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Review of a high school geometry course with
Geometry: A Guided Inquiry book;
its Home Study Companion; and
Geometer's Sketchpad software
I am reviewing here a "combo" package for high school geometry course, consisting of three different products: 1. Geometry: A Guided Inquiry textbook; 2. Home Study Companion to it, and 3. Geometer's Sketchpad software.
While these three will work together, I will describe each "part" separately in my review. Please use the quick links below to jump directly to the various sections, if you'd like. Naturally, the review about the book is the longest.
The book: Geometry: A Guided Inquiry
How Geometry: A Guided Inquiry handles proof
An axiomatic vs. discovery based geometry text
The Home Study Companion
The software: Geometer's Sketchpad
Geometry: A Guided Inquiry
by Chakerian, Crabill, and Stein
Geometry: A Guided Inquiry is a problem-centered textbook. Each chapter starts with a central problem, which acts as a starting point for developing important concepts and theorems.
The text reads like a "worktext": many of the problems worked on are an inseparable part of the overall instruction, because they will lead the student in a step-by-step manner to important theorems and results. This lets students become familiar with the concepts little by little, and constantly keep building upon previous knowledge, just as is normally done in mathematics. (Naturally, the book also contains plenty of "exercises" that simply practice a concept just presented.)
In essence, the student is guided in his geometrical inquiries by the questions presented, as if a teacher was guiding him by hand. For this reason, I feel this text is an excellent choice for anyone who is studying geometry by themselves, or without a teacher.
Let's look at an example.
An example of guided exercises leading to a theorem
On page 197, the student first learns and proves that if a point P is equidistant from the endpoints of a line segment, then P is on the perpendicular bisector of the segment.
The very next exercise has a chord AB of a circle with center P, and asks why the perpendicular bisector of AB passes through P. The next part has a picture showing two chords of a circle, and asks where the perpendicular bisectors of the two cords intersect. Another picture with two chords, and a question, "Why must the perpendicular bisectors of the two chords intesect? The student fills in a sentence: The perpendicular bisector of a chord passes through the __________ of the circle.
Soon follows a problem with a triangle and the perpendicular bisectors of its two sides. Questions ask, "Why is PA = PC? Why is PB = PC? Deduce that PA = PB."
In such a manner, the student is gently led to deduce a statement:
The perpendicular bisectors of the sides of any triangle all meet in a single point.
You can see all this for yourself in these two sample pages:
The chapters in the book
The Shortest Path. The "central" or opening problem for this chapter presents a camper in forest who needs to return to his campsite but first fetch a bucket of water from the river. The problem is, "At what point P on the river should he fill his pail in order to make the shortest possible path to put out the fire?"
The opening problem in chapter 1: The Shortest Path
This single problem leads to the development of several important concepts: distance from a point to a line, perpendicular lines, and reflection of a point across a line. Finally the exact solution AND its proof are studied, along with applications.
Below you can preview the pages 6-9 of the first chapter, which show the development of concepts (such as reflection and triangle inequality) towards solving the "shortest path" problem.
Tiling the Plane deals with polygons and angles, and of course solve the question as to which regular polygons tile the plane.
Triangles deals specifically with the SSS, SAS, and ASA congruence properties of triangles. This chapter also has some practices in proving theorems.
What is a Proof? is about proofs and some common pitfalls in devising proofs. Along the way, the chapter also develops properties of quadrilaterals, especially of parallelograms.
Constructions with Straightedge and Compass covers basic constructions, the special points of a triangle, and also some impossible constructions.
Area and Volume is like the title suggests: areas and volume of basic figures.
The Pythagorean Theorem
Perimeter, Area, and Volume of Similar Figures
Circles - this chapter has exceptionally many interesting projects.
Conic Sections - most of this chapter is devoted to the ellipse.
After each chapter's main worktext part (called Central), we find a Central Self Quiz, a review, a review self test, and projects, where the most interesting problems are found! There is also an algebra review at the end of chapters 1-7, and cumulative reviews at the end of chapters 3, 7, 9, and 12.
In the very end of each chapter you will find answers to the chapter's central self quiz, the review self test, algebra review, and most answers to the review.
The "projects" in the end of each chapter definitely have lots of interesting material. While they often deal with extensions of the main ideas of the chapter, you will also find explanations and explorations concerning mathematically important results, for example golden section, geometric mean, additional proofs of the Pythagorean theorem, the number of diagonals in a convex polygon, regular polyhedra, Gauss' theorem concerning which regular polygons can be constructed, and the problem about squaring the circle.
In a regular classroom instruction, these projects may get neglected or assigned only to the fastest students, but I would encourage anyone to really DELVE into them, as they often present the best and most fascinating parts of geometry -- all the while developing the student's reasoning skills.
How Geometry: A Guided Inquiry handles proof
Different geometry books handle the topic of proof in different manners, and teachers might prefer one approach over another, so I will now briefly describe how mathematical proof is approached in Geometry: A Guided Inquiry.
