Geometry theorem proving involves skills that are difficult to learn. Instead of working with abstract and complicated representations, students might start with concrete, graphical representations. A proof tree is a graphical representation of a formal proof, with each node representing a proposition or given conditions. A computer-assisted…

This is the teacher's edition of a text for the first year of a two-year high school geometry course. The course bases plane and solid geometry and trigonometry on the fact that the translations of a Euclidean space constitute a vector space which has an inner product. Volume 1 deals largely with affine geometry, and the notion of dimension is…

The author re-enacted the "Meno's" discussion between Plato and the slave about doubling the size of the square, and found that a six-year-old's answers paralleled those of the slave; neither had formal knowledge of geometry. Plato stated that learning is recollection of knowledge held before birth. (CHK)

This booklet, one of a series, has been developed for the project, A Program for Mathematically Underdeveloped Pupils. A project team, including inservice teachers, is being used to write and develop the materials for this program. The materials developed in this booklet include (1) elementary properties of the exponent, (2) properties of the…

High-ability Jamaican students in grades one, three, five, seven, and nine were asked to draw five three-dimensional objects (cuboid, pyramid, cylinder, cone, cube) from memory, and with the object visible; later they were asked to select the best sketch of each solid from among several presented. Drawings and selections were scored for…

The first chapter, Perpendiculars and Parallels (II), of the ninth unit in this SMSG series includes a discussion of the properties of triangles, circles and perpendiculars, parallels in space, perpendicular lines and planes, and parallel planes. The next chapter, on coordinate geometry, covers distance; midpoints; algebraic descriptions of…

A mathematical derivation is given, developing the cardioid as an epicycloid locus. Curve-stitched designs are given for a family of epicycloids. (MP)

The free-energy change, or binding energy, of an idealized bubble cluster is calculated on the basis of one mole of gas, and on the basis of a single bubble going from sphere to polyhedron. Some new relations of bubble geometry are developed in the course of the calculation. (BB)

A method for converting a three-dimensional coordinate system to a two-dimensional coordinate system is explained. (MN)

A class of triangles related to the Pythagorean triangles is described; theorems concerning this class are proved. (SD)

All lines which bisect the area of a triangle envelope a three-cusped curve made up of three hyperbolic arcs. (MM)