Hi all. I am three days into my new year. In Geometry this year, I decided to combine a version of standards-based grading (SBG) with project-based learning (PBL – love those acronyms!).

I shared my syllabus with the class the first day, both as a detail-oriented hard copy and as a more thematic zooming ‘Prezi’ presentation. Now, this will be a very non-traditional Geometry class, idiosyncratic in many ways to me and my mathematical/engineering interests (while still aligning with Maryland state standards). The first half of the course will be focused on patterns and motion, with units on symmetry+tessellations, fractals, and polyhedral patterns; following a one-week fast-paced review of measurement. The second half of the course will look in more detail at measurement and also construction: of triangles, of other polygons built via triangles, of circles, and of some familiar 3D shapes.

But even though some topics are based on my individual passions (although I had considered it before, I was inspired/motivated to actually do a fractals unit after hearing Bob Devaney speak at MathFest this summer), I believe in some ways this represents a truer picture of the real field of geometry than a curriculum driven by axiom/theorem/2-column-proofs where some of the things that go in the reason column are complicated names for something completely obvious (think ‘transitive property of equality’) while others are intuitive leaps that most high school students would never come up with. Some higher-level study of geometries focuses on symmetry and the invariants of a space (e.g. which transformations are isometries and leave an object/shape fundamentally unchanged, and which are not). I took a math course in college that focused on tiling theory (think your bathroom floor tiles or M.C.Escher’s tessellations). Fractals are a new and currently expanding area of study, unlike most of a secondary geometry course that is over 2000 years old. I’m hoping our unit on fractals will change the misperception that all math is static, unchanging, and always has an already-known-by-the-teacher right answer.

Some of my sequence’s focus on measurement and construction is influenced by my recent foray into learning and teaching engineering, while some of it is based on envisioning what real-life skills my students can learn via the course. Additionally, I admit that one personal goal in teaching this course is to convince students not to hate math, so this has also been a factor in deciding how I will teach geometry on this, my fourth go-round.

All of that is a roundabout justification of some of my choices of topics. Grading-wise, I decided to go 50/50 skills quizzes and projects. I am a big proponent of project-based learning, and I’m making some changes this year so that the geometry class will become more project-driven and less of, every week, oh here’s a project that sort of relates to what you’re learning. I believe that projects can assess applications of math, connections to other subjects, and a deeper contextual synthesis of multiple math understandings that is not reflected in discrete standards-based skills assessments. Perhaps one of the old hands at SBG can tell me how to refine standards based grading to get at these aspects. But since I don’t know how to incorporate this sort of learning into SBG, for this course I will weight the projects 50% and SBG skills assessments as the other 50%.

Anyway, here’s my Geometry Skills List. I would love feedback as to any topics I excluded that you consider essential to a students’ understanding of geometry, to their knowledge of the world, or to their grasp of future mathematics. I’d also appreciate hearing if any of the skills I listed are less important, or if any should be broken up into multiple skills or combined into a single skill.

- Student will measure lengths to the nearest 1/8 in or 0.1cm.
- Students will measure and draw angles accurate to the nearest degree.
- Students will classify angles by type.
- Students will identify the number of dimensions an object has.
- Students will use formulas to compute areas of various shapes.
- Students will compute perimeters of various shapes.
- Students will use formulas to compute volumes of various shapes.
- Students will reflect, rotate, and translate shapes with a high degree of accuracy.
- Students will identify reflectional, rotational, and translational symmetry.
- Students will classify plane tilings by the shapes around a vertex, and by symmetries.
- Students will use proportions to determine unknown sides in similar shapes.
- Students will demonstrate an understanding of fractals and self-similarity.
- Students will measure perimeter, area, and volume of fractals at various iterations.
- Students will count the edges, vertices, and faces of polyhedra.
- Students will use Euler’s Formula to predict aspects of unfamiliar polyhedra.
- Student will determine an appropriate unit of measurement given a situation.
- Students will convert between different units of measure by dimensional analysis.
- Students will use distance and midpoint formulas for lengths in the coordinate plane.
- Students will identify types of triangles, by angle and by side lengths.
- Students will construct triangles given side and/or angle measures.
- Students will use the congruence theorems to prove triangles congruent.
- Students will apply the Pythagorean Theorem to find unknown sides in right triangles.
- Students will use trigonometry to find unknown sides and angles in right triangles.
- Students will construct circles passing through 1, 2, or 3 points.
- Students will identify parts of a circle.
- Students will calculate lengths and areas of a circle and parts thereof.
- Students will use formulas to compute surface areas of various shapes.
- Students will use formulas to compute volumes of various shapes.

