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Origami Box Design as a Mathematical Modeling Activity in Grades Arnold Tubis Department of Physics (ret.) Purdue University West Lafayette, IN Curtis Center Mathematics and Teaching Conference, UCLA, March 2, 2013

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For further information, collaborative help with lesson plans, etc., please contact : Arnold Tubis

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Origami 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.

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Mental 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.)

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Forms of Life Simple 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.

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Forms of Beauty Simple folded patterns, ornaments, boxes, polygons, and polyhedra that demonstrate various aspects of symmetry.

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Forms of Knowledge Folding of simple shapes, geometric analysis of crease patterns, and folding sequences associated with basic concepts, constructions, and theorems of plane geometry.

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The 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.

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Contemporary 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.

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Van Hiele Levels (Continued) Deduction: systematic deduction of theorems from undefined terms, definitions, and axioms in the context of Euclidean geometry. Grades Rigor: systematic deduction of theorems from undefined terms, definitions and and axioms in generalized (non-Euclidean) geometries. University level.

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Basic Point : Students can successfully achieve geometric understanding at a given Van Hiele level only if he/she has achieved understanding at the lower levels.

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Origami Useful 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.

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Monographs 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.

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Origami 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 triangle (8.G.A.5) Area of a triangle (6.G.A.1) Pythagorean theorem (8.G.B.6, 8.G.B.7)

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Congruence of Vertical Angles

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Congruence of Alternate Interior Angles

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Sum of Angle Measures and Area of a Triangle

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Folding the Pythagorean Theorem

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Origami-Related Algebra Example: The Fujimoto Iterative Method for Dividing a Rectangular Strip in Thirds (Used in Folding Many Models)

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Origami 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.

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“Warm-up” Example not Requiring the Pythagorean Theorem for Analysis: the Magazine Box

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Folding a Simple Version of the Magazine Box

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Magazine Boxes Folded from the Same Size Starting Rectangle

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Magazine Box Crease - Pattern Analysis h l/2 w/2 W L w hem l = length of box face w = width of box face h = height of box w hem = width of hem L = length of starting sheet = l + 2h + 2w hem W = width of starting sheet = w + 2h

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The 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 an 8 ½” x 11" sheet of paper. Determine the hem 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 ".

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Basic Reference for the Rest of this Presentation: Arnold Tubis and Crystal E. Mills Unfolding Mathematics with Origami Boxes Key Curriculum Press, 2006

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The Traditional Japanese Open Box (Masu) from a Square

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Closed Masu from One Paper Square

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Nonstandard Height Open Masu Shallow Masu h < 1/2 s Tall Masu h > 1/2 s

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Variation of Masu Dimensions Folded from squares of same size. The larger the height, the smaller the face of the box.

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Rectangular-faced Masu

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Decorative Lids

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Two-Color Design (From the Folding Paper Exhibit at the Japanese American National Museum, LA)

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Decorative Masu Lids Preliminary Base with Sink Fold Waterbomb Base with Sink Fold

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More Decorative Box Lids Twist Base 1Twist Base 2

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Box from Preliminary Base Crease Pattern Upper Left Quadrant

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Box from Waterbomb Base Crease Pattern Upper Left Quadrant

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Box from Twist Base I Crease Pattern Upper Left Quadrant

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Box From Twist Base II s= l/2√2 – 2d – 2h

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Flower Box from Waterbomb Base Crease Pattern Upper left Quadrant s = l/2 √2 –d √2 -2h w = d/2 √2 = width of band

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Typical 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 Pattern Upper Left Quadrant

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Typical 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? Box Design Equation Crease Pattern Upper Left Quadrant

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Examples 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.]

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Some Mathematical Concepts and Techniques Involved in Studies of the Generalized Masu Designs Algebraic Equations Angles Area and Volume Arithmetic Bisection (line, angle) Calculator Math Comparison of theoretical and actual measure or box parameters Congruence (verified by folding) Fractions and ratios Graphical analysis Maxima/minima of box parameters Percent error Polygons (triangles, rectangles,... ) Pythagorean theorem Rectangular solid Spatial visualization Symmetry

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Starting Paper Shapes for N-Sided Generalizations of the Four-Sided Masu-type Models

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Triangular Masu From Regular Hexagon

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Design of a Triangular Masu

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Decorative Pentagonal Lid

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Tessellation – Inspired Design

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Summary Origami 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.

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A 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.

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Some Origami-Related Projects Funded by the National Science Foundation in 2012 Motion mechanisms in structures containing folds Self-folding polymer sheets Programmable origami in self-assembling systems Synthesis of complex structures via self folding Light-, heat-, and magnetic-sensitive self-folding materials Self-folding at the nanoscale level

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