12th grade math options

I just posted Isometries of the Plane on my Web site. It is a transformational geometry unit aimed at high school teachers, and/or students in grades 11-12. It does go beyond the Common Core State Standards for Math (CCSSM), and thus might make sense as part of a precalculus course or a 12th grade elective.

In theory, the  CCSSM only mandates content up to the end of grade 11, presumably to make room for Calculus in the senior year. In practice, I doubt all the Common Core topics will fit in grades 9-11 (read more on this here.), Be that as it may, 12th grade is up for grabs.

A standard alternative to 12th grade Calculus is a course in probability and statistics. If it weren’t for Common Core testing at the end of 11th grade, such a course could relieve some of the pressure on grades 9-11, and moreover would be of interest to a broader cross-section of the population than Calculus.

A non-standard alternative, intended for students who did poorly in grades 9-11, is a “Year 4” course which provides an overview of secondary school math for students hoping to go to community college after high school graduation. I wrote an outline for such a course, which can also serve as an introduction to large segments of my Web site.

I also created two other options for advanced, post-Algebra 2 electives: Infinity, and Space. I taught these classes in alternate years, between the early 1990’s and my retirement in 2012. Infinity included work on set theory and cardinal numbers, dynamical systems and chaos, fractals and programming. Space included some 3D geometry using Zome Geometry, some discussion of the 4th dimension, a touch of group theory, and a lot of work on geometric transformations and symmetry.

I resisted the private school obsession with multivariate calculus and linear algebra, two courses that are usually taught in high school as if to college students. That is so wrong, not only for developmental reasons, but also because students who take those courses will need to take them again in college, as they will not get any sort of credit for them. For some, they will be so turned off to math by this experience, that they will actually not take them again, and those high school courses will be the end of their mathematical careers.

Instead, both Space and Infinity were taught at a pace and in a style that is appropriate to teenagers, with an abundance of hands-on experiences and discussion. While they were quite challenging, they were not limited to the math superstars, and their impact was to make students realize math is not merely the servant of science, technology, and business: it is also fun, interesting, and beautiful in its own right, even on the most abstract of topics.

Anyway, what triggered this post was the fact that I finally uploaded a grades 11-12 unit on isometries that I developed mostly for my Space class, and tweaked for my summer Transformational Geometry workshop for teachers. Check it out! (Note: there’s a new page on my Web site that can serve as a teachers’ guide for the last few pages of this packet: Only Four Kinds of Isometries.)


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