Presentation for AMTNJ 2 Day Annual Math Conference October 2014

1. Norma Boakes, Ed.D.
Richard Stockton
College of New Jersey
Norma.Boakes@stockton
.edu
Friday October 24, 2014
AMTNJ 2 Day Conference
Barbara Pearl, M.Ed.
Atlantic Cape
Community College
bpearl@atlantic.edu

2. Notes  Notes  Notes
For a successful workshop experience:
What are your goals for taking this workshop?
1.
2.
3.
SHOOT FOR THE MOON
even if you miss,
you‘ll still land among the stars..
JUST FOR TODAY…
• Allow on-the-job concerns to be put aside today
and become a learner
• Interact positively with other participants
• Reflect on how to apply the new learning
back in your classroom
• Relax, have fun and enjoy!

4. Norma’s Origami Travels…..
• High school mathematics teacher
– Used Origami to help students “see” and touch
mathematics
• Doctoral student
– I focused my dissertation on learning how Origami
impacted students’ mathematics skills
– I created a set of “Origami-mathematics” lessons to
teach a group of 8th grade students
• College professor
– I created a course called “The Art & Math of Origami”
– I use Origami as a tool to teach about art,
mathematics, culture, and history of Origami
• International trainer
– Train primary and secondary teachers to be resource
teachers for other schools in the use of Origami as a
teaching tool

5. Barbara’s Origami Travels….
• M.A. Mathematics Education
– President of Pi Lambda Theta, Philadelphia Area Chapter
• Taught Pre-School thru High School integrating origami into
math lessons across the curriculum
• College Instructor at Atlantic Cape Community College
• International /National Trainer:
– Invited to present in China and Japan (Teacher/Parent/Student)
Origami Workshops
– Contributing writer and presenter for Japan Society, New York City,
Teacher Inservice
– Origami Exhibits: The Franklin Institute Science Museum and
Philadelphia International Airport.
– Participant in the John F. Kennedy, Artist as Educator – “Origami:
Unfolding the Secret”

6. What is Origami?
“Ori”- to fold
“Gami”- paper
It is literally the “art
of paperfolding”.

7. History
•ori= fold/ gami=paper
•History of Origami
• Map of Japan
• Famous Paper folders
Leonardo da Vinci (1452-1519)
Friedrich Froebel (1782-1852)
Lewis Carroll (1832-1898)
Lillian Oppenheimer (1899-1992)

10. Beautiful
mathematics
can be found
on the inside
too!
Bird Base
Frog Base

11. • Students benefit from the hands-on, student-centered
activity.

12. • Origami allows students to SEE and TOUCH
mathematics so they can understand it better.

13. Spain- “parajarita”
France- “cocette”
Germany- “Papierdrache”
England- “Hobby Horse”
• Teaches cultural diversity including creating an
awareness and appreciation of others.

14. • Origami is engaging and fun. Students and
adults alike enjoy folding. When do you hear
“fun” and math in the same sentence?

15. And yes, it’s in Common Core Math….
• 1.G.2…Compose two-dimensional shapes or three-dimensional
shapes to create a composite shape…
• 2.G.1… Recognize and draw shapes having specified attributes.
Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
• 3.G.1… Understand that shapes in different categories may share
attributes and that shared attributes can define a larger category.
• 4.G Draw and identify line and angles, and classify shapes by
properties of their lines and angles.
• 5.G.4 Classify two-dimensional figures in a hierarchy of properties.
• 6.G Solve real-world and mathematical problems including area,
surface area, and volume.
• 7.G Draw, construct and describe geometric figures and describe
the relationships between them.
• 8. G Understand congruence and similarity using physical models,
transparencies, or geometry software.

16. Consider also the Mathematical
Practices….
• 1 Make sense of problems and perservere in
solving them- “younger students might rely on
using concrete objects…to conceptualize”
• 3 Construct viable arguments and critique the
reasoning of others- “Elementary students can
construct arguments using concrete references
such as objects, drawings, diagrams, and
actions.”

