass prbability fr the experimental grup with Discvery teaching methd is p=.69 which is higher than the reference grup p=.. The glden rectangle is als applied t map the relatinship f regular pentagn and develped an interest fr engineering drawing. 01 Published by by Elsevier Elsevier Ltd. Ltd. Selectin Selectin and/r and/r peer-review peer-review under respnsibility under respnsibility f The Assciatin f The Science Assciatin Science Educatin and Technlgy Educatin and Technlgy Open access under CC BY-NC-ND license. Keywrds: Discvery teaching methd, engineering drawing, glden rectangle, regular pentagn, regular plygn 1. Intrductin Smene sketched a regular plygn as the mathematics research games. Sme f them were maybe difficult enugh fr a vcatinal high schl student, even university. The gemetric appearance is a significant tpic f engineering drawing curse which teaches students t draw sme regular plygns. * Crrespnding authr. Tel.: +886-9-70773; fax: +886-3-989099. E-mail address: changch@pst.savs.ilc.edu.tw. 1877-08 01 Published by Elsevier Ltd. Selectin and/r peer-review under respnsibility f The Assciatin Science Educatin and Technlgy Open access under CC BY-NC-ND license. di:10.1016/j.sbspr.01.11.0
8 Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 Nmenclature A t N apexes f regular pentagn L side length f regular pentagn L1 t L relative length in a regular pentagn N sample space R radius f a circumscribed circle a relative length in a regular pentagn b relative length in a regular pentagn n number f sides fr the regular plygn, event number r radius f an inscribed circle Theta inner angle f plygn Students are always taught t fllw the drawing prcedure and finish a regular plygn withut knwing the principle in Demnstrating methd. We expand a new teaching mdel abided by Discvery teaching methd which was been develped by Jerme S. Bruner. We will als investigate that the relatinship f glden rati is matched in a regular pentagn. Therefre, it is based n the gemetric drawing skill t emply a n-scale ruler and cmpass nly t sketch a regular pentagn. The glden rectangle can be invlved int a regular pentagn. Rark (19) had derived the relatin between the side length, radius f an inscribed circle and radius f a circumscribed circle. Richmnd (1893) had drawn an accurate regular pentagn but n calculating frmula included. The drawing prcedure was develped t sketch a crrect regular pentagn (Madachy, 1979 and Pappas, 1989a). Pappas (1989b) als fund the glden rectangle principle. Livi (00) had sketched a lgarithmic spiral frm the glden rectangle and called it as a glden spiral. They cntributed t the sketching methd f regular pentagn and glden rectangle, but the syllgism was seldm intrduced in the literatures. There was nt any study that used Discvery teaching methd t teach students hw t finish a regular pentagn drawing. We will derive a simple teaching mdel t apply in the engineering drawing technlgy field in this research.. The Study fr Drawing a Regular Pentagn Let s see Demnstrating methd fr drawing a regular pentagn. There are three mdels t sketch a regular pentagn. The length f ne side, radius f a circumscribed circle r radius f an inscribed circle is given that makes the specific regular pentagn separately..1. Mdel 1: Draw a Specific Regular Pentagn frm the Given Length f One Side The 1 st case f drawing a specific regular pentagn is frm the given length f ne side. Sketch it in Figure 1 and list the prcedure belw. Lengthen AB and make a vertical line BD which is perpendicular t AB, then BD = AB.
Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 9 Find the mid-pint C f AB, then put the centre f circle at pint C and use the length fcd as a radius t make an arc which intersects line AB at pint E. Set pint A as the centre f circle and make an arc with a radius f AE which intersects arc AD at pint F. The arc intersects the perpendicular divided line f AB at pint G. Use the length f GF as the radius, then set pint G and A as the centre f circle t draw tw arcs which intersects each ther at pint H. Cnnect five vertexes and finish the drawing f regular pentagn ABFGH. Fig. 1. Specific regular pentagn frm the given length f ne side.. Mdel : Draw a Specific Regular Pentagn frm the Given Radius f a Circumscribed Circle Fig.. Specific regular pentagn frm the given circumscribed circle
60 Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 The nd case f drawing a specific regular pentagn is frm the given radius f a circumscribed circle. Nw, sketch it in Figure and illustrate the steps belw. Get tw diameters which meet at right angles and make a perpendicular divided line f radius t get the mid-pint C. Set pint C as the centre f circle and make an arc with a radius fci which intersects the diameter at pint E (where pint I is ne the quarter psitin). Set pint I as the centre f circle and make an arc with a radius f IE which intersects circle at pint B. Therefre, we can get the length f ne side f regular pentagn as IB. Finally, use the same length f IB t get five apexes in rder and finish the regular pentagn IBHGD..3. Mdel 3: Draw an Apprximate Regular Pentagn frm the Given Radius f an Inscribed Circle The 3 rd case f drawing a specific regular pentagn by the given radius f an inscribed circle is nt fund in the literatures. The efficiency methd fr dividing an angle t five parts has nt been expanded. S, sketch an apprximate but nt a precius regular pentagn in Figure 3 and list the steps belw. Use a pin cmpass (a small wheel inside can be used t adjust tiny distance) and separate t five parts ( divided pints, S1~S) fr the right angle. Get tw parts f divided angle as degree 36 and intersects the tangent line which passes the quarter psitin f Q at pint A and B. It s based n AB as the length f ne side t sketch the ther sides in rder. We can get the apprximated regular pentagn ABCDE. Fig. 3. Apprximate regular pentagn frm the given inscribed circle 3. Use Discvery Teaching Methd t Guide Students Sketching a Regular Pentagn We use Discvery teaching methd t guide students drawing a regular pentagn and slving the teaching prblem withut knwing the gemetry f regular plygn. They are als derived three mdels f sketching a regular pentagn like Demnstrating methd. Hwever, we develp the mdels that the
Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 61 length f ne side, radius f a circumscribed circle r radius f an inscribed circle is given respectively. The teaching scheme will be fcused n the derivatin t guide students and make the right graph. 3.1. Mdel 1: Draw a Specific Regular Pentagn frm the Given Length f One Side Because each interir angle f a regular pentagn is 108, we start it frm the trignmetric functin f cs18 and sin18 which is derived by times angle frmula f csine functin. Refer t Figure 1 and express the length f diagnal line AG belw. L is the length f side AB. 3 cs90 = 16cs 18 0cs 18 + cs18 (1) + cs18 = 8 () 1 sin18 = (3) Refer t Figure 1 and express the length f diagnal line AG belw. L is the length f side AB. AC L / sin AGC = sin18 = = AG AG 1 AG = ( + )L () () It is based n Pythagrean Therem t derive the reductin f 1 + = that 1 and are the length f tw sides and is the length f hyptenuse. It culd be added with the riginal length f 1 and divided t tw parts which length is ( 1+ ). In Figure, fllwing are the steps we develp by Discvery teaching methd. The knwn length f ne side AB is L. Lengthen AB t pint D and let it satisfy BD = AB. Make a perpendicular divided line f AD, let BC equals twice f AB. S it satisfies AC L =. Pint A is set as the centre f circle and the length f AB is radius. It s used t draw an arc which intersects CA at pint E. ( AE = AB = L, CE = ( 1+ ) L ). Make a perpendicular divided line f CE and get the mid-pint H. The length f diagnal line is same asch = ( 1+ ) L /. Set IJ = L as the given length f ne side fr the regular pentagn and make a perpendicular divided line f IJ. Pint I is set as the centre f circle and the length fch is radius. Make an arc and intersect the perpendicular divided line f IJ at pint K which is an apex f regular pentagn. The length f diagnal line is IK = CH = ( 1+ ) L /. Set pint J as the centre f circle and the length f CH (length f diagnal line) is radius. And set pint I as the centre f circle and the length f IJ is radius, t. Make tw arcs and intersect the ther at pint L. The length f diagnal line is als JL = CH = ( 1+ ) L /. Same as the 7 th step, we set the centre f circle at pint I and the length f diagnal linech as radius. Tw arcs meet at pint M. Cnnect the apexes f IJMKL which is the regular pentagn.
