Home Study Companion: Geometry

Purchase HSC Geometry–Immediate Digital Download

Geometry: A Guided Inquiry is a very special textbook!

Of all the geometry texts I have used over the past 35 years, this one stands out as by far the richest, most intuitive, most interesting, and best integrated with the rest of the mathematics curriculum. This text is unique. (See the review on mathmammoth.com.)

• Most geometry textbooks present a long list of facts about geometric figures organized in a rigid logical order, working generally from simple to more complex. Applications of these facts may or may not be made clear to the student. Geometry: A Guided Inquiry is different.  It starts each chapter by posing an interesting geometric problem (puzzle), called the “Central Problem” for the chapter. Clusters of geometric facts are introduced, as needed, in the process of solving these problems. The usefulness and relevance of the new facts are therefore apparent from the moment they are first presented.
• Most geometry textbooks, especially those written under the influence of the “New Math” era of the 1960s, put heavy emphasis on precise use of technical vocabulary and mathematical notation. Geometry: A Guided Inquiry emphasizes the underlying geometric and mathematical ideas and works to help the student understand them intuitively as well as logically. Overemphasis on technical vocabulary and complex notation can actually stand in the way of understanding, so the authors use simplified vocabulary and notation wherever possible.
• Most geometry textbooks start each problem set with lots of routine, repetitive problems, gradually working up to an interesting problem or two at the end of the assignment. Geometry: A Guided Inquiry puts the best problems right up front! From the very beginning the student is given problems worth solving.
• Most geometry textbooks read like they were written by a committee following a prescribed agenda. Most in fact are! The life is squeezed out of the narrative in the process. Geometry: A Guided Inquiry has a distinct sense of authorship. The authors are good mathematicians, good teachers, and good writers. Their joy in the pursuit of mathematics shows through in their writing.

Geometry: A Guided Inquiry makes frequent use of compass, protractor and ruler activities, data tables, guess and check methods, model-building, and other techniques of intuitive exploration in preparation for general solutions.  Each chapter begins with a “Central Problem” that provides the focus and motivates the discussion in that chapter. The Central section presents all the essential new material. Along the way the student is led to a solution of the Central Problem, while exploring its connections with other topics. After the Central section is a Review section, and each of the first seven chapters are followed with a short Algebra Review that stresses algebra topics related to the current work.

Next comes the best part. Each chapter has an open ended Projects section with problems that are extensions to the material in the Central section, sometimes carrying the discussion in new directions. (The Project sections include some of the most interesting material in the text!) In a classroom setting, where students work at their own pace, the quicker students would work on the Project section while the slower students finish the Central and Review sections. In a home study environment the student should read through the whole Project section and work on as many of the project problems as possible within the time frame available. Students who find the work easy, rather than going faster, you should instead take more time and go deeper!

The textbook is available as a digital book (consisting of pdf files, chapter by chapter) that are now distributed together with the course materials. Since the format and style of this text will be new to most of you, here are some sample chapters:

Table of Contents / Chap 1 (Shortest Path Problem) / Chap 5 (Constructions)

Home Study Companion: Geometry

All of our courses are now distributed as digital downloads. The same version works for Windows, Mac, and Linux computers.

The image above illustrates Thales Theorem, possibly the very first theorem ever proved. It says that any angle inscribed in a semi-circle will be a right angle.

The Home Study Companion: Geometry course supplements the textbook in several important ways:

• It has a video introduction to each chapter which gives a chapter overview.
• It provides complete, worked out solutions (not just answers) to all problems in the Central and Project sections of the text.
• The pdf solution manual also contains additional commentary to supplement the presentation of the text. Be sure to read through the solution guide after working through the chapter to get this additional commentary.
• It provides a collection of hundreds of demonstrations using GeoGebra, covering most of the main concepts, and many additional explorations, in the Central and Projects sections of each chapter. Accompanying each demonstration, or cluster of demonstrations, is an activity guide that will help you get the most out of the GeoGebra explorations and teach you how to use GeoGebra as a tool on your own as well.  Download the Classic Edition, Ver 5  from the GeoGebra website.
• Geometry: A Guided Inquiry was written long before the current obsession with standardized testing, and it marches to a different drummer. It covers many fascinating topics you will see in no other high school Geometry textbook. The selection of topics in the text is excellent, but the authors’ choice of topics (in 1970) did not anticipate every choice of the various state standards committees at the end of the century. Therefore the Home Study Companion adds Extensions to the chapters, as needed, to cover these additional topics. (Students not affected by mandatory statewide testing can treat the extensions as optional topics, but they are actually good topics, so it is recommended that they be covered.)

