by Robert Duncan, Terry Mulhollan, Baburam Kharel
The goal of this project is to use origami to bring together the artistic, kinesthetic, and mathematic skills of students through a series of hands-on activities. The projects will begin with simple origami designs to build both student skill and confidence in their abilities. Rather than working problems out of a book, students will use these designs as a way to investigate the mathematical concepts inherent in them. This project is intended for use in a Geometry class, as many geometric concepts can be found in, or are directly involved in, the construction of almost all origami designs.
While the main focus of the lessons will be on discovering the math found in various origami designs, it also provides an excellent reference point for incorporating some of the history (and future) of origami and the cultures where it originated. The current and future uses of origami are even more interesting and important to students. Concepts derived from origami are used in many branches of science as well as higher mathematics, which are both areas of study we hope to encourage our students to pursue.
Finally, origami can often be more fun for students than simply working problems out of their books. Students are also encouraged to help their classmates with projects or learning new designs, which creates a positive cooperative learning environment. This project aims to do more than simply use origami as another visual teaching aid. It aims to change the change the entire structure of how material is taught and learned, using origami as the catalyst for doing so.
It can be difficult getting students interesting in mathematics and building their confidence in their own mathematical abilities. Presenting math in a using origami, something which isn't what students are used to, is one way to accomplish both of these goals at once. Not only can origami be an effective teaching tool, but it also has real-world applications in everything from your box of french fries and road maps to satellites orbiting earth and the proteins in our bodies. Through this project, students will work with a variety of origami designs while learning about the underlying math concepts used to create them.
The goal of this project is not only to find a new way to present the same old material, but also to expose students to the history and art of the cultures where origami originated. Origami is more than just the traditional art of folding paper to create any of a number of beautiful designs. Origami projects also act as a gateway to higher level concepts, such as those found in topology, architecture, engineering, and biology.
Asking teachers what sorts of problems they run into in the classroom will no doubt generate a variety of different responses. Apathetic students, finding ways to make math interesting, appealing to diverse learning styles, and dealing with students' lack of confidence in their abilities are some of the more common things that teachers often struggle to find ways to deal with. Our project aims to address (at least to some degree) all of these issues, while at the same time bringing a constructive learning approach to the classroom.
Our project is a unit of Geometry lessons on Area and Volume which is based around looking at, talking about, and doing origami. The actual material covered is not very different than in any Geometry class, but the material is presented in an entirely different way. The very first lesson deals with concrete spatial relationships by asking students how to create a square sheet of origami paper from a regular sheet of notebook paper. Most of the early lessons deal with introductory math and origami concepts to build student skill and confidence in both so that they will be prepared for more difficult designs and concepts later in the unit. In our experience using origami in the classroom, we have found that many students think it is something that they will never be able to do. But by starting with simple designs, the students were not only able to make them, but they also gained a measure of confidence in their own abilities.
The very nature of origami required practitioners to have good spacial reasoning abilities as well as to be coordinated. Using Howard Gardener's theory of multiple intelligences as a base, this means that using origami in math instruction combines elements of spacial, bodily/kinesthetic, and logical-mathematical learning styles. Since students have many diverse learning styles, it is a great benefit to be able to appeal to so many at once.
And not only does origami lend itself to these diverse learning styles, but it is also a skill in and of itself. Many students find that they are interested in it for its own sake and will fold designs on their own for themselves or their friends and family. The direct benefit of this in the classroom is hard to measure, but it is difficult to think that it can be anything but good for students to be so excited about the work they're doing in their math class.
The lessons generally consist of a design for the students to make, along with either one or a series of questions or goals which are associated with the design. Evaluation will be based on how well students are able to apply the things they have learned to solve the questions associated with the design. These questions can range from asking students to make a box and find its volume to having them estimate how large a paper bird they could fold from a 100' by 100' sheet of paper. This is just an example of how the lessons can go from simple application of mathematical knowledge to more abstract questions requiring students to reason about what steps must be taken and how to take them.
A major benefit of this project is the lack of resources necessary. The only real requirements are some paper and some time to learn the designs. There are many designs freely available on the Internet with easy to follow directions that anyone can master without much difficulty. There are also textbooks designed to teach math through origami, which would also be a great benefit.
Basic Folds, Fox Diagram
Area of Circles
Begin 2 day lesson:
Cube and Box diagram
Maximize and Minimize Surface Area
Review for Test
Test: Area of simple shapes
Intro to Volume
Modular Cube Diagram
Prisms and cylinders
Pyramids and cones
Truncated Pyramid Diagram
Find volume of truncated pyramid by solving for unknown lengths.
Volume of Spheres
Water bomb Diagram:
Density and Volume
Quiz: Volume and Density
Final Project: Concept Maps
Day 1: Design Concept Maps individually
Day 2: Work in groups to finalize concept map and prepare presentation
Day 3:Present Concept Maps
Unfolding Mathematics with Unit Origami……………. 40 units….17.95/ea…..$718.00
Unfolding Mathematics with Unit Origami poster……… 1 unit…....9.95/ea……$9.95
Origami Paper, 400 sheets………………………………. 1 unit..…17.95/ea…...$17.95
Miscellaneous supplies …………………………………………………………..$200.00
Copy machine toner
Color printer ink
The success of the project will be measured in terms of students’ achievement. Student’s knowledge and skills will be assessed throughout the project both individually and in groups using various tools such as supervision, home works, quizzes, journals, presentations, formal tests, etc.
At the beginning, students’ math foundation will be assessed with a quiz and a survey will be taken inquiring student’s view about origami and its connection with geometry. As we are teaching high school geometry employing origami, most of the mathematics content will be assessed individual by means of home works, quizzes, weekly reflective journals, and formal tests, considering the TEKS as a reference. Origami creations, presentations, on the other hand, will mostly be assessed in groups and a group progress log will be maintained. Each group will be provided a locker or a showcase to store/display their origami creations and the groups will keep a log on how each object relates to concepts in their geometry or math book. The log book will have to be signed by the teacher every week. They will be encouraged to read the related magazines or websites to come up with new creations and their interconnection with math/geometry. Every week a group with the best creation and/or its interconnection with math will be recognized as the group of the week and will be rewarded depending upon the availability of resources. Also their works will be displayed at the showcase.
At the end of the origami project students will work in teams to present a poster featuring their origami experience. The poster would include but not be limited to: the folded origami objects, blueprints of the folding sequence, a written description of the mathematical concepts revealed by folding or unfolding the object, and a paragraph identifying how origami was surprisingly used to design objects or processes completely unrelated to paper folding as solely an art form. Also an individual final test will be taken to make sure every body meets or exceeds the TEKS geometry standard.