https://en.m.wikipedia.org/wiki/Geometric_algebra
https://en.m.wikipedia.org/wiki/Clifford_algebra
For teaching geometric algebra, there are curricula and textbooks available...
http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/
https://arxiv.org/abs/1205.5935v1
http://www.faculty.luther.edu/~macdonal/GA&GC.pdf
http://assets.cambridge.org/052148/0221/sample/0521480221WS....
This looks a bit too high-level to inform my undergraduate teaching, but certainly I could learn something from this.
http://geocalc.clas.asu.edu/html/Evolution.html#References
under "Recent Books". Notably "Clifford Algebra to Geometric Calculus" and "New Foundations for Classical Mechanics" by Hestenes.
PD: if you ever attempt to read any of those two, I'd advise to go to the latest (corrected) reprint.
Less ambiguously called "Clifford algebras", though Clifford himself prefered the term geometric algebra.
I'm a math professor. To me "geometric algebra" refers to any use of geometry in mathematics to illuminate algebraic principles. There are very very many of these -- this article illustrates one good example. I try to incorporate intuition from geometry into my teaching whenever possible.
It seems that the two of you are referring to some particular unifying framework, and/or set of definitions and notation. Could I ask you to elaborate on what you mean?
Thanks.