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Harold Jacobs' Geometry

I've been considering a new text for a course in Euclidean Geometry that I teach for middle school teachers. I've been using Essentials of Geometry for College Students by Lial et al. The students seem OK with it, but I find it very boring. I supplement it with lots of my own exercises using Geometer's Sketchpad, paper folding, MIRA(tm), etc. to keep things interesting.

In looking for a replacement, the best book I have found so far is Geometry: Seeing, Doing, Understanding by Harold R. Jacobs. The latest (3rd) edition was published in 2003. Although I will probably use this book, I will transform many of the problems I assign from pencil, paper, ruler, and protractor to Geometer's Sketchpad. I would love it if the publisher W. H. Freeman would commission an update.

This is a high school text, but it is more challenging than Lial. The applications to "real life" are the most realistic and compelling that I have seen anywhere. I keep finding things that I didn't know, and ways of looking at geometry problems that I hadn't considered.

In one example on page 503 Jacobs shows a closed smooth curve bounding a convex region and consisting of circular arcs. One student said that the sum of the arc measurements must be 360 degrees, and the other doubts it because the curve is not a circle. From the nature of Jacobs' construction, it is easy to show that the sum of the arc measures is indeed 360 degrees. A good teacher could connect this with the fact that the sum of the exterior angles of a convex polygon is 360 degrees.

In another example, Jacobs gives an "Area Puzzle" where he guides students to prove a curious fact about triangle areas. If each vertex of a triangle (ABC in the figure below) is connected to a point 1/3 of the way from the next vertex (in CCW order, say) to the following vertex, and the intersections of these 3 segments (Cevians) are connected, an inner triangle (DEF) is formed. The area of DEF turns out to be 1/7 of the area of ABC. I have known this for some years, and even published a paper (with my brother Marshall and my nephew Michael) generalizing it to quadrilaterals and to ratios other than 1/3. The proof I used involved using analytic geometry to establish the result for a right triangle with vertices (0, 0) (1, 0), and (0, 1) and then arguing that the area ratio is preserved by affine transformations, so the result holds for all triangles.

Jacobs presents a neat synthetic proof that clearly shows where the strange ratio 1:7 comes from. He constructs 6 more triangles, each a translate of the central triangle, and then guides the student to show that the triangles can be dissected and reassembled to fill the original triangle. See the diagram below.


4 comments:

Live n Learn Teacher said...

How many lessons do you cover a week? And, were there any chapters you felt like you could skip so that you could make it to the end of the book?

Peter Ash said...

Remember, my course was for middle school teachers in a graduate course. We met 7 times, once every two weeks, for about 5 hours at a time. We covered Chapters 1 - 14 (two chapters per class meeting), and I gave some presentations for Chapter 15, Geometric Solids, but no exercises. I feel the last chapter, on non-Euclidean Geometry may be skipped. IMHO, it is better to cover as much as you can, rather than to make it to the end of the book. If you send me an email, I will send you the list of exercises I assigned. (peter.ash@mathfortherestofus.com) I will be happy to send you a copy of the problems I assigned.

Sarah Cates said...

Curious, since you are teaching this to adults, do you feel it still adequate for a high-schooler's geometry credit?

Peter Ash said...

Sorry for the long delay, Sarah. I don't check comments very often, I'm afraid. Yes, Jacobs' book is actually designed (very well) for high school students. The adults I teach have typically had very little geometry, so they are really starting out at the same place as high-schoolers. Also, since they will be teaching middle-schoolers, its nice that they get a chance to appreciate the attention to pedagogy in Jacobs' book.