This sixteenth unit in the SMSG secondary school mathematics series is the teacher's commentary for Unit 14. For each of the chapters in Unit 14, a guide to the selection of problems is provided, the goals for that chapter are discussed, the mathematics is explained, some teaching suggestions are given, the answers to exercises are listed, and…

Results of operations on the Pythagorean formula are interpreted pictorially to yield interesting art forms. (MP)

A nine-point plane coordinated by the field of integers modulo three is examined using motion geometry. The conceptual advantages of teaching this geometry are emphasized. (MP)

Discusses some theorems and properties of figures produced when circles are tangent to one another. (GA)

The development of square units for area and cube units for volume is discussed with emphasis that switching focus from one physical property to another is involved as opposed to simply multiplying units. (MN)

A classroom activity is described where students divided circular regions into equal sections and reassembled them to determine the formula for the area of a circle. Using an odd number of sections (Instead of an even number) changes the shape of the reassembled pieces but still gives rise to the same formula. (MN)

A combination of trigonometry and algebra is used to focus on the concept of congruent triangles. (JT)

Three different geometric constructions of the parabola are described.

A general historical background for the development of some common formulas concerning length, area, and volume is given through a discussion of various written records. (MN)

Designs and arrangements of pentominoes are examined. (SD)

The activities of a research working group are described. The investigators from several institutions coordinate their work on space and geometry. Underlying assumptions of the group and issues faced by it are described. (SD)

Three degrees of cognitive processing were tapped by four problem types when 60 children between four and eight years were administered a set of geometry tasks differing in complexity. Analysis revealed that the tasks differed in difficulty, task success was related to age, and a hierarchical sequence existed among the skills. (SKC)

Discusses: (1) how complex roots can be made visible; (2) a proof which supplies a fresh example of mathematical induction; (3) proving Heron's formula tangentially; and (4) income tax averaging and convexity. (JN)

How to determine when there is a unique solution when two sides and an angle of a triangle are known, using simple algebra and the law of cosines, is described. (MNS)

A twenty-week course was designed around investigating many properties of the torus. Topology, differential geometry, and topics from both real and complex analysis were included. (SD)