Teaching Tips

[Some of my clients have culture shock moving from Algebra 1 to Geometry. Be prepared for a change of pace.  Don’t reject the course just because the format changes.  The Foerster texts are based on a conventional daily lesson approach. The Geometry text is laid out in chapter-at-a-time workbook style, where a little chunk of material is presented, you work some problems, then go on to more material, etc. Alongside that I have provided lots of demonstrations of all major concepts using GeoGebra. Each chapter also has a large project section. It is intended to be more open ended. I have a video overview for the chapter, and a video solution guide for the Review problems at the end. Along the way I have a pdf solution guide to the Central and Project section problems. Start with a goal to spend X-amount of time on a chapter (the introductory video for the chapter provides suggestions), finish the Central section, then spend the rest of the time on the Projects section. Once you do the first chapter you will get into the swing of it.]

Each chapter of Geometry: A Guided Inquiry is divided into a Central section, a Review section, and a Project section, plus Algebra Review in the first few chapters. For each chapter, the Home Study Companion: Geometry has:

• A video introduction with chapter overview
• Pdf files with complete, worked-out solutions to every problem in the Central and Project sections of the text.
• (The pdf files contain additional commentary besides just the problem solutions.)
• A large collection of demonstrations using GeoGebra.
• A video solution guide covering the Review section.
• Extension sections covering extra topics included in many state standards.

Based on comments from users, we recommend that you:

1. Start by watching the short video introducing the chapter. This video gives an overview of the content and suggestions on how to proceed.
2. Set a goal for how long to spend on each chapter. A standard-length school year is about 185 school days, and there are 12 chapters. That divides out to approximately 15 school days (three weeks) per chapter. However, there are extension sections that have been added to Chapters 3, 4, 6, 8, 10, and 11. Allowing an average of 5 days for each extension section takes up 30 school days, leaving 155 school days to divide by 12. That comes out to about 2.5 weeks per normal chapter and 3.5 weeks for the chapters with extensions. This is just a rough guideline. Some chapters will undoubtedly take longer than others. Homeschooling does not have to conform to the traditional school year, so you may want to take a little more time. Adjust the pace according to your own time constraints and the quantity and perceived difficulty of the material. Work at a comfortable but persistent pace.
3. The GeoGebra demonstrations can be viewed at any time. Some of them can be understood on their own and can help motivate the material in the chapter. Others will make more sense after a certain point in the chapter. Each demonstration, or cluster of demonstrations, is accompanied by a pdf activity sheet. So preview them at the beginning and view them again as you progress through the chapter.
4. Print out the pdf solution guide for the current chapter. This turns out to be an important point. If they are printed out, they will be more immediately accessible and you are more likely to refer to them regularly. (However, some of the pdf files contain Internet hyperlinks that you may want to visit, so you may sometimes want to access them directly on your computer.)
5. Work through the Central section of the text as quickly as you are able, referring to the solution guide as necessary if you get stuck.
6. When you finish the Central section, go back and read through the entire pdf solution guide, both to check your work, and to digest the additional commentary that is included. This will serve as a good review before going further.
7. Do the Review section, then view the video solution guide. Rework any problems that were missed.
8. Do the other review, self test, and algebra review items. (Answers in the text.)
9. Take the remainder of the allotted time to work through selected problems from the Project section. (The Project section contains the most interesting material in the book, so don’t short-change it!) The method here is the same as for the Central section: view the demonstrations at any time, print out the pdf solution guide, work through as many problems as you can, and at the end, read through the entire pdf solution guide. It is best to try each project problem on your own first, but reading through the solutions of all the project problems at the end will still be of some benefit.
10. Regarding testing and grading, here is an essay I published in Homeschool Magazine on that subject: Assessment and Grading for Homeschoolers.  I highly recommend following this approach.

Many of the GeoGebra activity sheets include tutorials showing how the demonstrations were implemented in GeoGebra. These will help you become more familiar with the program and make it a useful tool to use with this and later mathematics courses.