Boxin the

Math
Box
in the
134
Mathematics Teaching in the Middle School
●
Vol. 15, No. 3, October 2009
Copyright © 2009 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Find out how
to make an
origami paper
box and explore
the algebra,
geometry,
and other
mathematics
that unfolds.
g
Hope College
Mary J. DeYoung
Vol. 15, No. 3, October 2009
●
Good problem-solving lessons should
be shared.
A simple square piece of paper
can provide an entry point into a
rich problem-solving session. A set
of simple origami steps transforms
the paper into an open box that
can hold mathematical surprises
for both students and teachers. An
origami lesson can engage students
in an open-ended exploration of the
relationship between the size of the
original paper and the volume of the
newly constructed box. The underlying mathematical concepts in this
problem include linear measurement,
area, volume calculations for rectangular solids, and proportional reasoning.
These concepts lead to operations
with fractions.
Mathematics Teaching in the Middle School
135
Fig. 1 The folding instructions prepare a flat piece of paper to become a net that is
then studied for resulting patterns.
Five-Sided Box Folding Instructions
1.Fold the square in half vertically and unfold, and then fold it horizontally
and unfold.
2.Fold each corner of the square in to meet the center.
3.Fold each of the original corners under so that they touch the midpoints of
their respective sides.
4.Turn the paper over to the back side. Fold the left and right edges in so
that they meet in the center of the paper.
5.a. Fold just the left flap over to meet the right edge.
b. Fold the small triangles on the top and bottom of the left side in to the
midline.
c. Fold the flap back over along the midline to meet the left edge.
a.
b.
c.
6.Repeat step 5 using the right flap so that the paper looks like the shape
below.
7.Crease where indicated by the dashed lines and pull the sides out while
pushing the top and bottom in to finish the box.
Adapted from AIMS Education Foundation 2000
136
Mathematics Teaching in the Middle School
●
Vol. 15, No. 3, October 2009
Groups of students who worked
on this problem relied on their prior
knowledge, found a connection to
the Pythagorean theorem, and used
irrational numbers in their solutions.
With guidance, this problem can be
made accessible to first-year algebra
students; it has also been intriguing to
college students and some adult learners. The student work was completed
by small groups of preservice elementary teachers who were enrolled in a
problem-solving course.
When I presented this problem
in my class of future teachers, they
were invited to follow along as we
each folded a small box from a square
of origami paper. (See fig. 1 for the
folding instructions.) All students
had started with origami paper that
was 5.5 inches along each side; the
students observed that the resulting
three-dimensional box had a fixed
volume. Recognizing that the size of
the paper and the volume of the box
were related, students were then presented with the task of investigating
this relationship.
To help focus their work, I asked
them to predict the corresponding
volume if we had started with a square
piece of paper that was 10 inches
along each side. They worked in
groups to find the volume for the larger box and derive a rule for a general
algebraic function that would enable
them to predict the output volume (V)
when the input variable was the side
length of the original square paper
(L). Many of these preservice teachers
were successful in formulating a correct function rule. When the groups presented their
solutions to the class, I was pleased
to see both the diversity of approaches
and the ownership of ideas by individual student groups. They cared about
their answers and were genuinely
committed to understanding how the
different approaches might connect
with each other. After building and unfolding the
box, many students observed that the
entire piece of paper was covered by
smaller squares in a diagonal pattern,
with isosceles right triangles around
the edges (see fig. 2a). The triangles
each had an area that was half of one
of the small squares. The net that
is outlined on figure 2b shows the
bottom and four rectangular sides of
the open box as it is positioned on
the unfolded paper. The square bottom face covers four of these smaller
squares. The rectangular side faces
each cover two of the small squares.
Most students also observed that the
side lengths of each smaller square
are equal to 1/8 of the diagonal of
the paper. Group A observed that the
volume of the resulting box is simply
Fig. 2 Unfolding the origami box elicited many observations and revealed the net’s
length, width, and height information visually.
(a)
(b)
Fig. 3 One group of students used the relationship between the side length l and the
diagonal length to determine the volume formula of the box created by origami paper of
size l × l.
V = l • w • h,
where the dimensions are
1  1  1 
 4 d  •  4 d  •  8 d  ,
and where d is the diagonal length of
the square paper.
2 • leg length.
The students worked to relate the
length of the diagonal back with the
L , the original square
side length 2(L) of
paper. Using their knowledge of the
L1theorem,
Pythagorean
  1 they
  1noted

2L
d  • d  • d  ,

that the hypotenuse
 4   4of any
 8isosceles

right triangle will be
1
 1
 1

V =
2L • 
2L • 
2L
4
 4
 8

1 21 • leg
1 length.
3
= • • • 2 • L3
4 4 8
1
Thus, the diagonal
of the original
= 2 L• 2, 2 • L3
square paper
will be 12L, where L
128
represents the
length of the
1 side
= 2L
LL3paper
2 . The students
square origami
64
substituted 12L for the diagonal
value in their original
1
1
 formula
for 1

