LIVERPOOL CONVENTION PACK 12-14 SEPTEMBER 2008 British Origami Society Autumn 2008 Convention Model Collection Liverpool Hope University Edited by Tung Ken Lam Mark Bolitho and Sue Pope 12–14 September 2008 Contents The Models 1 Simple 1.1 Albatross by Andr´es Sierra 1.2 Avion Simetrico by Nicol´as Delgado 1.3 Wild One by Michael Weinstein 1.4 Sharkie by Michael Weinstein 1.5 Stand Tall by Michael Weinstein 1.6 Simple Shirt by Gay Merrill Gross 1.7 Clothespin by Gay Merrill Gross 1.8 Ali’s Dish #2 by Nick Robinson 1.9 Sloth by Nick Robinson 1.10 Heffalump by Tony O’Hare 1.11 Latajaca Strzalka by Boleslaw Gargol 1.12 Bubble Drop by Miyuki Kawamura 1.13 Hungry Bird by Laura Kruskal 1.14 Baby Shoe by Swapnil Shinde 1.15 Squirrel by Yann Mouget 1.16 Hexa-Coaster by Loes Schakel 1.17 Angel by Tony O’Hare 1.18 Signpost by Max Hulme 1.19 Mobile Phone Case by Max Hulme 1.20 Octagonal Star by Gabriel Bland´on 1.21 3D Card with Infant Jesus by Zsuzsanna Kricskovics 1.22 Xmas Wreath by Zsuzsanna Kricskovics 1.23 Creche – Joseph by Zsuzsanna Kricskovics 1.24 Creche – Maria by Zsuzsanna Kricskovics 1.25 Creche – Infant by Zsuzsanna Kricskovics 1.26 Santa Claus with Bag by Loes Schakel Intermediate 2.1 Surprise! by Heinz Strobl 2.2 Crystal Star by Denver Lawson 2.3 Ocean Liner by Mark Bolitho 2.4 Two-sided Pinwheel by Hajime Komiya 2.5 8-Pointed Star Quilt by Paula Versnick 2.6 El Gato sin Botas by Patricio Kunz 2.7 Tulip Bowl by Boaz Shuval 2.8 Santa’s Boot by Assia Vely iv 10 12 13 14 15 16 18 20 22 24 26 28 29 30 32 34 39 40 41 42 43 44 45 46 50 51 54 56 58 60 63 CONTENTS 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 CONTENTS Inflatable Hen by Leyla Torres Bust of Horse by Omar Tapia Sparrow by A´ıda Urrutia Double Container by Dave Brill Cubo Espiral by Jaime Ni˜ no Swan by Jaime Ni˜ no ´ Rect´angulos Aureos Ortogonales by Jaime Ni˜ no Trilobite by S Elmer Lopez Ontiveros Fossil by Andr´es Sierra Striped Tetrahedron by John Montroll Chicken by Tom Defoirdt Manta Ray by John Morgan Stand by Marc Kirschenbaum Whale by Marc Kirschenbaum Dancing Bolero Dancers by A´ıda Urrutia Ram by Jim Adams A Tiling Problem by John Morgan Complex 3.1 Cowboy Hat by Owen Banton 3.2 Nonagami by Alex Bateman 3.3 Fly by Victoria Serova 3.4 Egret by Dan Robinson 3.5 Eagle Owl by Juan Landeta 3.6 Shark’s Attack by Fernando Gilgado Special Guest 4.1 Swan by Edwin Corrie 4.2 Bear by Edwin Corrie 4.3 Ashtray by Edwin Corrie 4.4 Pentagon by Edwin Corrie 4.5 Gondola by Edwin Corrie 66 68 70 72 75 78 80 82 86 89 92 94 95 96 100 102 106 115 116 122 125 130 132 143 154 155 158 161 164 166 Origami and Education 5.1 Paper-folding Polygons by Liz Meenan 5.2 Pull-Up Nets by Liz Meenan 5.3 Triangular Boxes from Rectangles – an Enrichment Project by Cornelius, Tubis & Andrisan 5.4 Open Triangular Box by Tubis & Cornelius 5.5 Triangular Box with Lid by Andrisan & Tubis 5.6 Promoting Problem-solving, Creativity and Communication in Mathematics Education by Pope & Lam 5.7 Origami in the Classroom by C Edison 5.8 Origami Art Therapy by T Kobayashi Index 168 169 180 182 187 191 196 202 204 208 v CONTENTS SIMPLE Simple Albatross Andr´es Sierra, Avion Simetrico Nicol´as Delgado, Wild One Michael Weinstein, Sharkie Michael Weinstein, Stand Tall Michael Weinstein, 10 Simple Shirt Gay Merrill Gross, 12 Clothespin Gay Merrill Gross, 13 Ali’s Dish #2 Nick Robinson, 14 Sloth Nick Robinson, 15 Heffalump Tony O’Hare, 16 Hungry Bird Laura Kruskal, 22 Latajaca Strzalka Boleslaw Gargol, 18 Baby Shoe Swapnil Shinde, 24 vi Bubble Drop Miyuki Kawamura, 20 Squirrel Yann Mouget, 26 CONTENTS INTERMEDIATE Hexa-Coaster Loes Schakel, 28 Angel Tony O’Hare, 29 Signpost Max Hulme, 30 Mobile Phone Case Max Hulme, 32 Octagonal Star Gabriel Bland´on, 34 3D Card with Infant Jesus Zsuzsanna Kricskovics, 39 Xmas Wreath Zsuzsanna Kricskovics, 40 Creche – Joseph Zsuzsanna Kricskovics, 41 