Origami Box Design as a Mathematical Modeling Activity in Grades 7 - 12

1. Origami Box Design as a Mathematical Modeling Activity in Grades 7 - 12 Curtis Center Mathematics and Teaching Conference, UCLA, March 2, 2013 Arnold Tubis Department of Physics (ret.) Purdue University West Lafayette, IN 47907 atu@physics.purdue.edu tubisa@aol.com

2. For further information, collaborative help with lesson plans, etc., please contact : Arnold Tubis tubisa@aol.com

3. Origami in K-12 Mathematics Education: a Brief Historical Survey Friedrich Froebel (1782-1852) 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.

4. 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.)

5. 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.

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

7. 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.

8. 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: 1880-1914. • Its decline was due, in part, to increasing dependence on pre-set, rigid standardized folding patterns.

9. 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.

10. Van 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.

11. 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.

12. 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.

13. 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.

14. 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)

15. Congruence of Vertical Angles

16. Congruence of Alternate Interior Angles

17. Sum of Angle Measures and Area of a Triangle

18. Folding the Pythagorean Theorem

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

20. 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.

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

22. Folding a Simple Version of the Magazine Box

23. Magazine Boxes Folded from the Same Size Starting Rectangle

24. Magazine Box Crease - Pattern Analysis w/2 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 h l/2 L w hem W

25. 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 ".

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

27. The Traditional Japanese Open Box (Masu) from a Square

28. Closed Masu from One Paper Square

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

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

31. Rectangular-faced Masu

32. Decorative Lids

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

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

35. More Decorative Box Lids Twist Base 1 Twist Base 2

36. Box from Preliminary Base Crease Pattern Upper Left Quadrant

37. Box from Waterbomb Base Crease Pattern Upper Left Quadrant

38. Box from Twist Base I Crease Pattern Upper Left Quadrant

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

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

41. 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

42. 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? Crease Pattern Upper Left Quadrant Box Design Equation

43. 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.]

44. 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 Some Mathematical Concepts and Techniques Involved in Studies of the Generalized Masu Designs

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

46. Triangular Masu From Regular Hexagon

47. Design of a Triangular Masu

48. Decorative Pentagonal Lid

49. Tessellation – Inspired Design

50. 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.