As explained above, this book often has a set of carefully crafted exercises leading to an important theorem, and these exercises build one upon another in logical succession, either preparing the student to understand the proof of the particular theorem or to build one for it.
Starting in chapter 2 (Tiling the Plane), students are often asked to complete parts of an argument. In some exercises students are also asked to give an argument or explain his answers using a certain fact. I find these bird-size steps towards more formal proof to be very well designed.
In chapter 3, after studying the SSS, SAS, and ASA properties of triangles, we encounter a section "Introduction to Proofs". In it, the student is again asked to fill in missing words or parts of sentences in paragraph-form proofs or supply reasons for a few two-column proofs. In othe problems the student is asked to write the proof with a hint, and finally without one.
The two pages below show an example of this.
Chapter 4 is titled, "What is a Proof?" It explains such basics as "If...then" statements, converses, and axioms, and lets students practice filling in proofs or writing their own in the the context of properties of parallelograms. We also meet Alex and Danielle, who seem to manage to write flawed proofs all the time - so the reader can learn from their mistakes.
In the rest of the chapters, proof is not forgotten; quite the contrary: the students are often asked to write proofs, many times with a hint. On page 346 we find an example of circular reasoning. I also especially liked problems where you have to find what is wrong with each picture, with the pictures purposefully drawn distorted. This practices simple logical reasoning not based on the appearance of the figure.
An axiomatic vs. discovery based geometry text
Geometry: A Guided Inquiry is NOT written in a strict formal axiomatic style à la Euclid. It does not start out by presenting a set of axioms in the very beginning and then develop all theorems from those.
However, from chapter 2 on, it IS written as a system of geometry that is developed from axioms. This is just somewhat "hidden" from sight by the explorative exercises that precede the axioms and the theorems alike. Also, the few axioms are only introduced as they are needed. The emphasis is more on "local" proofs than on a "global" axiomatic system. Since the only difference is that the axioms are not proved, from a student's point of view it may all just seem to be material or knowledge to be learned.
Personally I feel that the problem-solving approach of Geometry: A Guided Inquiry is first of all more student-friendly and "gentle" than a strict or "pure" axiomatic text. Even more importantly, it truly focuses on the content of geometry rather than being merely a logic course with a geometric backdrop. It DOES contains plenty of rigorous arguments and proofs and practice in writing them, but it is all in a context of solving problems, blended in with the explorative and other kind of exercises.
This is likely a more natural way of learning stuff the first time around. Reading and following an axiomatically built mathematical work is easier once you have a good idea of the material it covers. Personally I would additionally try to introduce students to Euclid's work and to the strictly axiomatic system just a little -- not only for the sake of logic but also for its historical significance. Chandler's Home Study Companion (see below) contains some material to familiarize students with Euclid's work.
Research shows that students' geometrical understanding proceeds from reasoning about shapes based on their appearance gradually into logical and abstract reasoning. Studying high school geometry strictly as an axiomatic system is probably not the most productive way for the vast majority of students. (Please see the article
Geometry and Proof by Michael T. Battista and Douglas H. Clements for more information.)
Home Study Companion
Home Study Companion - Geometry (HSC) by David Chandler really makes this book a home run for homeschoolers, because it provides complete, worked out solutions (not just answers) to all problems in the Central (the main worktext) and Project sections of the Geometry: A Guided Inquiry textbook. The book itself contains answers to its various review and self-test sections. With the Home Study Companion, you will always have help available should you get stuck while doing the problems from the worktext or the "projects".
Not only that, but the Home Study Companion includes a collection of nearly 300 interactive demonstrations using The Geometer's Sketchpad. These demonstrations cover most of the main concepts and many additional explorations of the Central and Projects sections of each chapter.
How do these demonstrations work? For example, let's say you're asked to prove that the diagonals of a parallelogram bisect each other. A demonstration of that would have a parallelogram and its diagonals, and then measurements of the two parts of each diagonal (in centimeters or inches). You would then change the shape of the parallelogram in Sketchpad and see those measurements stay equal to each other. That is not a proof, but it is an interactive demo that helps you understand the matter.
Here are some example screenshots. Click to enlarge. Remember the screenshots are static; in reality the demonstrations are dynamic.
Attempting to speficy a triangle by SSA
An interactive pantograph
A theorem about the parts of intersecting chords in a circle
Chandler has used lots of color and animations in these demonstrations to show that certain angles add up to 180° or that certain areas are equal, etc. Many of these demonstrations actually go through a whole proof step-by-step, with explanations on the side, and you need to click on buttons to see the various steps. And each time, you can stretch the figure or change the lengths or whatever by dragging some points in the figure, and it still works. I was quite impressed!
For example, included are finding the value of Pi through inner/outer polygon approximation, several animated proofs of Pythagorean theorem. Or, some projects in the book in chapter 2 ask a student to draw and find out if certain figures tile the plane. The Sketchpad demonstrations provided in the HSC facilitate this task tremendously because you can just copy and paste the figure and move it on the computer workspace, instead of drawing copies of it on paper.
