About GEOMETRY by Harold R. Jacobs

Notes by Peter Renz

Geometry is intuitive and appealing with useful results but it can be demanding, and it has tricky twists and turns. The challenge is to build intuition and develop. the curiosity and persistence needed to push through tough problems and avoid pitfalls. Working with Jacobs on this book was a chance to fit the pieces of geometry into a coherent whole.

Particular examples are the core of this book, especially those in its exercises, as Don Albers notes in his Foreword. Every chapter offers a new crop of puzzles and applications. By following the steps of the exercises, you can always see what is happening, but in seeing why may be a challenge.

As I struggled, I remembered the NCTM Standards recommendations about flexibility. Use the tools best suited to a particular problem; choose your approach — direct geometric arguments, transformations, matrices, graphs, technology, whatever. This is fine advice for me, but the students are just starting out. Many times I found an easy approach but sometimes there were difficulties. Perhaps you will find the Royal Road I missed.

Euclidean geometry can be regarded as a system developed from postulates or as the geometry of the coordinate plane with the usual distance formula. Either way, the same properties hold. These properties are built into geometric software; they underlie computer representations, three-dimensional graphics, etc. If students understand that things must fit together, they will be able to check ideas for correctness using whatever tools are at hand — starting with paper, pencil, and brain power.

My experience is that looking things up and asking others for ideas are excellent problem-solving techniques. But nothing beats knowing the answer from school. A reasonable repertoire of standard results and formulas is useful, which is one reason a Formulary was added in this edition.

These notes outline key and motivating ideas for students and provide background for teachers, as well as outlining how to crack some tough nuts for those who want to prove some of the conjectures arising from the exercises.

Reading Geometry or teaching from it, you are bound to find questions and answers not touched on in these notes or in Jacobs’s Teacher’s Guide. Harold Jacobs and I would be happy to read your comments and suggestions and to try and answer any questions you may have. We can be reached care of W. H. Freeman and Company by mail or electronically.

Peter Renz