Presentation on theme: "Department of Physics (ret.)"— Presentation transcript:
1Department of Physics (ret.) Curtis Center Mathematics and TeachingConference, UCLA, March 2, 2013Origami Box Design as a Mathematical Modeling Activity in GradesArnold TubisDepartment of Physics (ret.)Purdue UniversityWest Lafayette, IN 47907
2For further information, collaborative help with lesson plans, etc For further information, collaborative help with lesson plans, etc., please contact :Arnold Tubis
3Origami in K-12 Mathematics Education: a Brief Historical Survey Friedrich Froebel ( ) invented the kindergarten, and he and his followers introduced the wide-scale use of paper folding (in kindergarten and beyond) as a tool for mental development and informally introducing some of the basic elements of geometry.
4Mental development begins with the observation of concrete objects and gradually expands into comprehension of abstract ideas. The possibilities of the square (sheet of paper) may be regarded as practically endless in the development of instructive and interesting forms. It is customary to divide these forms into three great classes: life, knowledge, and beauty. The latter two forms are the ones with the deepest mathematical relevance.(Dr. Albert Elias Maltby – an American follower of Froebel; Pennsylvania, 1894.)
5Forms of LifeSimple origami models with which children develop their folding skills, sharpen their observations of the special characteristic features of different things around them, and make simplified models of real objects.
6Forms of BeautySimple folded patterns, ornaments, boxes, polygons, and polyhedra that demonstrate various aspects of symmetry.
7Forms of KnowledgeFolding of simple shapes, geometric analysis of crease patterns, and folding sequences associated with basic concepts, constructions, and theorems of plane geometry.
8The Rise and Demise of the Froebel Classroom Paper Folding Program Paper folding in the classroom spread from Germany to the rest of Europe, Great Britain, Japan, and North and South America.Its height of popularity:Its decline was due, in part, to increasing dependence on pre-set, rigid standardized folding patterns.
9Contemporary Reincarnation of Some Froebelian Ideas: Van Hiele Levels of Geometric Understanding (1957)Visualization: classification of shapes by holistic appearance. Grades K-2.Analysis: recognition of figures in terms of their properties (e.g., square – 4 equal sides, 4 congruent angles). Grades 3 – 5.Abstraction: demonstration, understanding, and simple informal proofs that certain properties of geometric figures may imply other properties (e.g., isosceles triangle - congruent base angles). Grades 6 – 8.
10Van Hiele Levels (Continued) Deduction: systematic deduction of theorems from undefined terms, definitions, and axioms in the context of Euclidean geometry. Grades 9-12.Rigor: systematic deduction of theorems from undefined terms, definitions and and axioms in generalized (non-Euclidean) geometries. University level.
11Basic Point:Students can successfully achieve geometric understanding at a given Van Hiele level only if he/she has achieved understanding at the lower levels.
12OrigamiUseful for promoting the analysis and abstraction Van Hiele levels of geometric understanding.Useful as a platform for modeling studies (main point of this presentation).Strengthens students’ working knowledge of geometric concepts and techniques.
13Monographs on K-12 Origami Math (Excluding Many on Polyhedra) Row, Tandalam Sundara (1893, 1905)Olsen, Alton T. (1975)Johnson, Donovan A. (1976)Jones, Robert (1995)Serra, Michael (1997)Youngs, Michelle and Tamsen Lomeli (2000)Tubis, Arnold and Crystal E. Mills (2006)Chinese book: Paper Folding and Mathematics (2012)Except for Tubis and Mills (2006), these books mainly focus on geometric constructions and the demonstration of properties of geometric figures via folding.These mostly focus on geometric constructions and demonstrations of geometric results via folding.
14Origami and the Van Hiele Abstraction Level –Some Illustrative Examples, with Associated CCSS–M Congruence of vertical angles(7.G.B.5, 8.G.A.5)Congruence of alternate interior angles(8.G.A.5)Sum of interior-angle measures of a triangleArea of a triangle (6.G.A.1)Pythagorean theorem (8.G.B.6, 8.G.B.7)
19Origami-Related Algebra Example: The Fujimoto Iterative Method for Dividing a Rectangular Strip in Thirds (Used in Folding Many Models)
20Origami Box Design as a Mathematical Modeling Activity (HSG-MG.A.3) Relatively simple designs, but with considerable utilitarian/aesthetic qualities, thus increasing synergy of learning to apply geometric concepts and techniques with the general pleasure of folding interesting forms.Models that can be folded in a reasonably short amount of class time.Model crease patterns used in the design process are clearly associated with many CCSS–M.