The interactive demonstrations add another dimension to the study of geometry by themselves, but your student can go even further: simply encourage him to learn Geometer's Sketchpad well enough to duplicate the demonstrations, including with color and animations. Computer-oriented kids will surely love such challenge.
In addition to the GSP demonstrations, the Home Study Companion also includes some additional lessons to supplement the main textbook. These extensions bring the content of
the text into alignment with the California State Standards. The extensions included are: Isometries and Symmetry, More on Proof, Surface Areas of Prisms and Pyramids; Introduction to Trigonometry;
Surface Areas of Cylinders, Cones, and Spheres; and More on Coordinate Geometry.
Geometer's Sketchpad (GSP) is a dynamic geometry software. This means that once you construct a drawing, you can dynamically move its parts and see what changes, and what does not. It also has the capabilities for animations, coordinate systems, fractals, graphing functions, and much more - it's a very versatile program.
Really the best way to understand how it works is to experience it. If you don't yet know what dynamic geometry software is all about, check some online dynamic geometry constructions real quick, and then come back here.
Prior to Sketchpad, I had used another dynamic geometry software while in university (about 12 years ago). So I had some knowledge of how things would work.
However, I decided to go through the GSP Learning Guide provided on the CD. It contains "tours" where you will construct certain things step-by-step. Doing these "tours" will teach you the basics of using Geometer's Sketchpad.
I found the guide vere well planned and enjoyable. It had me construct a square, learn a theorem about quadrilaterals, and even construct an animated kaleidoscope! Personally I would perhaps add an exercise where the student needs to construct two circles with identical radius, since those are so commonplace in many basic constructions that are studied in high school.
By the way, you shouldn't restrict the usage of Sketchpad for only viewing the demos in the Home Study Companion. Your student or child should definitely learn the simple basics of this software, and do SUITABLE construction/drawing exercises from the book with it.
To put myself to the test, I also tried doing a few of the exercises from the Geometry: A Guided Inquiry using Geometer's Sketchpad:
- Review, p. 205. Problem 3: Construct a 6-leaved pattern. I did this first on paper, so I knew how to do it before I started Sketchpad. My paper construction wasn't very accurate; the lines didn't pass through the various points exactly, plus I had all kinds of auxiliary lines all over the place, some of which I tried erasing, but it sure looked messy. It hought maybe it'd look better with Sketchpad.
6-leaved pattern done in SketchPad. I could drag the points and see the pattern enlarge.
So here's a screenshot of my construction in Sketchpad. I had to do it twice; my original didn't stand the "drag test" because I had used a "copy and paste" of a line segment instead of making a longer line segment and finding its midpoint. But it was fun! And the result looks way neater than the pencil and paper version. Here's the GSP file if you'd like to download it.
In fact, I decided to try do it using a regular image editing program as well. These always have a circle drawing function. I drew the figure in a vector image program (Corel):
6-leaved pattern done in CorelDraw (click to enlarge)
I do such work all the time, but the fun part in working with Sketchpad is that your circles have a center point, you can "snap" objects to other objects, you can construct things by their relation to others, such as being perpendicular, and then "drag" the points and change it dynamically. In regular image editing programs you pretty much just "eyeball" many of the things or use a coordinate system and measurements - which of course works when you are not planning to dynamically change the figure in order to observe what changes and what doesn't. The two are for different purposes. (However, I'd say that learning the basic geometric constructions is a great help for anyone who plans to do extensive work with computer images or graphic design.)
Another one I tried was problem 1 on page 212. This one is about an alternative method for bisecting a line segment. At first I tried copying and pasting a certain circle in my construction, which worked for THAT particular situation, but then when I dragged my original point that defined the original line, the construction fell apart, because I needed the two circles to maintain the same radius (or as the problem states, "Using the same compass setting..."). I was able to do this right (so it stands the drag test) only after finding and using the construction "circle by center+radius". A lesson learned!
Your work with Sketchpad doesn't have to be limited to your geometry textbook, either. At the Geometer's Sketchpad resource website you can find tons of projects to do and sketch galleries to explore.
Overall, I feel this combo is very comprehensive, yet very suitable for a self-learning environment. You get a very logical and enjoyable textbook where exercises lead step by step to theorems or to proofs, solutions, interactive demonstrations, and a full-fledged geometry software to "play" with.
It is thus an excellent choice for doing high school geometry with a "guided inquiry" method. I especially applaud the problems of the book, which are very varied and make geometry enjoyable, and the quality of the demonstrations in the Home Study Companion.
One thing some people might find lacking in the text is that it won't work well as a reference book. You will not find all the main theorems on topic X clearly laid out in boxes with example pictures, followed by proofs. Instead, this is a "learn-by-doing" book. The important theorems and results are studied AND proven in a very natural surrounding: while doing problems.
However, I would venture to say that this type of learning is longer-lasting than the traditional "theorem-proof-exercises" model.
Review by Maria Miller, MSc, author of HomeschoolMath.net
The sample pages are images scanned from the book, reproduced here by permission from the publisher.
G.D. Chakerian, Crabill, Stein,
1998, Denver, CO: Morton Publishing Company. www.morton-pub.com.
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