Creche – Maria Zsuzsanna Kricskovics, 42 Creche – Infant Zsuzsanna Kricskovics, 43 Santa Claus with Bag Loes Schakel, 44 Intermediate Surprise! Heinz Strobl, 46 Crystal Star Denver Lawson, 50 vii Ocean Liner Mark Bolitho, 51 CONTENTS INTERMEDIATE Two-sided Pinwheel Hajime Komiya, 54 8-Pointed Star Quilt Paula Versnick, 56 El Gato sin Botas Patricio Kunz, 58 Tulip Bowl Boaz Shuval, 60 Santa’s Boot Assia Vely, 63 Inflatable Hen Leyla Torres, 66 Bust of Horse Omar Tapia, 68 Sparrow A´ıda Urrutia, 70 Cubo Espiral Jaime Ni˜ no, 75 Trilobite S Elmer tiveros, 82 Lopez On- Double Container Dave Brill, 72 ´ Aureos Swan Jaime Ni˜ no, 78 Rect´angulos Ortogonales Jaime Ni˜ no, 80 Fossil Andr´es Sierra, 86 Striped Tetrahedron John Montroll, 89 viii CONTENTS SPECIAL GUEST Chicken Tom Defoirdt, 92 Manta Ray John Morgan, 94 Whale Marc Kirschenbaum, 96 Dancing Bolero Dancers A´ıda Urrutia, 100 Stand Marc Kirschenbaum, 95 Ram Jim Adams, 102 A Tiling Problem John Morgan, 106 Complex Cowboy Hat Owen Banton, 116 Egret Dan Robinson, 130 Nonagami Alex Bateman, 122 Eagle Owl Juan Landeta, 132 Special Guest ix Fly Victoria Serova, 125 Shark’s Attack Fernando Gilgado, 143 CONTENTS Swan Edwin Corrie, 155 Pentagon Edwin Corrie, 164 ORIGAMI AND EDUCATION Ashtray Edwin Corrie, 161 Bear Edwin Corrie, 158 Gondola Edwin Corrie, 166 Origami and Education W/2 W/2 h H Paper-folding Polygons Liz Meenan, 169 Open Triangular Box Tubis & Cornelius, 187 Pull-Up Nets Liz Meenan, 180 Triangular Box with Lid Andrisan & Tubis, 191 “My Abuelita Taught Me a Bird” My Experiences of Teaching Origami in the Classroom Christine Edison August, 2008 I have been teaching for five years, the majority of it high school mathematics Mathematics, in the U.S.A., is minimally comprised of Algebra, Geometry, and Algebra with Trigonometry The majority of my time has been spent with low income, high risk students The first day I taught as a certified teacher in a CPS school we constructed a skeletal octahedron and found surface area, volume, and identified vertices and faces The students, ranging in age from 16 to 18, were amazed Creating a manipulative that also looked “raw” gave students a physical connection to Platonic Solids That day was, simply put, fun The students did a wonderful job, were excited, and I was hooked Origami gives students a wide array of skills that go hand in hand with mathematics Spatial skills, sequencing, and problem solving are just some of the benefits A great surprise was how it affects the student’s willingness to accompanying worksheets/bookwork If I constructed an origami project based on the specific skill set and created practice sheets that directly applied to construction and the skill objective of the day the number of students who completed their work skyrocketed, sometimes with 100% completion In the low income urban setting that is not the norm Inclusion classes are classes in the U.S where students with special needs are mainstreamed Different learning modalities are often discussed, but some tend to be put aside in the classroom The ten times that I have had students construct skeletal dodecahedrons in the classroom using Tom Hull’s PHiZZ unit the students that were the best at construction were by and large special needs, and were able to help others, often a first for them One student, Darius*, came after school and told me “I’m stupid, but today I wasn’t Please make me smart tomorrow.” While it isn’t feasible to origami everyday, Darius’s behavior and effort in class changed A small success for him created a large change in his demeanor and he ended up passing the class Sometimes the benefit was not academic, but social A lot of my boys had gone to jail and they went in and out during the school year One student who came out of lockup midyear was angry and non-responsive, coming to school high most days I decided to teach a wreath