High School Mathematics has often been referred to as abstract.  The higher level the mathematics class, the less concrete, and the more abstract it is.  Typically, algebra introduces abstract reasoning, followed by geometry, which also expects students to be able to reason.  Not all students are ready to make the transition to generalize, model and analyze mathematical situations, as NCTM describes, as one of the purposes of algebra.

Too often a Geometry classroom is filled with postulates, theorems and proofs; in that order.  Who owns this plethora of If-Then statements?  Euclid?  The textbook author or publisher?  The Common Core State Standards Initiative? The state’s department of education who restates the CCSS?  The school district who creates the planning and scheduling timeline?  The teacher?  Wouldn’t it be amazing if the ownership was put on the students?

In the Common Core State Standards for Mathematics, the point is made that understanding mathematics is equally as important as being able to perform the procedures.  Where or when or how exactly is this understanding going to be developed?  The Standards of Mathematical Practice encourage mathematical strategies that students can use to gain understanding of mathematical ideas.  A teacher’s job is to teach the students the strategies, so he/she can create meaning in each topic covered.  One way a teacher can do this is to provide opportunities for students to observe, analyze and discover!

I am a believer in the pedagogical strategy of constructivism, an educational theory influenced by Jean Piaget, Maria Montessori, Lev Vgotsky, Jerome Bruner and other educational philosophers. Constructivism, specifically in mathematics, means students develop mathematical ideas and principles themselves, rather than being taught a number of formulas and rules.  I believe student ownership in mathematics encourages reasoning and creates life-long learners and mathematicians.  Using concrete models is one way for students to construct mathematical ideas.  I propose using origami to do so.  Because so many lessons for origami in mathematics already exist, I propose a Philadelphia theme to reach across the curricula to local sites, culture and history.

Philadelphia is an old city, among the first colonized by William Penn and the early founding fathers.  We have a rich history and we remain a vibrant center for culture, art and academia.  My hope is to use origami lessons to connect hands on learning to Philadelphia themes.

This unit is intended for high school students.  The lessons can be used separately and taught throughout the school year.  Some appeal to direct Geometry or Algebra topics; others are geared to integrate Philadelphia history with origami folding while utilizing and enriching mathematical ideas.

The Objectives of the Lessons will include:

• Students will answer questions before, during and after watching the documentary Between the Folds by Vanessa Gould.
• Students will fold and with one cut, create a 5-pointed star to simulate its historical significance and examine the mathematics involved.
• Students will investigate the properties and characteristics of a pentagon and pentagram.
• Students will create an Alexander Calder themed origami mobile, write about the mathematics and the Calder inspired characters.
• Students will self-reflect on their Calder mobile project by answering questions.
• Students will create a pop-up greeting card to celebrate Pi Day.

• Students will use hands-on paper folding activities to develop/increase patience and perseverance and attention to detail.
• Students will work cooperatively to create origami objects.
• Students will work cooperatively to discover geometric properties of polyhedral
• Students will use a KWL chart to assess their knowledge of Betsy Ross and the first U.S. flag.
• Students will research Alexander Calder Philadelphia themes in order to create a mobile.

Lesson 1: Introduction to Origami: Documentary Between the Folds

Learning Objective: At the end of this lesson, students will be able to explain how origami connects with many fields of study, including mathematics.  They will be able to list the roles and or purposes origami serves.

Materials:

• DVD or Stream Between the Folds (54 minutes)
• Discussion Questions (appendix 1)
• Origami Paper (I use deli Patty Paper which is readily available at our school)
• Directions for an easily folded origami object.

Procedures:

1. Before beginning the documentary, make sure students are familiar with what origami is. This could be a pre-viewing homework assignment or a Do Now.
2. Watch the documentary Between the Folds and answer/discuss questions. The questions could be answered by students as they watch the movie, or discussed as they watch, stopping at various points, or discussed after watching.  Because the video is 54 minutes long, this may have to be done over two class periods or skip some of the movie.
3. Fold something for goodness sakes! Start with something easy a crane or a frog or a butterfly.  This could be an exit ticket, a do now, or a homework assignment.  Chances are there is at least one student who would be willing to demonstrate to the class or work with the student teams.

Notes:

• If a teacher does not have a DVD or is unable to stream the documentary, there is a trailer: https://www.youtube.com/watch?v=tGxsvmJg18c (5:34) and highlights (from website) https://www.vanessagould.com/between-the-folds (2:00) that provide some of the origami story and origami creations.  At a minimum, either or both of these should be used to introduce the ever-growing world of origami.
• There is a more recent origami documentary released by NOVA, The Origami Revolution: http://www.pbs.org/wgbh/nova/physics/origami-revolution.html
• I use deli patty paper for most of the origami folding we do. It creases well and doesn’t tear easily.  It measures 5.5” by 5.5” and comes in boxes of 1000.  It is available from many math supply companies and some restaurant supply stores

Lesson 2: The Nation’s First Flag

Learning Objective:  At the end of the lesson, students will be able to fold and with one cut, create a five-pointed star.  Students will be able to give the historical significance of Betsy Ross’ storied contribution to our country’s first flag.  Students will be able to identify the properties and characteristics of a pentagon and pentagram.