V =in terms
L • 
2 L of• the 2diagonal
2L
the volume
4
 4
 8

to find the correct
volume
formula,
1 1 1
3
3
as follows: = • • • 2 • L
4 4 8
1
3
=
•2 2 •L
128
1 3
Vol. 15, No. 3, October 2009
●
Mathematics Teaching in the Middle School
137
Fig. 4 Although using correct mathematics to determine the individual dimensions of the paper box, this group of students made an error when calculating
their values to generate an expression for the volume of the paper box.
the box height to find the box length,
which was, again, a correct step in the
process. However, when they substituted these individual values into their
rule for volume
(V = l • w • h),
1  1  1 
 4 d  •  4 d  •  8 d  ,
2 • leg length.
2L,
2L
L
1
 1
 1

V =
2L • 
2L • 
2L
4
 4
 8

1 1 1
3
= • • • 2 • L3
4 4 8
1
3
=
•2 2 •L
128
1
= L3 2
64
triangle. They used a lowercase l to
represent the paper’s original side
length (see fig. 3).
l 2 + l 2 = 2l 2 = 2 l 2 = l 2
A nearly correct solution came
2x
x isdiscussion.
from group C’s
Their work
is in figure 4. They determined that
the diagonal
(hypotenuse)
of a square
1 23
2
V2 =
L =2 2 l2 =l 2
+
=
2
l
l
l
L denotes the side length of the origiof length
64
nal square piece of paper. 1
= (10 )3 2
Group B used a similar approach
x 64
is 2x
with the diagonal by remembering the
1
(1000 ) the height
2
Pythagorean theorem. The drawand correctly=determined
614 3
ing of the triangle shows students’
of the boxVto=be 1/8
of the diagonal
≈ 22 .L
0972in.3
knowledge of the hypotenuse-tolength, as seen64
by dividing the diagoleg relationship for any 45°-45°-90°
nal in figure=2a.1 They
(10 )3 then
2 doubled
64
1 No. 3, October 2009
138 Mathematics Teaching in the Middle School
1 ● Vol.
115,
= 6 4( 5(1000
.5 in.)) 22
L 2=
8
8
3
they made a common mistake. Instead
of squaring the length of the box (to
multiply length × width), they multiplied the box length by 2 instead. Their
individual pieces were correct, but their
final volume formula was incorrect.
Groups that relied on direct
measurements were limited by their
imprecise rulers but were still able to
make reasonable estimates. One group
measured the height of the final box
and found it to be 1 inch, which then
led them to use 2 inches, or twice that
distance, for the length and width of
the box (see fig. 5). They correctly
reasoned that increasing the paper size
from 5.5 inches to 10 inches would give
proportional values (also doubled) for
the length, width, and height of the new
larger box. Their volume estimate of
1.8 × 3.6 × 3.6 ≈ 23.56 in.3
is reasonably close to the calculated
value that they discovered by evaluating the function found by the other
groups for volume for side length L =
10 inches.
Fig. 5 Students using measurement, rather than algebraic and Pythagorean calculations,
arrived at a reasonably accurate volume for the paper box.
l 2 + l 2 = 2l 2 = 2 l 2 = l 2
x is 2x
1 23 l 2 = 2 l 2 = l 2
l 2V+ l=2 =
L 2
64
1
= x (is10 )23 x 2
64
1
= (11000 ) 2
6
V =4 L3 2
≈ 2264
.097 in.3
1
= (10 )3 2
The initial height
64measurement of 1
inch corresponds 1closely to the value
) 2
1
1= (1000has
of 0.9723
45 in.) 2 been
= ( 56.which
L 2inches,
8 and 8can
rounded,
calculated
from
≈ 22be
.097
in.3
≈ 0 .for
9721/8
3 in.
the earlier rule
of the diagonal
value found by groups A and B.
2
1
1
L 2 = ( 5 .5 in.) 2
8
8
≈ 0 .972 3 in.
As students unwrapped the box,
2
they opened up their mathematical
understanding in several ways. Initially, they relied on their knowledge
of rectangular solids and visualization
skills to find the dimensions of the
box that were hidden in the fold lines
of their unfolded paper. All students
showed their mathematical thinking
in relating linear, area, and volume
measurements. Further, they grappled
with proportional reasoning as they
made sense of how changing the paper size affected other measurements.
Their discussions of possible rules
for the side length-volume relationship allowed some groups to review
their knowledge of right triangles and
the Pythagorean theorem. Another
important aspect of their work was
the manipulation of functional rules
that included the value of 12 and
related irrational values. Note that the
written work of the college students
shows a high degree of comfort in
manipulating irrational numbers;
middle school students who successfully find a function rule will likely use
decimals to express their calculated
values for the volume. This practice
might lead to a helpful discussion of
appropriate rounding techniques.
The importance of proportional
reasoning should also be emphasized. Most students need to observe
many applications to deepen their
Vol. 15, No. 3, October 2009
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Mathematics Teaching in the Middle School
139
Deriving
Another
Method
One rather simple, yet elegant,
method did not occur to the students. The final fold lines divide the
original square piece of paper into
24 small squares and 16 triangles
that are half squares, or 24 + 8
= 32 small squares. Each square
will thus have an area that is 1/32
of the paper, or 1/32 of L2. If each
small square has an area of
L2
2
32
,
L
, side length will be the
then the
32
L2 2 or
L
square root of that value,
.
L, which simplifies to
,
32
4
2
32
L2
L
, which simplifies to
.
32 2
4 2
L L 
L 2
L,
L
2 •  , L2=
, which
simplifies to
.
32
2  is,2the2 height
 4length
This
side
of
the
 L 
32
4
2
L
Folding boxes of different
value,
2• 
, that32
= Twice
box.
 4 2  2 2 L2
sizes allows students to
L
= 2l • wsimplifies
•h
to
. think about proportion.
 ,VLwhich
L
L =
L
232
•
 , simplifies
 , Lwhich
L  4  2toL  .
2
2
4
2


