My Math Education Blog

"There is no one way"

Thursday, August 17, 2017

More on Geometric Construction

(Previous posts on this topic.)

I suspect that by far the most common introduction to geometric construction in US classrooms is a presentation by the teacher (or textbook) on various compass and straightedge construction techniques. "This is how you construct a perpendicular bisector. This is how you construct an equilateral triangle." And so on. "Now memorize these techniques, because you will be tested on them." Fortunately, some teachers take this further. Best case scenario:
  1. students learn how to prove that the constructions are correct
  2. students are asked to apply the techniques they learned in increasingly challenging construction challenges (e.g. "construct a square with a given side" after learning how to construct a perpendicular bisector)
  3. students are asked to create beautiful symmetric designs using those techniques
Such a scenario is certainly preferable to stopping after introducing the basic techniques, and it is far preferable to completely avoiding the topic. But I believe it is not optimal. In this post, I will try to present mathematical and pedagogical arguments for a somewhat different approach.

First of all, I would like to discuss the underlying mathematics. The essential mathematical concept underlying geometric construction is not the use of straightedge and compass. Interesting versions of construction have been developed for straightedge and the collapsing compass, and for the compass alone, not to mention for pedagogical tools such as patty paper, Plexiglas mirrors, and of course interactive geometry software. There are even interesting challenges involving only a (two-sided) straightedge which allows one to readily create parallel lines.

The essential concept underlying geometric construction is that of intersecting loci.  The locus of a point is the set of all possible locations of that point, given the point’s properties. The locus can be a line, a circle, or some other curve. If one knows two loci for a certain point, the point must lie at their intersection. In other words, given a figure, an additional point can be added to it in a mathematically rigorous way by knowing the locus (location) of the point in two different ways. Geometric construction is the challenge of finding such points and, in some cases, using them to define additional parts of the figure.

For example, the standard straightedge-and-compass construction of the perpendicular bisector is based on this theorem: PA = PB if and only if P is on the perpendicular bisector of AB. A compass is used to find all points at a distance AB from A (a circle centered at A, with radius AB.) That is one locus. Similarly, the circle centered at B with radius AB is the locus of points at a distance AB from B. These circles intersect at two points. Each of these points is equidistant from A and B, and therefore must be on the perpendicular bisector of AB.

The essential construction question is: given this figure and these tools, construct these additions to the figure in a mathematically rigorous way. In other words, it is a puzzle to solve, not a recipe to execute. In this view, the student is not a programmable machine. The student is a thinking human being. The pedagogical question becomes: what tools are most effective if we want to use geometric construction to teach geometric concepts in part through student problem-solving? To think about this, you need to assess the "overhead" a given tool entails, in other words the learning curve it requires. And you need to weigh this against the educational benefits of using the tool. As is usually the case in math education, there is no one way. I will present an approach that has worked for me.

In my view, one should start with compass, straightedge, and patty paper. (Patty paper is inexpensive tracing paper. Its use in geometry class was pioneered by San Francisco teacher Michael Serra.) The reason for including patty paper in this initial phase is that it makes it easy to copy line segments and angles. Of course one can do that with compass and straightedge, but it is too laborious and complicated for beginners. Including patty paper makes it possible to do interesting things right away. This phase of the work is mostly intended to get at some basic ideas. The physical challenges in using real-world compasses suggest that one should quickly transition to interactive geometry software if that is at all possible.

One way to do this is to use GeoGebra, which is free and works on every platform, including smart phones and tablets. GeoGebra is a huge and powerful program, and can be intimidating. A good strategy for introducing it is to start with a few of its tools, and give students time to explore it without a particular goal in mind, other than developing familiarity with the application. I used to work at a school where every student had a laptop, and the initial interactive geometry homework was to create something interesting in Cabri, using any tools at all. (Cabri is the interactive geometry application we used before GeoGebra. The same would work with GeoGebra) When students shared their creations, there were invariably some stunning images, and students developed a positive attitude towards the software. After that, I introduced whichever tools were needed for the work at hand (in particular, the crucial compass tool.) I did not find it necessary to hide any tools. This can be done in Cabri and in GeoGebra, but in this context I believe it is counterproductive. If there was a particular activity where I wanted to restrict students to using certain tools, I just told them which ones were allowed for that activity. Not hiding any tools almost guarantees that some students will discover additional tools, develop some curiosity about them, and teach them to others. Nothing wrong with that!

Once students are using interactive geometry software, many powerful labs and lessons become possible: construction challenges on the one hand, and informal explorations on the other hand. In fact, the availability of geometric transformations tools in those applications opens up many other possibilities, but I will have to save those for a future post.

-- Henri

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Links

I share my construction unit here: 8th grade | 9th-10th grade.

In the approach to proof in a transformations-based geometry course which I am developing with Lew Douglas, an informal unit on construction is very helpful in laying the groundwork for many foundational theorems. The reason is that proving many of those requires the use of what we call the construction postulates. Read more about this here.

And once again, here are my previous posts on geometric construction.

Wednesday, August 16, 2017

Errata

According to Merriam-Webster, the word errata means "errors" in Latin, but it is used in English to mean corrigenda which in Latin means "corrections". So there you have it: errors can be corrected — student errors, teacher errors, and (ahem) curriculum developer errors.

My books, great as they are, do contain errors. Some are small errors which I should have caught myself, or which an editor should have caught, but they got past us. Some are more substantial, and undermine the usability of a lesson, or of a particular problem.

For my (free download) Geometry Labs book, I have a section for Connections, Corrections, Extensions, and Revisions at the bottom of its home page some time ago. When I create or receive an alternate version of a particular lab, I make it available there.