Materials:

• The story of George Washington and the secret committee of the Continental Congress who ordered the first flag from Betsy Ross. http://www.ushistory.org/betsy/index.html
• Erik Demaine’s Folding and Unfolding: The Fold-and-Cut Problem http://erikdemaine.org/foldcut/ (go to Try it out yourself by printing one of our several examples and print out the Fancy Star.) Appendix 2
• Pentagram investigation questions. Appendix 3
• Origami Paper (I use Patty Paper which is readily available in our school).

Procedures:

1. Begin the lesson by finding out what students know about the U.S.’s first flag. I suggest using a KWL.  (I give each student a post it and have them write a K, W, and/or L and then post it on the KWL chart at the front of the room.)
2. Impart the story of Betsy Ross and the first flag. The teacher can give a detailed or peripheral account or can assign students via jigsaw to read the separate pages from ushitory.org and report back to their teams. A small writing assignment can be given to ensure students have learned the story.
3. Copy and distribute Erik Demaine’s cut-and-fold Fancy Star. The reading Folding and Unfolding: The Fold-and-Cut Problem, can also be assigned.  Students should fold and cut the star just as Betsy Ross did for George Washington.
4. Students will fold a pentagon http://www.origami-instructions.com/origami-pentagon-base.html and complete the investigation questions. This may be done on a second day. The internet investigations (#7 & #8) may be assigned for homework.

Notes:

• There are other methods for folding and cutting a 5-pointed star from a rectangular sheet and a square sheet. Search online or use those given at ushistory.org.
• The idea of history vs. story may be a new one to your students. Pursue this if
• There is a link to Erik Demaine’s MIT lecture on The Fold-and-Cut Problem if you’d like to show that to the class or assign to individuals seeking enrichment.
• I use deli patty paper for most of the origami folding we do. It creases well and doesn’t tear easily.  It measures 5.5” by 5.5” and comes in boxes of 1000.  It is available from many math supply companies and some restaurant supply stores.

Lesson 3: Three Generations of Calders in Philadelphia Art

Lesson Objectives: At the end of the lesson students will be able to describe the artistic contributions of Alexander Milne Calder, his son Alexander Stirling Calder, and his son Alexander “Sandy” Calder.  Students will have made a mobile with origami pieces inspired by the work of any or all of the Calders.

Materials:

• Article: Generations of Alexander Calder Art a Trinity of Blessings for Philadelphia by Susie Perloff May 13, 2014 https://whyy.org/articles/generations-of-alexander-calder-art-a-trinity-of-blessings-for-philadelphia/
• Web article: Three Generations of Calders in Philadelphia (Adapted from Public Art in Philadelphia by Penny Balkin Bach (Temple University Press, Philadelphia, 1992).http://www.associationforpublicart.org/apa-now/story/three-generations-of-calders-in-philadelphia/
• Calder Mobil Project Guidelines (appendix 4)
• Calder Mobil Project Grading Rubric (appendix 5)
• Calder Mobil Project Self-Reflection Questions (appendix 6)
• Barbara Pearl’s Instructions on how to make a mobile (appendix 7)
• Erik Demaine’s Fold-and-Cut Swan (appendix 8)
• A completed mobile for display
• Materials for the mobile:
  • paper for the origami models (patty paper?)
  • needle
  • invisible thread or fishing line
  • glue
  • thin soft wire
  • wire cutters

Procedures:

1. Students will be introduced to the Alexander Calder Philadelphia legacy by reading the Susie Perloff Article: Generations of Alexander Calder Art; A Trinity of Blessings for Philadelphia and the Web Article: Three Generations of Calders in Philadelphia (Adapted from Public Art in Philadelphia by Penny Balkin Bach). You can use your choice of reading strategies or assign it for homework before you introduce the project.
2. Introduce the project to the class with the project guidelines. After going over the goals, expectations and requirements, answer any questions the students may have.  Return to these guidelines throughout the project time period to ensure students are staying on task and understand the outcome.  Distribute the project rubric and self-reflection questions and answer any questions students may have.  Decide how long you will allow for the project and how many class periods students will have to work with you and/or their peers on the project.  Also clarify which materials you will supply for the class and which materials the students will be responsible for themselves.  It is advisable to have a finished mobile to show the class exactly what end result you expect.
3. Create check-in times for students to show you their progress and ask questions that may arise. Give advice to those students who are doing too much or too little.  Allow students opportunity to discuss their projects among themselves as well.
4. On the due date, allow the students to present their mobiles to the class and share the mathematics they learned and their Calder connections.
5. Allow one or two days between the due date of the Calder mobile and the due date of the self-reflection questions. This will give students opportunity to compare their creative process with a classmate’s.
6. Do your own reflection on the project. Will you do it again.  What changes will you make?  Record these now so you will remember for next time.

