I taught geometry for decades, starting in the 1980’s, and loved it. I’m reasonably good at manipulating algebraic symbols, but I don’t especially enjoy it. In contrast, I am happy to spend plenty of time on visual puzzles, and I am enthusiastic about sharing that passion with colleagues and with students.
Early in my high school career I happened to come across an outstanding geometry textbook, which had a powerful impact on my teaching: Geometry: A Guided Inquiry. One thing I learned from it was that to lay a foundation for introducing proof you need to start by involving students in interesting activities and problems. Over the years, I attended several talks by G.D. Chakerian, one of the authors of the book. Among other things, he argued for the importance of geometry: it is intrinsically worthwhile, and not merely a vehicle to teach logic. As he put it, geometry is a part of math where we look at and think about “the big picture”, literally and figuratively.
In the early 1990’s, at the end of the school year, one of my geometry students asked why my department offered Algebra 2 and did not offer Geometry 2. I didn’t have a good answer, so I created an advanced geometry course which I called Space. (Read about it here.) One component of that course was transformational geometry, which turned out to be a great high school topic, and became even better when I started using interactive geometry software (Cabri and then GeoGebra.)
The Common Core State Standards for Math (CCSSM, 2010) called for an introduction to geometric transformations in eighth grade, and making that foundational:. the triangle congruence and similarity conditions (SSS, SAS, etc.) would no longer be postulates — they would be theorems, derived via a transformational argument. This was an excellent decision mathematically, because of the many connections it facilitates. (I will return to those later on in this post.) It was also an excellent decision pedagogically: the conditions for triangle congruence are complicated and are just not obvious to students. In contrast, transformations are more intuitive and more accessible.
However, the transition to a transformational approach has been challenging. Some curriculum materials do a terrible job of it, emphasizing transformations in the Cartesian plane instead of presenting a geometric viewpoint. (And worse, limiting themselves to reflections in the x- and y-axes, and to 90° rotations and dilations centered only at the origin.) Moreover, many teachers believe that a transformational approach is “not rigorous”, perhaps because the CCSSM do not spell out a rigorous path to the congruence criteria, or even so much as mention the possible use of transformations and symmetry in proofs further along in the course.
What Do Teachers Need to Know?
To see some forward motion towards transformational geometry taking root in the curriculum, one key ingredient is teacher understanding. Teachers should of course understand the math that is involved in teaching a geometry course, but it is even better if their understanding extends beyond their most immediate need.
After retiring from the classroom in 2013, I developed some basic introductory materials on transformations (for grades 8-10). I already had some more advanced materials from my Space course (grades 11-12). But I did not have the materials needed to fill the gap in the CCSSM. My late colleague Lew Douglas worked with me to address this gap for an audience of teachers and curriculum developers.
In the rest of this post, I will outline a transformational geometry syllabus for teachers, linking to the various materials on my website. (Everything there is available to download for any non-commercial use.)
Before engaging with proof, it is best to start with a hands-on introduction, for example using the activities I link to below. (These lessons are intended for middle schoolers. If the content is already familiar, a teacher should just look them over, but it’s not a bad idea to quickly work through the pages indicated below.)
These applets may be helpful:
Filling The Common Core Gap
The CCSSM defines congruence and similarity of geometric figures on a foundation of geometric transformations. However the document does not spell out how to do that. I presented an approach based on the construction assumptions:
- Two distinct lines meet in at most one point.
- A circle and a line meet in at most two points.
- Two distinct circles meet in at most two points.
These assumptions are readily accepted by students. To use them effectively in proofs, students should have some experience with compass and straightedge construction, possibly with the help of interactive geometry software.
Beyond that, assuming that reflections preserve distance and angle measure is enough to get us to the triangle congruence conditions in a logically rigorous way. I outline how in this opinionated document for teachers and curriculum developers. The document includes possibly useful footnotes to establish context.
Lew Douglas found a way to get to the triangle similarity conditions by adding a simple assumption about dilations. This is outlined in Version 2 of the document.
Beyond this, the CCSSM seems to suggest going back to the traditional approach of proving everything else on the foundation of congruent and similar triangles.
A More Ambitious Take
But the CCSSM does not seem to rule out a more ambitious interpretation, which would involve the use of transformations throughout the course, in combination with the traditional approach to proof. Lew Douglas and I mapped this out in a strictly rigorous way in this 50-page booklet. Again, this is intended for teachers and curriculum developers.
Among other things, we suggest using symmetry definitions for special triangles and quadrilaterals. (Of course, symmetry needs to be defined in terms of isometries.) This has implications for the proofs of those figures’ properties, but it can also be pursued in parallel to the traditional approach, as an alternate way to think about them. Here is a worksheet we have used to introduce this idea to both teachers and students.
Here are some related topics that belong in high school after the Geometry course, in Algebra 2 or beyond. This content is either part of the CCSSM, or is sufficiently close to it that it may fit in many schools’ math program.
Computing transformations (this is material I have taught successfully in grades 11-12)
Thinking about graphs as geometric objects (this is intended for teachers, but some version may work with students)
(The traditional definition of symmetry involved equal angles and proportional sides, and thus could not be applied to curves.)