since Winter Break was coming Roberto1 actually took the paper, which was a change He completed the activity and more importantly told me about his grandmother teaching him Names changed Origami in the Classroom Christine Edison, 202 Origami Art Therapy Toshiko Kobayashi, 204 x Triangular Boxes from Rectangles Cornelius, Tubis & Andrisan, 182 Promoting Problemsolving, Creativity and Communication Pope & Lam, 196 5.3 TRIANGULAR BOXES BY CORNELIUS ET AL W W2 cos θ2 = ORIGAMI AND EDUCATION − h2 4H q W h L Figure 1: Diagram for determining the relationships between L, W, H, θ, and h h θ is the apex angle of the box: sin θ2 = W Folding Procedure – Open Box The first author had originally designed right-apex-angle (45◦ –45◦ –90◦ ) and equiangular/equilateral (60◦ –60◦ –60◦ ) pie containers from arbitrary rectangular sheets of paper The second author noted that the folding procedure could be generalized so that a box with an arbitrary apex angle could easily be folded using well-defined landmarks The folding steps for the general case are shown immediately after this paper For a starting rectangle of length L and width W , the apex angle θ is set by the choice of the parameter h in step 2, with sin θ2 = 2h W For students not familiar with trigonometric concepts, one can say that the angle θ2 is determined by the choice of the ratio, 2h W Some familiar 2h W ◦ choices would be ( θ2 ◦ = θ 2(2 = 30◦ , θ = 60◦ ), and 2h W = = 0.707 to d.p = 45 , θ = 90 ) The folding procedures in step establish the value H of the height of the box so that each of the box walls involves two layers of thickness (except added thickness where tabs are inserted in steps 11 and 12) Now that H is determined, the remaining steps of the construction involve the transfer of the initially formed angles to the central area of the rectangular strip The ends of the strip are reserved for the box walls ◦ ◦ For θ = 180 = 36 , the entire bottom of the box is exactly five layers thick, and for smaller apex angles, extra pleating of the layers of the bottom of the box is required This extra pleating is not shown in the folding steps, and in any case, the box becomes progressively less sturdy and more difficult to fold (and hence less satisfactory as a practical container) for apex angles θ < 36◦ The relationships of L, W, h, θ, and H are shown in figure 1, with The case of θ = 90◦ is easily handled by not initially making the valley crease in step and instead: 1) bringing the left-hand edge of the top layer along the lower edge in step and valley folding to make an angle of 45◦ (half of a right-angle), 2) valley folding the new flap so as to bisect the 45◦ angle, and 3) making the valley crease of step so that the horizontal crease line passes through the new position of the point that originally was the upper-left corner of the top layer in step These steps are much easier to perform than to describe in words 183 5.3 TRIANGULAR BOXES BY CORNELIUS ET AL ORIGAMI AND EDUCATION W θ W2 cos = − h2 (1) 2 The last form of (1) would be appropriate for students who know about the Pythagorean theorem, but who have not studied any trigonometry Some students might find the following rationale for (1) useful: Consider the folded box of step 12 and imagine a tiny insect crawling lengthwise from one end of the starting rectangular strip to the other Clearly, the insect must travel twice H along the two layers of the base walls and twice H along the edge formed by the two walls, which meet at the vertex angle θ Finally, the insect will travel a distance equal to the height of the triangle (from base to the vertex-angle point), which is given by the right-hand sides of (1) These three distances must sum to L L − 4H = Folding Procedure – Box with Attached Lid The extension of the folding procedure to the case of a lidded box was suggested by the third author, and its