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Topos Uranos
English

Final review of Algebra and Geometry II

Summary

This entry is not one more lesson, but the final account of the course: it recalls the seven learning objectives that the syllabus set at the outset, shows in a single figure to which of them each lesson of each unit was assigned, gathers objective by objective the first attempt at the exit assessments of the units, and closes the journey through the two Algebra and Geometry courses by pointing to where it continues.

Learning objectives

  1. State the seven global objectives of the course and recognize in which units and lessons each one was worked on.
  2. Distinguish the objectives that have a unit of their own from the cross-cutting ones, and explain how the latter are exercised.
  3. Decide, on the basis of the self-assessment by objective, of the figure and of the challenges of each synthesis, which units and which lessons are worth reviewing before moving on to Calculus I.

What the course set out to do

The course was born with a general aim: to carry the numbers, the functions and the plane geometry built in Algebra and Geometry I to trigonometry, to complex numbers in polar form, to matrices and, finally, to three-dimensional space, 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 seven global learning objectives, which should be kept in view with the numbering by which the figure cites them:

  1. Prove by explicit deductive reasoning, distinguishing what is deduced from what is admitted.
  2. Apply trigonometry to angles and triangles, prove identities, solve trigonometric equations and describe polar curves.
  3. Study the geometry of the plane with angles between lines, rotations of axes and transformations expressed with trigonometry, complex numbers and matrices.
  4. Operate with complex numbers in polar and exponential form, and interpret their operations geometrically.
  5. Solve linear systems with matrices, determinants and inverses, and recognize groups, rings and fields.
  6. Represent lines and planes in space and calculate angles, distances, areas and volumes with the dot product, the cross product and the scalar triple product.
  7. Model situations from science, technology and everyday life, and communicate in writing setups, calculations and proofs.

Each lesson of the course also stated its own specific objectives. The lessons do not say which global objective each of them serves; that correspondence is a reading made from outside, with the same method as in Course I. Each specific objective is assigned to one main global objective and, if it touches them, to other secondary ones; each counts in full toward its main objective and by half toward each secondary one, and the lesson is joined as main to the global objective that gathers the most weight and as secondary to the others that receive some. The syntheses follow a criterion common to the two courses: their main objective is objective 1, because their core reconstructs the deductive chain of the unit. Finally, two lessons whose problems are mostly setup problems, the one on the angle between two lines and its applications and the one on linear systems, have objective 7 as their main objective. 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 its main objective, and the seven objectives float around the column, held up by the lessons that feed them.