What Are the Differences between the Two Courses “Geometry after Algebra 1” and “Geometry after PreAlgebra”?

My course “Geometry after Algebra 1” includes and balances three pillars: geometric computations, geometric proofs and geometric constructions, while the course “Geometry after PreAlgebra” focuses on only geometric computations and omits the other two.

In “Geometry after Algebra 1”, students need to understand proofs of theorems and derivations of formulas. They need to know how to apply the theorems and formulas to do computations, to prove geometric arguments and to construct geometric figures. Algebra used in this course includes systems of linear equations, quadratic equations, rational equations, and a little of trigonometry.

In “Geometric after PreAlgebra”, students are presented with geometric relations and formulas without proofs. They mainly learn how to use the relations and formulas to compute geometric quantities. The prealgebra used in this course includes arithmetic, proportions, square roots, and linear equations and inequalities in one variable.

Take similar triangles as an example to illustrate the differences. In “Geometry after Algebra 1”, students need to know how to prove triangles are similar, how to prove properties of similar triangles, how to use triangle similarity, and why trigonometric ratios make sense. In “Geometry after PreAlgebra”, students just need to know what similar triangles are and how to use proportions for similar triangles to find a missing length or area.

Finally, I give a short introduction to geometric constructions since some parents ask what they are. A geometric construction is to construct a geometric figure using only a compass and a straightedge, where the straightedge is used to draw a straight line but can not measure a length. For example, a regular polygon with 17 sides (17-gon or heptadecagon) can be constructed (Gauss was the first who figured it out and proudly requested a regular heptadecagon be carved on his tombstone), while an angle trisection (construction of an angle measured one third of a given arbitrary angle) is impossible. Constructibility of a geometric figure is deeply rooted in algebra.