17. From the Mathematical Progressions
document….
Common core standards were built on progressions, narrative statements
describing the flow of a topic across grade levels based on research on
learning math.
• “Students can learn to use their intuitive understandings of measurement,
congruence, and symmetry to guide their work on tasks such as solving
puzzles and making simple origami constructions by folding paper to make
a given two or three-dimensional shape.” referring to Grade 1, p.9
• “More advanced paper-folding (origami) tasks afford the same
mathematical practices of seeing and using structure, conjecturing, and
justifying conjectures. Paper folding can also illustrate many geometric
concepts.” referring to Grade 3, p.13
• “Students also analyze and compose and decompose polyhedral solids.
They describe the shapes of the faces, as well as the number of faces,
edges, and vertices. They make and use drawings of solid shapes and learn
that solid shapes have an outer surface as well as an interior. They develop
visualization skills connected to their mathematical concepts as they
recognize the existence of, and visualize, components of three-dimensional
shapes that are not visible from a given viewpoint” referring
to Grade 6, p.18

18. A little Origami 101- Types of Origami

19. Traditional Origami-a
single sheet made to look like animals &
objects

20. Unit Origami-made
with several pieces of origami
paper then tucked together to make a
new form

21. Action Origami-origami
that moves such as a flapping
bird

22. Unique Material Origami-origami
made from paper other than
the typical square cut paper

23. Our Workshop Focus
• Learning how to teach mathematics through
Origami, what I call “Origami-Mathematics”
lesson
• Models we will do together (time pending)
– Box
– Leaping Frog
– Octagon Star
– Origami Booklet

24. When folding a model it helps when you
know the terminology and the visual cues.
It’s just like math. You learn symbols and pay
attention to what you see to help do
problems.
Valley Fold
Mountain Fold
Find your one page reference in your packet

25. Visuals are so powerful that eventually you can
even follow this!

26. Origami isn’t
just for squares.
Vocabulary Concepts
rectangle 4
width
l
e
n
g
t
h
quadrilateral
parallel lines
1 2
1
2
1 2 3 4
¼ ¼ ¼ ¼
triangle
octagon
horizontal
V
E
R
T
I
C
A
L
line of symmetry
perpendicular lines
vertex 

27. Number Grid
p c
seven
SIX
IX
even
Xll

28. PRINCIPLES
of
PAPER FOLDING
Patience
Precision
Practice

29. Model 1- Origami Box
See
packet

30. Guidelines to Brainstorming
1. Say everything that comes to mind
2. Repetition is OK
3. No judging (positive or negative)
4. Expand on others’ ideas
Minds are like parachutes
they function best when open.
If you always do what you have always done,
then you’ll always get what you’ve always got.
If your heart  is in it,
the sky’s the limit.

31. Model 2- Leaping Frog Type: unique material & action,
Difficulty: easy
Gr. 2 or above
Math concepts:
Angle measure & relationships, shapes &
spatial relations
CCSS-M
Strand: Geometry
2.G.1, 3.G.1, 4.G.1-3, 5.G.3-4
This is a favorite
because it really
jumps. It works
great with index
card paper or a
business card.
See
packet

32. • Before you fold your card, what mathematical terms could you
use to describe it?
• Once you make the creases using adjacent corners of the card,
what kind of line segments were formed?
• What kind of angles are formed then?
• What could you say about the measure of two adjacent right
angles?

33. • Once you mountain fold you form a third line segment (Step 3). Do
you recognize any of the angles formed here?
• Can you find a set that are supplementary?
Could you find the exact measures of the angles without a protractor?

34. Once you do the squash-fold (Step 6),
what kind of shapes are formed?
Can you identify each of the angle
measures of each of them?
Is there a more specific name you can
give to the triangle?
What special terms are associated with
an isosceles triangle
• When you fold the base angles of
the isosceles right triangle up, what
have you formed (Step 7)?
• What can you say about them?
• How does the area of the small
triangles compare to the one from
the previous step? [

35. When you fold the sides into the middle
(shown at Step 8), what new shapes do you
have?
How do they compare in size?
If you ignore all the folds and look at the figure
as a whole, what is it?

36. What to do with the completed
model:
• Explore the polygons visible in
the finished figure
• Have a hopping content.
Measure the heights of the
hops. Try experimenting with
different kinds of paper.
• Unfold the model and explore
the math visible in the folding
lines by darkening them w/a
pen.
• Research unique facts about
the frog like the largest (size
of football), smallest (eraser
on a pencil), jumping
strength, etc.