generalization for any vertex angle θ was worked out by the second author The folding steps are shown after this paper For a θ lidded box, we need to use the portion, W cos θ = W cos , of the length L for the triangular faces of the lower and lid portions of the box The remainder, L − W cos θ2 , of the length is left for the two sets of walls that intersect at the apex angle θ and the bottom box wall with a connecting hinge If both sets of the intersecting walls were two layers thick and the wall-hinge combination three layers thick (see steps 18 and 19), then the box height H would be required to be 17 of [L − W cos θ2 ] Although folding techniques exist for division into an odd number of equal parts, we wish to avoid this type of division To this, we make one set of the walls that intersect at the apex angle θ two layers thick (just as in the open box) and the other three layers thick (See steps and for the forming of these walls.) Then H is required to be 18 (instead of 71 ) of [L − W cos θ2 ] and hence very easy to construct (see step 5) (See also the discussion at the end of the next section.) The relationship between L, W, H, θ, and h for the lidded box in this case is thus W2 θ − h2 (2) =2 The folded open and lidded boxes constitute very useful and attractive containers Their sturdiness may be enhanced by the insertion of cardboard or cardstock into the wall portions (This procedure, however, may be anathema to origami purists.) L − 8H = W cos Discussion First, consider the open box Since 0◦ < θ < 180◦ , or equivalently, < h < in (1), we see that L− W2 − h2 = 4H > W θ cos = L − 2 184 W , (3) 5.3 TRIANGULAR BOXES BY CORNELIUS ET AL ORIGAMI AND EDUCATION so that the full range of the apex angle will be allowed if L > W However, this last inequality is not a restriction on the proportions of the starting rectL angle If L < W , then W > 2L > , in which case we simply interchange the labels “length” L and “width” W Thus, for an arbitrary starting rectangle and specified apex angle θ, a foldable isosceles triangular box exists We assume in the remainder of this paper that L > W A number problems are suggested by the box construction and analysis At the lowest level, students may be simply asked to design a box with specified H, apex angle θ, and W (twice the length of the walls forming the apex angle) At a somewhat higher level, they may be asked to determine the value of θ or h that maximizes the area, of the triangular face of the box, for given values of L and W A= [Answer: θ = 90◦ or h W W2 sin θ = h W2 − h2 (4) √ = ] More advanced students in the high school may choose to apply the calculus of maxima and minima to this problem (a discussion of this is beyond the scope of the present paper), but other students could simply estimate numerically the value of θ or h for which A is maximal A much more challenging problem would be to determine the value of θ or h that maximizes the volume, of the box, for given values of L and W : V = V = hH W2 W2 W θ H sin θ = sin θ[L − cos ] 32 2 h W2 − h2 = [L 4 W2 W2 − h2 − + h2 ], 4 (5) For the lidded box, L − W cos θ =L−2 W2 − h2 = 8H > (6) and the full range of apex angle will be allowed if L > W As already discussed for the case of the open box, this inequality is not a restriction on the proportions of the starting rectangle, and a foldable lidded isoceles triangular box with arbitrary apex angle θ from an arbitrary starting rectangle exists Student problems, similar to the ones presented for the open box, may be posed 185 5.3 TRIANGULAR BOXES BY CORNELIUS ET AL ORIGAMI AND EDUCATION We finally note that if the open-box wall opposite the vertex angle θ is allowed to be of single-layer (instead of double-layer) thickness, then in (1), L − 4H is replaced by L − 3H on the left-hand side In the case of the lidded box, it is easily seen that the walls of the bottom that intersect