For my (free download) Algebra: Themes, Tools, Concepts, I have a section for Sample Lessons. Most of them are versions of lessons from the book that I edited to make them work better. Sometimes that involved rephrasing the problems, but often it was just a matter of breaking the lesson up into smaller, more manageable chunks.

My Algebra Lab Gear books (published by Didax) have a number of small errors here and there. I have corrected the ones I found, and posted the corrigenda on a new errata page.

If you find an error in anything I've written, please let me know! I ought to correct the errors. If you have a way to improve a given lesson or lab, please send me your version. If I like it, I'm happy to include it on my site, and of course credit you.

--Henri

Thursday, May 25, 2017

Taxicab geometry

A few weeks ago, I led a workshop on taxicab geometry at the San Jose and Palo Alto Math Teacher Circles. Taxicab geometry is based on redefining distance between two points, with the assumption you can only move horizontally and vertically. So the taxicab distance from the origin to (2, 3) is 5, as you have to move two units across, and three units up. This has all sorts of geometric consequences which are fun to explore on grid paper (and with GeoGebra.)

The workshop was largely based on Labs 9.1 and 9.6 from my (free) book Geometry Labs. I quite enjoyed leading those sessions, as I learned a lot both about the topic at hand, and about how to best present it. The approach in Geometry Labs was based on separating a very basic introduction from further enrichment questions. The reason is that the basics are helpful as a contrast to Euclidean distance on the coordinate plane, and help students better understand the latter, while the further explorations were unlikely to be pursued in very many classes.

However in a session for teachers, or for a math club or an 11th or 12th grade elective, there is no reason for that separation. Moreover, in preparing those sessions, I got interested in other questions, including taxicab parabolas. The result is that I created a stand-alone worksheet on Taxicab Geometry that incorporates much of the material from Geometry Labs, somewhat reorganized and edited, and to which I added some interesting extensions. Download it here, and learn a lot about taxicab geometry! (If you use activities from Geometry Labs, be sure to periodically check the Geometry Labs home page for any new connections, corrections, extensions and revisions.)

This bizarre figure is the answer to one of the problems on the worksheet. Can you figure out what was being asked?



Here's a hint:

Calvin

What prompted this post is that last Saturday, I attended the San Francisco Math Teachers Circle's last meeting of the school year. The session was led by Dr. Cornelia Van Cott, a math prof at USF. After an exploration of taxicab circles and ellipses, we got to think about what circles would look like in other metrics. This was both entertaining and thought-provoking. Each metric had a fun name: the elevator metric, the post office metric, the teleportation metric. Unfortunately, the resulting circles were not as satisfying as the elegant circles of Euclidean or taxicab geometry. Cornelia explained that to be as mathematically productive as taxicab distance, a metric needs to be a norm, i.e. satisfy additional constraints. She wrote about this in an award-winning article for the February 2016 issue of Math Horizons (MAA's journal for undergraduates.) The title of the article is "A Pi Day of the Century Every Year", because different norms lead to different values for π, and thus, you could get a value like 3.1418, which would be perfect for next year.

What blew me away was Dr. Van Cott's explanation of how you can reverse the whole thing. Instead of defining the norm, and seeing what a unit circle would look like, you can go the other way: define your unit circle, and build the norm from it. So taxicab geometry would be derived from a square with vertices at (1, 0), (0, 1), (-1, 0) and (0, -1) (or in fact any square centered at the origin!) Other geometries could be derived by starting from any convex figure that is symmetric around the origin, e.g. a 2n-gon or an ellipse. Apparently, such geometries are called Minkowski geometries. Read the article!

If this sort of exploration is interesting to you, you should join or start a Math Teachers Circle in your area!

--Henri

Sunday, May 7, 2017

Polyarcs

My early forays as a curriculum developer date back to my days as a K-5 math specialist in the 1970's. A key insight of my young self was that activities intended for students were that much more worthwhile if they were also interesting to me. I learned to view with suspicion activities that were boring to me and other adults, but supposedly good for children to slog through. Thus started a career-long search for low threshold, high ceiling problems and puzzles. (That way of describing such activities originated in the 1980's Logo movement, but that is another story.)

Among my first creations were pentomino puzzles whose difficulty spanned the whole range from Kindergarten to adult. (If you don't know what pentominoes are, visit Geometric Puzzles in the Classroom before reading further.) Those puzzles were collected in books and cards that remained in print for an astounding 30 years or so. Alas, those are no longer available, as they have fallen victims to test prep obsession, to the gobbling up of smaller publishers by mega-corporations, and to the latter's lack of interest in actual children's education. If and when time allows, I will make those puzzles available online on my Web site, as I did for the not-unrelated SuperTangram puzzles*.

Anyway, what prompted this post was this message from Meghan, a teacher in Santa Cruz:
I've been using your website like crazy this year. I'm teaching Geometry and used your transformation materials, and mixed in a little bit from your Space class. I used many of your puzzle problems to create a presentation project: each group had a different spatial/area puzzle including the polyarcs. One particularly artistic student drew these, which I thought you'd appreciate.
Polyarcs were my own invention, which you can read about here. Here are Aviva's polyarc creations:


Fun stuff! I'd love to see other student polyarc art! Start them off with this worksheet. (Prerequisites: they need to know how to find the area and perimeter of a circle.)

Meghan's project assignment also included these options, among others: Tiling Rectangles with Polyominoes, and various SuperTangram puzzles*.

Back in the day, I used geometric puzzles in my elementary school classes as activities in a weekly "math lab" session, and as menu options for students who finished other work early. The logistical key was that each student had a folder which listed all the options, and where they checked off the puzzles they had successfully solved.

--Henri

* Alas, I am the only source of plastic supertangrams. If you sell math manipulatives and are interested in selling them, get in touch!