Notes:

• Some students/classes will need more guidance than others. Be prepared to make suggestions for Calder characters.  Students can use swans, frogs or turtles from the Swann Memorial Fountain, they can use anything Shakespearean from the Shakespeare Memorial, they can use any of the zodiac symbols from the Zodiac Sundial, they can use any of the circus animals or performers from Sandy Calder’s circus.
• Students may also need guidance when considering the mathematics they used for their Calder mobile. Did they have to consider weight and/or surface area when creating balance in their mobile?  Were they able to scale their origami models by using different size paper?  Were they interested in preserving orientation or symmetry when creating their mobile?
• I use deli patty paper for most of the origami folding we do. It creases well and doesn’t tear easily.  It measures 5.5” by 5.5” and comes in boxes of 1000.  It is available from many math supply companies and some restaurant supply stores.  It has no color and is semi-transparent, but students can decorate as they wish.

This project can be done almost exclusively at home or in class or a mix of both.  Perhaps use the shorter periods during standardized or before or after a break.  It should be FUN!

[Please see PDF attached above for additional lesson plans & appendices]

Brooks, Martin G, & Brooks, Jacqueline Brennon (1999) The Courage to Be Constructivist, ASCD Journal Educational Leadership November 1999 | Volume 57 | Number 3 The Constructivist Classroom Pages 18-24

Gould, Vanessa (Director & Producer) (2009) Independent Lens: Between the Folds, available to watch on Amazon Prime, available to buy on shoppbs.org

trailer: https://www.youtube.com/watch?v=tGxsvmJg18c  (5:34)

highlights (from website) https://www.vanessagould.com/between-the-folds (2:00)

Pearl, Barbara. (1994).  Math in Motion Origami in the Classroom. www.mathinmotion.com.

Demaine, Erik, Demaine, Martin, & Lubiw, Anna (2016) Erik Demaine’s Folding and Unfolding, The Fold-and-Cut Problem. Retrieved from http://erikdemaine.org/foldcut/

Balkin Bach, Penny (1992) Three Generations of Calders in Philadelphia (Adapted from Public Art in Philadelphia by Penny Balkin Bach (Temple University Press, Philadelphia, 1992). Retrieved from http://www.associationforpublicart.org/apa-now/story/three-generations-of-calders-in-philadelphia/

Perloff, Susie (2014) Generations of Alexander Calder art a trinity of blessings for Philadelphia by Susie Perloff May 13, 2014 Retrieved from https://whyy.org/articles/generations-of-alexander-calder-art-a-trinity-of-blessings-for-philadelphia/

Independence Hall Association (2018) Betsy Ross and the American Flag.  Retrieved from http://www.ushistory.org/betsy/index.html

Origami Pentagon Base (2014) Retrieved from http://www.origami-instructions.com/origami-pentagon-base.html

Sabuda, Robert (2018) Pop-Up Basics Retrieved from http://wp.robertsabuda.com/make-your-own-pop-ups/

Annotated Bibliography Resources for Teachers

Origami Resources

https://origamiusa.org/

https://www.origami-resource-center.com/money-origami-things.html

Origami at the Philadelphia Airport based on work by Barbara Pearl, author of Math in Motion. http://www.phl.org/Arts/Pages/Archivedexhibitions/origami.aspx

Franco, Besty & Varner, Diane (1999) Unfolding Mathematics with Unit Origami. Emeryville, CA. Key Curriculum Press.

Hull, Thomas (2013) Project Origami: Activities for Exploring Mathematics. Boca Raton, FL. CRC Press.

The Effect of Origami-Based Instruction on Spatial Visualization, Geometry Achievement and Geometric Reasoning. International Journal of Science and Mathematics Education, Authors: Arici, S; Aslan-Tutak, F.

Montrell, John (2012) Origami and Math, Simple to Complex. New York, NY. Dover Publications, Inc.

Mitchell, David (2015) Mathematical Origami: Geometrical Shapes by Paper Folding. St Albans, Herts, UK. Tarquin Publications.

O’Rourke, Joseph (2011) How to Fold It, The Mathematics of Linkges, Origami, and Polyhedra. New York, NY. Cambridge University Press.

Betsy Ross Resources

dollar bill 5-pointed stars

http://make-origami.com/money-5-pointed-star/

Alexander Calder Resources

city hall virtual tour – sculpture by Alexander Milne Calder  http://www.phila.gov/virtualch/virtual_tour.html

Calder Foundation http://www.calder.org/

http://www.theartstory.org/artist-calder-alexander.htm

comedy & tragedy book fold!  costs $3.41

https://www.etsy.com/listing/205058533/folded-book-art-tragedy-comedy-drama?ref=shop_home_active_4

comedy & tragedy dollar bill fold

https://www.pinterest.com/pin/484559241128280238/?lp=true

CCSSI Mathematics High School Geometry

Understand congruence in terms of rigid motions

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Make geometric constructions

12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices,
paper folding, dynamic geometric software, etc.).

Visualize relationships between two-dimensional and three- dimensional objects

4. Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Apply geometric concepts in modeling situations G-MG

6. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★
7. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★
8. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

Mathematical Practices

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.