6 Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 Fig.. Regular pentagn drawing frm the given length f ne side fr discvery teaching methd 3.. Mdel : Draw a Specific Regular Pentagn frm the Given Radius f a Circumscribed Circle Refer t the circumscribed regular pentagn in Figure and gt BFI = 36. The frmula will be derived belw. We can invlve eq. () and eq. (3) int the fllwing equatins. cs BFI = cs36 cs36 = cs 18 1+ BF = R sin BF = IF 18 1+ = (6) We can get the fllwing result. ( 1 + BF = ) R (7) The length f R is radius f circumscribed circle. It is als based n Pythagrean Therem t derive the length f BF frm eq. (7). Beginning steps are similar t Figure which the ther sketching prcedure listed belw.
Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 63 Draw a circle with the radius f AB (equal R in the left picture), and let the length f OI equal R in the right ne. The relatinship f CH = ( 1+ ) R / is similar t Figure. Use pint I (centre f circle) t prtray tw arcs intersect the circle at pint J and K which are tw apexes. Set CH as the length f regular pentagn, pint J as the centre f circle and JN as the radius t prtray an arc. The arc intersects the circle at pint L. Pint K is set as the centre f circle and the length f JN is radius. It s used t draw an arc which intersects the circle at pint E. Cnnect apexes f LMKNJ and get an accurate regular pentagn. Fig.. Regular pentagn drawing frm the given circumscribed circle fr discvery teaching methd 3.3. Mdel 3: Draw an Accurate Regular Pentagn frm the Given Radius f an Inscribed Circle Fig. 6. Accurate regular pentagn drawing frm the given inscribed circle frm discvery teaching methd
6 Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 MOB = 36 We will analyze this mdel frm a half central angle in Figure 6. Derive the length f a and b belw by trignmetric functins. It is an accurate drawing which is different frm Figure 3. 1+ cs36 = MB tan36 = OM (8) a = b = ( r 1) r (9) Because the length f b is easier t develp than a fr Pythagrean Therem in Figure 6. It is als derived the reductin f 1 + = initially. We will guide students t sketch it by relatinship f b and the radius f inscribed circle. Cntrast Figure 6 and design the drawing prcedure as fllwing. Draw a tangent line AB fr circle O. The tangent pint is M, and the radiusom equals r. Make a line segment FG n the right side f Figure 6 which is equal t the radiusom. Draw a segment GH which is perpendicular t FG. The length f GH is r. Cnnect pint F and H, and the length f FH is r. Set pint F as the centre and FG as the radius f circle t make an arc which intersects FH at pint I. We can find the frmula IH = ( 1) r is same with the value b in eq. (9). Back t the left side f Figure 6, pint O is set as the centre and IH is the radius f circle which intersects the tangent line at pint A and B. The tw apexes f regular pentagn are als decided, and the length f ne side is cnfirmed. Refer t the length f ne side in previus step; draw the apexes as pint C, D and E in rder. We cnnect pint ABCDE and get an accurate regular pentagn.. Glden Rectangle Fund in a Regular Pentagn Fig. 7. The glden rectangle (glden rati) is clsely related t the length f a regular pentagn
Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 6 There is a certain rati f ( 1 + ) / in eq. () and eq. (7). We can say that it is really a magic rati because its reciprcal is ( 1+ ) /. It shws that the rati is clsely related t the length f a regular pentagn. The illustratin is fund in Figure 7. The majr length f the regular pentagns ( A1 A A and 3A A B in Figure 7) shws as 1BB3BB belw. The length f segment A in Figure 7 is the reference length which equals L. 3 A 3 L1 = B3B = L 0.38L 1+ L = A B = B3 A = L 0.618L L = A B = B A = A A = A B = A B = L 3 3 1+ L = A A = L 1.618L 3 1 1 (10) The cmparisn between each length is listed belw. L + L = L 1 3 3 L + L = L (11) Use tw expressins f A A = 1. 