What is an “algebra”?

You learn something new every day.

Here’s how one teacher defines an “algebra”:

 An algebra is essentially a set of objects that can be both added and multiplied, with the two operations fitting together via the distributive property.

In other words, that means it’s a set of things where if you add (which really also includes subtraction) or multiply (which, ditto, includes division), the thing you get out is still in that set.  If you take a number, and add another number, you get out a number.

That property also holds true for: expressions with variables, including polynomials; matrices; functions; even transformations on the plane!

This allows me to see how some things we teach in algebra that sort of seem tangential–like matrices, for example, and also some of the geometry–make a whole lot more sense. We are figuring out what rules work on an algebraic system, starting with the system of real numbers that students have learned about all through elementary and middle school, and seeing how it also applies to other sets of mathematical objects.

Mr. Honner cues up a couple other “big ideas” in algebra. I’m rewriting those slightly, based on some feedback from Grant Wiggins, while trying to get somewhat student friendly:

  • Organizing sets into binary relationships allows us see when there is and is not equivalence (e.g., equality and inequality)
  • Some of those relationships can be matched perfectly to each other (e.g. functions, in a big simplification)
  • Those relationships can be mapped onto a simple coordinate plane (e.g. the Cartesian coordintes)

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