I look forward to discussing these with you all. By the way, there’s still time to influence my decision and change the list–any but the first few that are on tomorrow’s quiz *:^)*

That doesn’t look like a year’s worth of geometry to me. Some of the skills (like measuring to the nearest degree) are mechanical, not conceptual. I doubt that I could ever draw angles to the nearest degree—I had a slight hand tremor as a child.

Tossing in trigonometry when almost no relevant geometry has been taught seems weird to me. The scaffolding needed for constructing circles passing through 3 points seems to be completely missing. “Various shapes” is a bit too vague.

I think that your dismissing proofs is going to result in the students never having seen a proof before college. If you are doing fractals without teaching fractal dimensions, you are just doing pretty pictures, not math.

Thanks for sharing your thoughts! A few comments and questions:

1) Given that projects will also be taking place, and the fact that the course is only one semester long (90 minute periods), I do not wish to create a list of hundreds or even scores of skills. I am a firm believer in depth over breadth. 32 skills would be my goal, maybe just because it is a power of 2, but probably also because that fits the model I’m planning to use of reassessment and weekly quizzes, and the number of weeks the class will last [18, figuring on 16 weekly skills quizzes, separate from midterm and final].

2) OK. Good point; maybe nearest 5 degrees is better. That sounds weird; perhaps that can be rephrased to “accurate to a tolerance of +/-2.5 degrees”. I disagree that measuring is solely mechanical, since it is the foundation of geometry (‘Earth measurement’) and also where we get the dimensions of an object to use in area and volume calculations. We do discuss other historical and bodily measurement systems, besides the arbitrary but common inches/centimeters/degrees units that I assess.

3) What geometry do you consider relevant to (right triangle) trigonometry? Similar figures and Pythagorean Theorem are already in my plan/list.

4) Remember that not all the scaffolding for a skill will be assessed, even if taught. Also, some skills will be part of a project and assessed thereby instead of in an SBG quiz. With that in mind, what skills do you have in mind that would help build up to constructing a circle through three points?

5) How does area/perimeter “of triangles, quadrilaterals, and other polygons” sound? Volume/Surface Area “of prisms, pyramids, cylinders, cones, and spheres”.

6) Great catch on the fractal dimensions; I should definitely include that. Any thoughts on giving students a better conceptual understanding of log(self-similar copies) / log(scale factor) [or the equivalent as a single log expression without change-of-base] for students who have never seen logs before? Or just plug-and-chug plus comparison talk of regular [non-Hausdorff] dimension and what it means to have an in-between dimension?

7) I did not intend to dismiss proofs, only a certain style of presenting proofs thatt is overly formal (in my view) and popular in textbooks. My course is probably lighter in proofs than a standard geometry course, but I do certainly use and teach them. Students research various proofs of the Pythagorean Theorem (again, a project grade, so doesn’t show up in my skills assessment list). Students will do a few triangle congruence proofs, but also look at the triangle congruence theorems of SSS, SAS, ASA, & AAS from the perspective of what conditions are sufficient to uniquely construct a triangle. I intend to walk the students through a proof that there are only five platonic solids, though I don’t expect them to come up with it all on their own. And throughout the course, I will emphasize justifying your reasoning, of which proofs are merely a more formal [and also more airtight :-)] version.

This may be a rhetorical question, but, given that proofs key to all areas of study throughout higher mathematics(not just geometry), why do we throw all the proofs into geometry and none in algebra or pre/calculus? I would love to teach a proof-heavy number theory course (though I probably never will at the high school level). But at the same time, I wish to convey some of the exploratory and pattern-finding essence of mathematics that often comes before proof and is just as important in my experience doing research mathematics.

Thanks again for your input, and I look forward to hearing your and others’ additional comments.

Personally, I think that proof would be better taught in a class that covered logic and mathematical induction than in geometry (I’ve taught such a class for college engineering students, called “Applied Discrete Math”).

I’d recommend you look at Ruscyk’s

Introduction to Geometryhttp://www.artofproblemsolving.com/Store/viewitem.php?item=intro:geometryIt provides a very thorough intro to geometry with a proof style that is less rigidly formal and more problem-solving, thus more like what real mathematical proofs look like. My son went through geometry twice at different schools. The first time self-paced though Ruscyk’s book, and the second time through a very traditional 2-column proof class. I think he learned much more the first time. Warning: some of the exercises are really challenging, so it may be worthwhile to get the solution manual as well, which does a very good job of explaining how to do each problem.

I’m afraid I can’t give you much help on what should or shouldn’t be in a geometry class from personal experience, as I’ve not done geometry for over 40 years.

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