# Math in Origami

by Robert Duncan, Terry Mulhollan, Baburam Kharel

 Introduction DescriptionThis project is designed to present math to students in a way that is both interesting and hands-on. The nature of origami makes it both a very visual and very tactile discipline, which helps appeal to diverse learning styles. But more than just a different way to present math, this is a way to broaden students' minds on exactly what math is, as well as to expose them to an ancient and beautiful art they might otherwise have not known about. Another key to this project is that origami isn't nearly as difficult to do as some people think. So when students realize they are able to create something, that sense of accomplishment will often carry over to the more math intensive areas of the class. Some origami diagrams also require several people to fold or put together, which helps students learn to work together to accomplish a goal, as well as to help eachother learn. This project unit has been designed to cover the topics of area and volume as presented in a high school geometry class. The entire unit is designed to last 4 weeks. Driving QuestionWhat does origami have to do with math? Goals and ObjectivesIntroduce students to Origami-Teach simple paper folding techniquese and terminology-Have students fold their own origami creations-Increase student awareness of cultures where origami originiatedHave students discover the math hidden in origami-Find lengths, angles, shapes, and solids found during or after folding various origami diagrams-Learn the properties of the areas and volumes of these shapes-Apply previously learned theories to origami designed.Increase student interest and confidence in mathematical-Use origami as a way to get students excited to learn-Increase student confidence by showing them that they can fold designs on theire own RationaleIt 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 roadmaps 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. BackgroundThe basic origami folds and diagrams are very simple, but it is important that the teacher be very familiar with them in order to be able to teach them to the students. There are a variety of websites designed to instruct people at all levels of origami skill. These are useful for the teacher themselves as well as a source of ideas for helping teach students the basics of origami. Origami itself has a rich history in cultures all over the world. The word'origami' is Japanese, and although it is true that much of the paper folding that people are familiar with has its origins in asian cultures, it has existed in various forms in many different cultures all over the world. Even today, things like pop-up books and paper airplanes involve many of the same ideas and methods as traditional origami.Origmi isn't limited to art either. In recent years it has found a number of practical applications in fields such as topology, architecture, and aerospace engineering. Paperfolding principles have even been used to create an axiomatic plane geometry system.Much more information on both the history of and instruction techniques for origami can be found at the following sites:http://www.paperfolding.com/ - Possibly the best all around origami resource on the internet. Contains links to MANY more origami sites.http://www.origami.as/ - Homepage of Joseph Wu, as well know and very skilled origami artist. Contains instructions and articles on current happenings in the world of origamihttp://www.origami.com/model.html – Extensive, searchable database of origami diagrams in PDF format. Standards AddressedTEKS:34. Geometry (One Credit). ```(a) Basic understandings. (1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences. (2) Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; shapes and figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them. (3) Geometric figures and their properties. Geometry consists of the study of geometric figures of zero, one, two, and three dimensions and the relationships among them. Students study properties and relationships having to do with size, shape, location, direction, and orientation of these figures. (4) The relationship between geometry, other mathematics, and other disciplines. Geometry can be used to model and represent many mathematical and real-world situations. Students perceive the connection between geometry and the real and mathematical worlds and use geometric ideas, relationships, and properties to solve problems. (5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, algebraic, and coordinate), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities to solve meaningful problems by representing figures, transforming figures, analyzing relationships, and proving things about them. (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problem-solving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof. (b) Geometric structure: knowledge and skills and performance descriptions. (1) The student understands the structure of, and relationships within, an axiomatic system. Following are performance descriptions. (A) The student develops an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. (B) Through the historical development of geometric systems, the student recognizes that mathematics is developed for a variety of purposes. (C) The student compares and contrasts the structures and implications of Euclidean geometries. (2) The student analyzes geometric relationships in order to make and verify conjectures. Following are performance descriptions. (A) The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. (B) The student makes and verifies conjectures about angles, lines, polygons, circles, and three-dimensional figures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. (3) The student understands the importance of logical reasoning, justification, and proof in mathematics. Following are performance descriptions. (B) The student constructs and justifies statements about geometric figures and their properties. (C) The student demonstrates what it means to prove mathematically that statements are true. (D) The student uses inductive reasoning to formulate a conjecture. (E) The student uses deductive reasoning to prove a statement. (4) The student uses a variety of representations to describe geometric relationships and solve problems. Following is a performance description. The student selects an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (c) Geometric patterns: knowledge and skills and performance descriptions. The student identifies, analyzes, and describes patterns that emerge from two- and three-dimensional geometric figures. Following are performance descriptions. (1) The student uses numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles. (3) The student identifies and applies patterns from right triangles to solve problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples. (d) Dimensionality and the geometry of location: knowledge and skills and performance descriptions. (1) The student analyzes the relationship between three-dimensional objects and related two-dimensional representations and uses these representations to solve problems. Following are performance descriptions. (A) The student describes, and draws cross sections and other slices of three-dimensional objects. (C) The student uses top, front, side, and corner views of three-dimensional objects to create accurate and complete representations and solve problems. (2) The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. Following are performance descriptions. (A) The student uses one- and two-dimensional coordinate systems to represent points, lines, line segments, and figures. (C) The student develops and uses formulas including distance and midpoint. (e) Congruence and the geometry of size: knowledge and skills and performance descriptions. (1) The student extends measurement concepts to find area, perimeter, and volume in problem situations. Following are performance descriptions. (A) The student finds areas of regular polygons and composite figures. (B) The student finds areas of sectors and arc lengths of circles using proportional reasoning. (C) The student develops, extends, and uses the Pythagorean Theorem. (D) The student finds surface areas and volumes of prisms, pyramids, spheres, cones, and cylinders in problem situations. (2) The student analyzes properties and describes relationships in geometric figures. Following are performance descriptions. (A) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties of parallel and perpendicular lines. (B) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of polygons and their component parts. (C) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of circles and the lines that intersect them. (D) The student analyzes the characteristics of three-dimensional figures and their component parts. (3) The student applies the concept of congruence to justify properties of figures and solve problems. Following are performance descriptions. (A) The student uses congruence transformations to make conjectures and justify properties of geometric figures. (B) The student justifies and applies triangle congruence relationships. (f) Similarity and the geometry of shape: knowledge and skills and performance descriptions. The student applies the concepts of similarity to justify properties of figures and solve problems. Following are performance descriptions. (1) The student uses similarity properties and transformations to explore and justify conjectures about geometric figures. (2) The student uses ratios to solve problems involving similar figures. (3) In a variety of ways, the student develops, applies, and justifies triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples. (4) The student describes the effect on perimeter, area, and volume when length, width, or height of a three-dimensional solid is changed and applies this idea in solving problems.``` Assessments and Final Project Anchor Video Concept Map Project Calendar Lesson Plans Letter to Parents Assessments Resources Modifications Grant