Fractions, Tilings, and Geometry (IAS/PCMI Teacher Program)

by Bowen Kerins, Darryl Yong, Al Cuoco

Designed for precollege teachers by a collaborative of teachers, educators, and mathematicians, Fractions, Tilings, and Geometry is based on a course offered in the Summer School Teacher Program at the Park City Mathematics Institute.The overall goal of the course is an introduction to non-periodic tilings in two dimensions and space-filling polyhedra. While the course does not address quasicrystals, it provides the underlying mathematics that is used in their study. Because of this goal, the course explores Penrose tilings, the irrationality of the golden ratio, the connections between tessellations and packing problems, and Voronoi diagrams in 2 and 3 dimensions. These topics all connect to precollege mathematics, either as core ideas (irrational numbers) or enrichment for standard topics in geometry (polygons, angles, and constructions).But this book isn't a ``course'' in the traditional sense. It consists of a carefully sequenced collection of problem sets designed to develop several interconnected mathematical themes. These materials provide participants with the opportunity for authentic mathematical discovery--participants build mathematical structures by investigating patterns, use reasoning to test and formalize their ideas, offer and negotiate mathematical definitions, and apply their theories and mathematical machinery to solve problems.Fractions, Tilings, and Geometry is a volume of the book series IAS/PCMI--The Teacher Program Series published by the American Mathematical Society. Each volume in this series covers the content of one Summer School Teacher Program year and is independent of the rest.

157 pages, Paperback

Published January 25, 2018

Reviews

[Tom Schulte · Rating 5 out of 5]

"Ostensibly designed for precollege teachers, this teach-the-teacher course based on one offered in the Summer School Teacher Program at the Park City Mathematics Institute has applicability to teaching undergraduates as well. This introduction to non-periodic tilings in two dimensions and space-filling polyhedra is architected around engaging and enlightening group activities to benefit of the secondary education instructor. This content is ready classroom capsules to enliven first-year college courses. Topics covered range from tessellations including Penrose dart and kite tilings and Voronoi diagrams to the golden ratio, packing problems, and more. Very well designed to offer participants authentic self-discovery through an orderly succession of motivation and revelation through guided, hands-on investigation of patterns, the finished material can be directly applied to aid students in honing reasoning, learning to formalize ideas, and grow in mathematical sophistication while having fun solving problems." ...

[Look for my entire review at MAA Reviews]