=32
V = l•w•h
4 2
 •
 •
 2 2  2 2  4 2






L
L
 L L L
the
= equal
•  , the length
L•3l • w and
2 •  will
= both
proportional understanding. They

V
=
•2h


=
L
L
2
2
2
2
4





2 2box. Thus, the volume
4 22•  of the
 width
to realize that the dimensions
 3  = 32, 2 L   L   need
L 
2  by:2 2=
L4given
can be
•
•
of
the
=

 
 
 constructed box are all one2
2
2
2
4
2






32
2
dimensional
numerical values (or
V = l•w•h 2
3
linear), just as the original square’s
 VL = l • w
 =•Lh L  L 
=
•
 •2L   L  side length is one dimensional. When
2  2 2  L2 32
2
=  1 3•  4  2• 
 the original size of the square in3 V =2 2 L
 22 2   4 2  creases (or decreases), the lengths of
L
64
3 2
1 =3
all the one-dimensional fold lines will
V=
L 322 =2 L
64
change proportionally.
32 2
Multiplying the numerator
1 3and the
Middle school students would
2 by 12
V =showsL this
2 funcdenominator
benefit from folding boxes of three
64
2
tion to be algebraically
equivalent
different sizes, and allowing them to
to the function
1 3
physically handle several examples.
V=
L 2
1
They will then be more likely to
64V =
L3 2
articulate how increasing or decreas64
that was derived in other ways. 140
ing the size of the original square will
change the length, width, and height
of the resulting box. Middle school
Mathematics Teaching in the Middle School
●
Vol. 15, No. 3, October 2009
students might make two significant
realizations:
1. The base of the box appears to
always be a square.
2. The value for the height of the box
appears to always be half the side
length of the square base.
These relationships are important
to their future thinking about the
problem.
THINKING OUTSIDE THE BOX
Graphing the functional relationships would be a valuable additional
activity. Students whose understanding is at the level of measuring the
various lengths could build boxes from
Table 1 Table values reinforce the one-half relationship discussed, namely, that height
= 1/2 length.
Measured Edge
Length of Paper
Square
Measured
Height of the
Folded Box
Measured
Length of the
Folded Box
Volume Found
by
V=l•w•h
5.5 inches
1 inch
2 inches
4 cubic inches
10 inches
?
?
?
different-sized squares and measure
the dimensions to produce a series of
ordered pairs. The two coordinates
would refer to the edge length of
paper and the height of the folded
box. A linear function will result when
these coordinate pairs are graphed.
Note that the dependent variable
could also be the length or width
of the folded box instead of height.
Students could build a table relating
their measured values to the simple
calculation of volume (see table 1).
Discussing the table values should
reinforce the one-half relationship
discussed, namely, that height is
1/2 of the length. The table leads
naturally into graphing the volume
as a function of the square’s original
edge length, which results in a cubic
graph. Advanced students who can
successfully formulate the volume
rule could use graphing to compare
calculated values to measured values.
This origami problem grabs the
attention of students. Most enjoy the
challenge of both the folding activity
and formulating the algebraic rule.
The integration of problem solving
with visual geometry and algebra
makes this a valuable activity for students at many levels.
REFERENCE
AIMS Education Foundation. “FiveSided Box.” Paper Square Geometry:
The Mathematics of Origami, pp. 3−4.
Fresno, CA: AIMS Education Foundation, 2000.
Mary J. DeYoung, mde
[email protected], began
her mathematics teaching career in the middle
school; she currently
teaches preservice elementary teachers at
Hope College in Holland, Michigan. Her
interests include problem solving and active learning for students of all ages.
2009 NCTM
Regional Conferences
and Expositions
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• Nov. 18–20
Oct. 21–23 •
Nov. 4–6
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professional development you can’t find anywhere else.
Register today! For additional information, visit
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2010 Annual Meeting in San Diego, California
April 21–24, 2010
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