Many students think of geometry as the study of shapes -- points, lines, two-dimensional shapes (angles, triangles, rectangles, etc.) and three-dimensional figures (solids). But there is a fundamental skill taught that pertains to all such shapes, and carries over to many other branches of mathematics. What is that skill?
The study of geometry teaches the student to think rationally and logically, and to use formal methods to develop proofs of theorems based on fundamental postulates and previously proven theorems. To be able to start with the end in mind and find pertinent information to logically construct proofs spills over to many of the physical and social sciences as well. The study of Geometry introduces and develops such skills.
How does Algebra 2 build on, and enhance, concepts learned in Algebra 1. particularly the factoring, solution, and graphing of linear equations and functions, quadratic equations, and equations of higher order?
Algebra 2 typically includes a review and enrichment of concepts learned in Algebra 1. It goes on, however, to begin to prepare the student for Pre-Calculus and Calculus by introducing additional, and increasingly sophisticated expressions including rational numbers, radicals, exponents, and logarithms. Analytical Geometry concepts are introduced, including conic sections such as circles, ellipses, parabolas, and hyperbolas. Core concepts, such as the Fundamental Theorem of Algebra and the Binomial Theorem are frequently also taught.
What are the characteristics of Algebra that distinguish it from other branches of mathematics? How does it fit in, as a building block, for further study, and, aside from academic considerations, how is it practically applied on a daily basis?
Algebra serves as a "gateway" from arithmetic, taking it from the specific to the general. It offers abstract approaches to solving entire families of problems, rather than the individual problems themselves. In addition, Algebra offers a bridge between mathematical expressions, in terms of variables, constants, and operators, and the graphical depictions of those expressions. Thus, it provides a means of understanding mathematical concepts both analytically and graphically. The applications of Algebra to our daily lives are endless, providing ways of understanding problems in such diverse areas of construction, engineering, medicine, finance...the possibilities are endless...