21“Warm-up” Example not Requiring the Pythagorean Theorem for Analysis: the Magazine Box
23Magazine Boxes Folded from the Same Size Starting Rectangle
24Magazine Box Crease - Pattern Analysis w/2l = length of box facew = width of box faceh = height of boxw hem = width of hemL = length of starting sheet= l + 2h + 2w hemW = width of starting sheet = w + 2hhl/2Lw hemW
25The Design Equation Provides the the Basis for Many Exercises What size paper is required for folding a Magazine Box of length 4", width 3", height 2", and a hem width of 1"?A Magazine Box with:length = width = 2 x height is folded from an8 ½” x 11" sheet of paper. Determine thehem width.Determine the size of paper required for a Magazine Box bottom and fitted lid, with length 5", width 3", height of bottom 3" and lid rim 1 ½ ".
26Basic Reference for the Rest of this Presentation: Arnold Tubis and Crystal E. MillsUnfolding Mathematics with Origami BoxesKey Curriculum Press, 2006
27The Traditional Japanese Open Box (Masu) from a Square
40Flower Box from Waterbomb Base s = l/2 √2 –d √2 -2hCrease PatternUpper left Quadrantw = d/2 √2 = width of band
41Typical Mathematical Exercises Identify the polygons –triangles, squares, rectangles , trapezoids, etc. – in the crease pattern.Identify, and verify by folding, the line and point symmetries of the crease pattern.Identify, and verify by folding, the angles in the crease pattern.Crease PatternUpper Left Quadrant
42Typical Mathematical Exercises Obtain the Box Design Equation for s (edge length of box face) in terms of h (box height), d (sink parameter), and l (edge length of starting square).Determine the width w of the bands on the box in terms of the sink parameter d.If you want to fold a nonstandard height box with d = 1" and s = 4", what is the smallest possible value of l?Crease PatternUpper Left QuadrantBox Design Equation
43Examples of “Challenge” Problems Determine the values of box height h, edge length s of box face, and sink parameter d, that correspond to the maximum possible height of a box folded from a starting square of edge length l.Find the largest possible volume of a nonstandard height box with 0.5" –wide bands folded from a starting square with l = 12". [Use calculator math or calculus.]
44Some Mathematical Concepts and Techniques Involved in Studies of the Generalized Masu Designs Algebraic EquationsAnglesArea and VolumeArithmeticBisection (line, angle)Calculator MathComparison of theoretical and actual measure or box parametersCongruence (verified by folding)Fractions and ratiosGraphical analysisMaxima/minima of box parametersPercent errorPolygons (triangles, rectangles, )Pythagorean theoremRectangular solidSpatial visualizationSymmetry
45Starting Paper Shapes for N-Sided Generalizations of the Four-Sided Masu-type Models
50SummaryOrigami box designs provide a rich framework for the integrated learning and application of a number of CCSS–M.Origami as an art form is being pursued by many students, with vast resources readily available from the internet. Its practice tends to reinforce the working knowledge of these CCSS–M by associating them with the general development of origami skills.The association of CCSS–M with areas of skill development is key to achieving a practical working knowledge.
51A Skills-Accumulation Approach to Knowledge So much of school is knowledge based.No useful skills are connected to the accumulation of that knowledge as part of the learning process.In mathematics instruction, tilting the weight of instruction to a skills–accumulation approach makes more sense than the approach where accumulation of (bits of) knowledge alone is emphasized.Hal Torrance, Connecting Art to Mathematics: Activites for the Right Brain, 2002, 2011, 2012.
52Some Origami-Related Projects Funded by the National Science Foundation in 2012 Motion mechanisms in structures containing foldsSelf-folding polymer sheetsProgrammable origami in self-assembling systemsSynthesis of complex structures via self foldingLight-, heat-, and magnetic-sensitive self-folding materialsSelf-folding at the nanoscale level