All these connections are part of the reason we should celebrate the introduction of transformational geometry into the secondary school curriculum. In addition to the topics listed above, note that geometric transformations are functions, and can serve to introduce many ideas (inverse function, composition, function notation, etc.) in a different context, one which can complement the usual approach, and may work better for many students.
Going for More Depth and for Enrichment
The remainder of this syllabus consists of non-CCSSM topics which should work well in the sort of courses suggested for grades 11-12 by NCTM’s Catalyzing Change in High School Mathematics. Even in the absence of such courses, studying this content will help teachers get a more in-depth appreciation for the beauty and power of transformational geometry. In my view, this should take priority over the topics listed under Later Courses, as it is more foundational and more geometric.
This is all content I have taught successfully at various grade levels. Some of it exists in classroom-ready worksheets, and some is geared to teachers and curriculum developers.
Isometries of the Plane. I developed this unit for my Space course (grades 11-12). It covers a lot of ground, from a precise definition of the transformations, to the composition of isometries, and to an overview of a proof of the Fundamental Theorem of Isometries: for any two congruent figures in the plane, one is the image of the other in one of four possible isometries: reflection, translation, rotation, or glide reflection.
Introduction to Abstract Algebra. I have taught this in various forms at all levels, K-12. This version is appropriate to middle school and high school. It is helpful as a prerequisite to the last stretch in the proof of the Fundamental Theorem. Moreover, since operations are perhaps the main topic overall in K-12 math, an introduction to their underlying structures via finite groups can only be a good thing.
To teach all this effectively, teachers should familiarize themselves with the content of these pages:
The latter parallels the development in “Isometries of the Plane”, and can serve as a resource to answer some of the questions posed in those worksheets. The inclusion of many interactive GeoGebra figures provides a helpful visual support for the proof.
Finally, symmetry offers a mathematical context with connections to art and design which makes it interesting to a wide range of students. It is shameful that there is so little work on it in the K-12 math curriculum. I offer some symmetry activities for all grades on this page. Those can enrich the program in almost any math class, and in any math circle or math club.
After posting the above, I discussed these issues with other educators. Thus these additional thoughts.
Glide Reflection Early On
In the post, I postpone introduction of the glide reflection until the end, under More Depth and Enrichment. The reason is that the glide reflection is not mentioned in the CCSSM. At second thought, that is not a good enough reason, and it’s probably a good idea to introduce it early on, soon after the other rigid motions. I spell out some arguments about the importance of the glide reflection in the article cited above. But let me expand on one of those arguments, and try to convince you the glide reflection is a valid topic for the geometry course.
Consider two congruent figures in the plane. If they have the same orientation (clockwise vs. counterclockwise), one is the image of the other in a translation or rotation. Of those two, the rotation is the general case, and the translation is an exception. Consider two points in one of the figures, A and B, and the corresponding points in the other, A’ and B’. Draw the perpendicular bisectors of AA’ and BB’. If those lines meet in a point, it is the center of rotation. If they are parallel or identical, we have a translation, and the two figures “face” exactly the same way. Clearly, that is the exception.
If the two figures have opposite orientation, one is the image of the other in either a reflection or a glide reflection. Let M and N be the midpoints of AA’ and BB’. In either case, the line MN will be the reflection line. We only have a reflection is the two figures are “directly across” from each other. That is the exception. The typical situation will require translating one of the figures to get it there. In other words, the reflection is the exception, and the glide reflection is typical.
That is a strong argument in favor of introducing the glide reflection early on: it is as common as a rotation, and far more common than either a translation or a reflection. It is a perfectly legitimate topic for geometry class. Ninth and tenth graders need not follow the reasons I outlined above, though that logic is certainly within the reach of 11th and 12th graders and of course of math teachers. (See Isometries of the Plane, in the lesson titled “Isometry Specifications” for the basic geometry underlying the above reasoning.)
Introducing the glide reflection early on is the approach taken by Scott Steketee, Daniel Scher, and their colleagues in their Web Sketchpad activities. (As far as I know, Web Sketchpad is the only environment where the glide reflection is a native tool. Back when I used Cabri, I was able to create a tool for it, but I have not succeeded in doing that in GeoGebra.)
Will This Work with Students?
I wrote this post as a sort of syllabus for preservice and inservice education for math teachers. I believe the same basic outline would work with high school students at the appropriate grade levels. However, there is one caveat: the level of rigor does need to be different. In Transformational Proof in High School Geometry, the pamphlet I co-authored with Lew Douglas, we marked many basic assumptions and theorems with asterisks. Those are results that are so obvious that it is in fact an advanced concept that they need to be proved. It is pedagogically catastrophic to insist on a formal approach to such results in a high school class.
One Axiom or Three?
The CCSSM suggest that rotations, translations and reflections be seen axiomatically as isometries. In another departure from Common Core orthodoxy, we suggested instead that it is a good idea to explore the composition of two reflections early on. Doing this makes it possible to easily prove that rotations and translations are rigid motions, starting from one axiom: reflections preserve distance and angle measure. This is not too difficult for a ninth or tenth grader to grasp. It is is not only more elegant, but it helps students see transformations as mathematical objects, and it goes well with the introduction of the glide reflection, as it too is a composition of transformations. Finally, it is a much more concrete introduction to function composition than the one they will see in Algebra 2.