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Topos Uranos
English

Final review of Algebra and Geometry I

Summary

This entry is not one more lesson, but the final account of the course: it recalls the eight learning objectives that the syllabus set at the outset, shows in a single figure how each unit and each of its lessons has contributed to them, gathers objective by objective the first attempt at the exit assessments of the units, and closes the journey by pointing to where it continues, in the course Algebra and Geometry II and in the subjects that this first course opens up.

What the course set out to do

The course was born with a general aim: to construct, from a few axioms about the natural numbers, the number systems, the algebra of polynomials, the elementary algebraic and transcendental functions and the analytic geometry of the plane, proving every result that is used, so that whoever finishes it knows not only how to calculate, but also where each calculation comes from. The lesson on the syllabus unfolded that aim into eight global learning objectives, which should be kept in view with the numbering by which the figure cites them:

  1. Prove with explicit deductive reasoning, distinguishing what is deduced from what is admitted.
  2. Construct the number systems, from the Peano axioms to the complex numbers in rectangular form, with the language of sets, relations and functions.
  3. Prove and apply the arithmetic of the integers: divisibility, primes, greatest common divisor, congruences, positional notation and counting.
  4. Transform algebraic expressions and solve equations, inequalities and systems, justifying each transition.
  5. Analyze real functions and represent their graph.
  6. Represent loci of the plane by means of equations and recognize the curve that an equation defines.
  7. Solve problems with the exponential, logarithmic and hyperbolic functions, deducing their properties from their definitions.
  8. Model situations from science, technology and everyday life, and communicate in writing the setting up of problems, calculations and proofs.

Each thematic lesson of the course also stated its own specific objectives, and each of them served directly one or several of these eight; the introductory lessons, on the other hand, also stated orientation objectives (describing the unit, placing its history) that contribute to none. The syntheses, by a criterion common to the two Algebra and Geometry courses, are joined to objective 1, because their core reconstructs the deductive chain of the unit. If this is so, the whole course takes the form of a spine: the head is the general aim; the column, the order of the units, each with its segment; each lesson is a node of that segment, joined to the global objective it most directly serves, and the eight objectives float around the column, held up by the lessons that feed them.