618L and A B1 = A B1 = L t make a figure f glden rectangle shwed as the right ne in Figure 7.. Results and Cnclusins.1. Results We select grade ne students fr engineering backgrund f Natinal Sua Marine & Fisheries Vcatinal High Schl in Taiwan. There are seventy-six students separated tw parts f experimental grup with Discvery teaching methd and reference grup with Demnstrating methd. All f students must be taught hw t make a perpendicular divided line and knw the gemetry f a regular pentagn in class. In rder t satisfy the essential requirements in trignmetric functins, the class f teaching sine and csine functin is arranged fr the students f experimental grup t ensure that they have ability t calculate the simple size f a regular pentagn enugh. The test is designed as tw stages t g n. In the first stage, shw them the drawing prcedure t sketch three mdels f regular pentagn step by step fr reference grup. But, teach them the gemetry f regular pentagn and guide t finish the drawing by Discvery teaching methd fr experimental grup. It is essential t cnfirm that everyne culd sketch all types f drawing by these tw different teaching mdels. In rder t avid affected the test results due t desired psychlgy, the nd stage test isn t ntified t all students. The nd stage test is held tw weeks later. We dn t ffer any drawing prcedure t all students. Let the students f reference and experimental grups t sketch three types f regular pentagn which are drawn befre. All f the auxiliary line shuld be retained t check the crrectness. We dn t limit the drawing time t sketch them. It was estimated whether they pass just fr the crrect drawing steps r nt. The results are listed in Table 1.
66 Chin-Hsiang Chang / Prcedia - Scial and Behaviral Sciences 6 ( 01 ) 7 66 Table 1. Results fr demnstrating and Discvery teaching methds (sample space, N=38) Drawing Mdel Fig.1 Fig. Fig.3 Average Fig. Fig. Fig.6 Average Pass Member (n) 3 17 0.7 8 9 6.3 Pass Prbability (p).60.79.7..737.763.79.69.. Cnclusins Figure 1, and 3 (refer t Demnstrating methd) are crrespnding Figure, and 6 (refer t Discvery teaching methd) develped frm the same mdel f regular pentagn. The cnclusins frm Table 1 are listed belw. The pass prbabilities fr Figure 1 and Figure are higher than Figure 3, we infer the cause abut nt familiar with using the graphic instrument f pin cmpass fr mdel 3 in demnstrating mdel. Figure 3 and Figure 6 are the mst difficult ne in drawing a regular pentagn, s they get lwer prbability. The pass prbability fr Discvery teaching methd is higher than fr Traditinal ne. It can be illustrated that Discvery teaching methd applied in engineering drawing was gd. Because the pass prbability fr Figure 6 isn t high enugh, we analyze that gemetry f trignmetric functins is difficult fr sme students. The pass prbability just derived ut the gemetry f regular pentagn crrectly is p=.763 in experimental grup. It indicates 1/ students culd nt calculate the gemetry frm trignmetric functins. But it had high average prbability (p=.907) fr the students wh can derive ut the gemetry f regular pentagn crrectly. It shws students will almst pass when they culd deduce the gemetry. The teaching mdel can als be used t the ther engineering gemetric appearance f sme regular plygns (fr example, regular heptagn (n=7) etc.) fr future. References Cxeter, H. S. M. (1969). Intrductin t Gemetry ( nd ed.). New Yrk, U.S.: Wiley. Livi, M. (00). The Glden Rati: The Stry f Phi, the Wrld's Mst Astnishing Number. New Yrk, U.S.: Bradway Bks. Madachy, J. S. (1979). Madachy's Mathematical Recreatins. New Yrk, U.S.: Dver. Pappas, T. (1989a). The Pentagn, the Pentagram & the Glden Triangle - The Jy f Mathematics. San Carls, CA: Wide Wrld Publ. / Tetra. Pappas, T. (1989b). The Glden Rectangle - The Jy f Mathematics. San Carls, CA: Wide Wrld Publ. / Tetra. Rark, R. J. (19). Frmulas fr Stress and Strain (3 rd ed.). New Yrk, U.S.: McGraw-Hill. Wells, D. (1991). The Penguin Dictinary f Curius and Interesting Gemetry. Lndn, England: Penguin.