at angle θ must be of at least double-layer thickness but that the corresponding walls in the lid portion need only be single-layer thick Thus in (2), L − 8H may be replaced with L − 6H or L − 7H on the left-hand side As was previously discussed, these cases would require division into three (prior to division into six) or seven equal parts so that the case considered in the body of the paper is the most straightforward In all of these cases, the condition for the existence of boxes with the full range of apex angle θ is again L > W Conclusion A novel class of paper-folded isosceles triangular boxes from arbitrary starting rectangular sheets of paper has been presented It is illustrative of a large number of other context-rich origami-based classroom activities that may provide useful practical exercises in the application of concepts and analytical techniques of geometry and, if appropriate, trigonometry and even calculus References [1] Carter, J.A., and Ferrucci, B.J (2002) “Instances of Mathematics within Mathematics Content Texts for Preservice Elementary School Teachers,” in T Hull (ed.), Origami3 , Third International Meeting of Origami Science, Mathematics and Education, A K Peters, Natik, MA, pp 337–344 [2] Cornelius, V., and Tubis, A (2002) “Using Triangular Boxes from Rectangular Paper to Enrich Trigonometry and Calculus,” in T Hull (ed.), op cit, pp 299–305 186 5.4 OPEN TRIANGULAR BOX BY TUBIS & CORNELIUS ORIGAMI AND EDUCATION Figure Open isosceles triangular box from an arbitrary rectangle W L h h/2 h is the apex angle of the box [sin(h/2)=2h/W] 4H 2H H 187 H 5.4 OPEN TRIANGULAR BOX BY TUBIS & CORNELIUS ORIGAMI AND EDUCATION Figure Open isosceles triangular box from an arbitrary rectangle continued Completely unfold the model 1st Valley Fold 2nd Valley Fold H The Folds involve all layers Turn over the model end to end 188 5.4 OPEN TRIANGULAR BOX BY TUBIS & CORNELIUS ORIGAMI AND EDUCATION Figure Open isosceles triangular box from an arbitrary rectangle continued Open up the model, but leave end fold in place 10 H 11 H Tab Pocket Tab Lift back and insert tabs into pockets 189 5.4 OPEN TRIANGULAR BOX BY TUBIS & CORNELIUS ORIGAMI AND EDUCATION Figure Open isosceles triangular box from an arbitrary rectangle continued W/2 W/2 h H Complete L-(w/2)cos(h/2)=4H (W/2) cos(h/2)=(W2/4-h2)½ 4H h h/2 h is the apex angle of the box [sin(h/2)=2h/w] Figure Diagram for determining the relationships between L, W, H, h, and h 190 5.5 TRIANGULAR BOX WITH LID BY ANDRISAN & TUBIS ORIGAMI AND EDUCATION Figure Isosceles triangular box with an attached lid from an arbitrary rectangle W L h h/2 h is the apex angle of the box [sin(h/2)=2h/w] (Turn over end to end) Pinch 8H 4H H H 191 5.5 TRIANGULAR BOX WITH LID BY ANDRISAN & TUBIS ORIGAMI AND EDUCATION Completely unfold the model Ignore creases made in steps and on this side H H Use the left-hand-side crease as a guide for right-hand one H Three layers thick Two layers thick H H Page 192 5.5 TRIANGULAR BOX WITH LID BY ANDRISAN & TUBIS ORIGAMI AND EDUCATION H 10 H H H 11 H 12 Folds involve all layers Turn over, end to end 13 Page 193 5.5 TRIANGULAR BOX WITH LID BY ANDRISAN & TUBIS ORIGAMI AND EDUCATION Triangular Box Continued 14 Open up the model, but leave end folds in place H H H 15 Use a table edge as an aid in making the mountain folds 16 Page 194 5.5 TRIANGULAR BOX WITH LID BY ANDRISAN & TUBIS ORIGAMI AND EDUCATION 17 Tab Pockets Tab Insert the tabs into the left-side pockets Close the attached lid on the inside or the outside of the bottom section of the box W/2 18 h W/2 H Complete L-W cos(h/2)=8H Page 195 5.6 PROBLEM-SOLVING BY POPE & LAM ORIGAMI AND EDUCATION Using Origami to Promote Problem-solving, Creativity and Communication in Mathematics Education ∗ Sue Pope & Tung Ken Lam Introduction The use of origami in mathematics education has long been known: e.g [7], [2], [3] and [11], originally published 100 years ago However, such paper folding activities are generally little known and not widespread [8] If origami is used at all in mainstream lessons, it tends to be for demonstrating a small number of geometric principles and ideas [1] Otherwise, origami is seen purely as a ‘fun end of term’ activity Furthermore, the pedagogic approach in both cases is for the teacher to instruct students one step at a time There are few opportunities for creativity and problem solving; students are hardly ever encouraged to ask why methods work There are other teaching and learning strategies besides students following a teacher’s instructions Some presented here are based on those advocated by Wollring [12] 1.1 Whole class teaching and challenges Only the very simplest folds are introduced to the whole class Students then investigate the properties of the resultant shapes and justify their findings Students are challenged to develop their shape into something more interesting 1.2 Students in groups are challenged to make an existing origami object Each group has two examples of a folded object Students are advised to dismantle just one object and figure out how it is made As the objects are modular (i.e made of more than one piece) once they have decided how one unit is made they can then work together to produce the units they need By having one intact model they can figure out how to reconstruct the model Each of these activities can be followed up by asking students to communicate their findings by preparing posters One aim of these posters is to communicate to other groups of students how to make the same object In order to emphasise visual and geometric understanding, the constraint of using as few ∗ To appear in the forthcoming 4OSME Proceedings 196 5.6 PROBLEM-SOLVING BY POPE & LAM ORIGAMI AND EDUCATION Figure 1: Crease lines on A paper, and trapezium with angles marked words as possible was given The use of step folds allows students who cannot draw neatly with pencils and rulers to produce quality work A third origami activity does not involve instructing students or giving students existing origami objects: 1.3 Students are challenged to design shapes to have given properties For example, students might be asked to produce the largest octagon from a square Alternatively, students may be shown a tiling and asked to reproduce it by folding paper units In contrast with students following a teacher’s instructions, these activities allow students to think for themselves and to solve problems A benefit for the teacher and students is that students can work at their own pace This avoids the frustration of some students having to wait for others to catch up, and some students struggling to keep up The poster making activity ensures students probe the structure of an origami design Another advantage is that students can work at their own level For example, there are a number of methods for finding the centre of an equilateral triangle Some may use their visual judgement; others might be able to work out a more precise folding method Examples of the strategies Examples of these teaching and learning strategies activities are now shown They are drawn from a number of contexts including in-service training, mathematics masterclasses and summer schools for gifted and talented children 2.1 Whole class teaching and challeges: Folding a hexagram Students were asked to fold the paper in half along the long mirror line, unfold, and to then fold one corner of the shortest edge onto the original crease to make a new crease through the adjacent corner (figure 1) Students were asked to find out all they could about the resultant quadrilateral 197 ... British Origami Society Autumn 2008 Convention Model Collection Liverpool Hope University Edited by Tung Ken Lam Mark Bolitho and Sue Pope 12–14 September