Curtis Center Mathematics and Teaching
Conference, UCLA, March 2, 2013
Origami Box Design as a Mathematical
Modeling Activity in Grades 7 - 12
Arnold Tubis
Department of Physics (ret.)
Purdue University
West Lafayette, IN 47907
atu@physics.purdue.edu
tubisa@aol.com
For further information, collaborative help
with lesson plans, etc., please contact :
Arnold Tubis
tubisa@aol.com
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.
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.)
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.
Forms of Beauty
Simple folded patterns, ornaments, boxes,
polygons, and polyhedra that demonstrate
various aspects of symmetry.
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.
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.
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.
Van Hiele Levels (Continued)
Deduction: systematic deduction of theorems
from undefined terms, definitions, and axioms
in the context of Euclidean geometry. Grades 912.
Rigor: systematic deduction of theorems from
undefined terms, definitions and and axioms in
generalized (non-Euclidean) geometries.
University level.
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.
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.
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.
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)
Congruence of Vertical Angles
Congruence of Alternate Interior
Angles
Sum of Angle Measures and Area
of a Triangle
Folding the Pythagorean Theorem
Origami-Related Algebra Example:
The Fujimoto Iterative Method for Dividing a
Rectangular Strip in Thirds (Used in Folding
Many
Models)
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.
“Warm-up” Example not Requiring the
Pythagorean Theorem for Analysis: the
Magazine Box
Folding a Simple Version of the
Magazine Box
Magazine Boxes Folded from the
Same Size Starting Rectangle
Magazine Box Crease Pattern Analysis
w/2
l = length of box face
w = width of box face
h
h = height of box
w hem = width of hem
l/2
L
L = length of starting
sheet
= l + 2h + 2w hem
W = width of starting
sheet = w + 2h
w hem
W
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 ".
Basic Reference for the Rest of this
Presentation:
Arnold Tubis and Crystal E. Mills
Unfolding Mathematics with Origami Boxes
Key Curriculum Press, 2006
The Traditional Japanese Open Box
(Masu) from a Square
l
2
s=
4
Closed Masu from One Paper Square
Nonstandard Height Open Masu
Shallow Masu
Tall Masu
h < 1/2 s
h > 1/2 s
s = l 2 - 2h
2
Variation of Masu Dimensions
Folded from squares of same size.
The larger the height, the smaller the face of
the box.
Rectangular-faced Masu
(
l = 2 sw + sl + 4h
2
)
Decorative Lids
Two-Color Design (From the Folding Paper Exhibit at
the Japanese American National Museum, LA)
Decorative Masu Lids
Preliminary Base with Sink Fold
Waterbomb Base with Sink Fold
More Decorative Box Lids
Twist Base 1
Twist Base 2
Box from Preliminary Base
s=
l 2
- 2d - 2h
2
Crease Pattern
Upper Left Quadrant
Box from Waterbomb Base
l 2
s=
- d 2 - 2h
2
Crease Pattern
Upper Left Quadrant
Box from Twist Base I
l 2
s=
- d 2 - 2h
2
Crease Pattern
Upper Left Quadrant
Box From Twist Base II
s= l/2√2 – 2d – 2h
Flower Box from Waterbomb Base
2
s = l/2 √2 –d √2 -2h
w = d/2 √2 = width of band
Crease Pattern
Upper left Quadrant
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
s = l 2 - d 2 - 2h
2
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
s = l 2 - d 2 - 2h
2
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.]
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
Starting Paper Shapes for N-Sided
Generalizations of the Four-Sided Masu-type
Models
Triangular Masu From Regular
Hexagon
Design of a Triangular Masu
Decorative Pentagonal Lid
Tessellation – Inspired Design
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.
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.
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