37. Model 3- Octagon Star
Gr. 3 or above
Math concepts:
area, shape,
symmetry, spatial
relations
CCSS-M
Strand: Geometry
3.G.1, 4.G.1-3, 5.G.1-
3
See
packet
Type: unit origami, Difficulty: beginner

38. Fold the paper in half and unfold.
When you fold the sheet in half, what shape do you make? How
do you know? How does the area of the rectangle compare to
the area of the square?

39. Rotate the paper 90 degrees and fold in half again and unfold.
What shapes are formed after this step? How does their area
compare to the original? What can you call the fold lines in the
square? Does the figure have rotational symmetry?

40. Fold the top two corners down.
*When you fold the corners down, does the resulting shape
remind you of a real object?
*What shapes make it up?
* How does the area of the triangles compare to the rectangle?
*If you ignore inner shapes what polygon is this outer figure?

41. Fold the white sides together to form the diagonals shown to
the right.
*Tip- Fold point A to point B. It’s easier to see that way and do
one side at a time.
A
B
Stop fold here.

42. Rotate the piece 90 degrees
and fold in half.

43. Push the fold to the
inside so that a
parallelogram is
formed.
*What kind of shape do you
have now? How do you know?
*What kind of symmetry does
the figure have?
*Can you tell how the area of
the parallelogram compares
to the original square?

45. Things you can do
with the completed
model:
• Explore what
polygons are
present
• Review concept
of
interior/exterior
angles

46. • Discuss regular
vs. irregular
polygons
• Find side length,
perimeter, area,…
• Explore angle
measures in the
parallelograms
formed and the
central angles
visible in the final
shape shown
here

47. A B
C
D
G

48. a
b
c
d
e

49. Model 4-
Origami Booklet

50. What questions could you ask?
Use the template in your packet to write prompt questions for each step.

54. More inspirations….

55. Reflection
 Did you achieve your
goals and objectives?
 If not, is there anything else you could
have done differently?
 What steps will you take to implement
some of the strategies you learned
today? 1. I will be able to…
2. I will…
3. I will…

56. Visit Math in Motion:
www.mathinmotion.com
Books available by Barbara Pearl
Unfolding the
Common Core State
Standards for
Mathematics
thru Origami
Norma Boakes and
Barbara Pearl
Pending Spring 2015

58. References
• Common Core Standards Writing Team. (2013,
March 1). Progressions for the Common Core
Math Standards in Mathematics (draft). Tucson,
AZ: Institute for Mathematics and Education,
University of Arizona.
• National Governors Association Center for Best
Practices & Council of Chief State School Officers.
(2010). Common Core State Standards for
Mathematics.Washington, DC: Authors.

59. Taking advantage of the math of Origami… an Origami math lesson in action….
Math vocabulary:
-parallel lines
-perpendicular lines
-Angles- acute, obtuse, right
-Right triangle
-Quadrilaterals
-Symmetry
-Area

60. • When you make the
valley fold in Step 1,
what can you say
about the fold line
formed?
• What kind of triangle is
formed and how do
you know for sure.
How does the area of
each triangle compare
to the original square?

61. • When you do the two additional
folds in Step 2, what kind of
shapes do you have now?
• Where do all the fold lines meet?
• What kind of angles can you find if
you darken in the line segments?
• Describe any special relationships
with the line segments formed.
See the copy of
my Origami
lesson for these
along with the
answers.

62. • What kind of shapes do
you see once you fold
the corners in (shown
at Step 4)?
• Can you find parallel or
perpendicular lines
anywhere?
• Once you squash-fold your model,
what shapes do you find?
• How does the area of the colored
triangle compare to the two smaller
white ones?
• Can you still find parallel or
perpendicular line segments?

63. • With the last fold done, what shape
is the colored base of the boat?
• Does it have a special name?
• If students are ready they can discuss the difference
between congruent and similar triangles using the
sails formed in the final step.
• Have students open up the sailboat completely and
look at the fold lines formed. Darken them and see
what they observe about the lines and/or angles.
• Look for more kinds of polygons in the folding steps
(ex. Step 4 is a pentagon.)