Norma Boakes, Ed.D.

Richard Stockton College of New Jersey

Norma.Boakes@stockton.edu

Barbara Pearl, M.Ed.

Atlantic Cape Community College

bpearl@atlantic.edu

Friday October 24, 2014AMTNJ 2 Day Conference

SHOOT FOR THE MOONeven if you miss,

you‘ll still land among the stars..

What are your goals for taking this workshop?1.

2.

3.

For a successful workshop experience:

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!

Notes Notes Notes

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

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”

What is Origami?

“Ori”- to fold

“Gami”- paper

It is literally the “art of paperfolding”.

•ori= fold/ gami=paper

•History of Origami

• Map of Japan

• Famous Paper folders

History

Leonardo da Vinci (1452-1519)

Friedrich Froebel (1782-1852)

Lewis Carroll (1832-1898)

Lillian Oppenheimer (1899-1992)

Beautiful mathematics can be found on the inside

too!

Bird Base

Frog Base

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

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

• Teaches cultural diversity including creating an awareness and appreciation of others.

Spain- “parajarita”France- “cocette”Germany- “Papierdrache”England- “Hobby Horse”

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

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.

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.”

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

A little Origami 101- Types of Origami

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

objects

Unit Origami-made with several pieces of origami

paper then tucked together to make a new form

Action Origami-origami that moves such as a flapping

bird

Unique Material Origami-origami made from paper other than

the typical square cut paper

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

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

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

Voca

bula

ry C

on

cep

ts

Origami isn’t

just for squares.

rectangle 4

width

l

e

n

g

t

h

quadrilateral

1 21

2

1 2 3 4

¼ ¼ ¼ ¼

triangle

octagon

horizontal

V

E

R

T

I

C

A

L

perpendicular lines

parallel lines

lin

e o

f sy

mm

etry

vertex

Number Grid

seven

p c

IX

SIX

even

Xll

PRINCIPLES

PAPER FOLDING

Patience

Precision

Practice

of

Model 1- Origami Box

See packet

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.

Model 2- Leaping Frog Type: unique material & action, Difficulty: easy

This is a favorite because it really jumps. It works great with index card paper or a

business card.

See packet

Gr. 2 or aboveMath concepts:Angle measure & relationships, shapes & spatial relationsCCSS-MStrand: Geometry2.G.1, 3.G.1, 4.G.1-3, 5.G.3-4

• 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?

• 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?

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? [

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?

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.

Model 3- Octagon Star

See packet

Gr. 3 or aboveMath concepts:area, shape, symmetry, spatial relationsCCSS-MStrand: Geometry3.G.1, 4.G.1-3, 5.G.1-3

Type: unit origami, Difficulty: beginner

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?

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?

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?

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.

Rotate the piece 90 degrees and fold in half.

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?

Things you can do with the completed model: • Explore what

polygons are present

• Review concept of interior/exterior angles

• 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

A BC

D

G

ab

c

d

e

Model 4-Origami Booklet

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

More inspirations….

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…

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

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.

Math vocabulary:

-parallel lines

-perpendicular lines

-Angles- acute, obtuse, right

-Right triangle

-Quadrilaterals

-Symmetry

-Area

Taking advantage of the math of Origami… an Origami math lesson in action….

• 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?

• 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.

• 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?

• 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.)