by Robert Duncan, Terry Mulhollan, Baburam Kharel
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. (a) Basic understandings. (1) Foundation concepts for high school mathematics. As presented in Grades K8, 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 realworld 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 handheld 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 problemsolving, computation in problemsolving 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 threedimensional 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 threedimensional 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 (454590 and 306090) 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 threedimensional objects and related twodimensional representations and uses these representations to solve problems. Following are performance descriptions. (A) The student describes, and draws cross sections and other slices of threedimensional objects. (C) The student uses top, front, side, and corner views of threedimensional 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 twodimensional 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 threedimensional 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 threedimensional solid is changed and applies this idea in solving problems.




