7
|
NATIONAL ASSESSMENT GOVERNING BOARD
Algebra
Algebra was pioneered in the Middle Ages to
solve equations easily and effciently by
manipulating symbols, rather than by using the
earlier geometric methods of the Greeks. The two
approaches were eventually united in the analytic
geometry of René Descartes.
As algebra became more widely used, its formal
structure began to be studied, resulting in the “rules
of algebra”—a compact summary of the principles
behind algebraic manipulation. Similar thinking
produced a simple but fexible concept of function
and also led to the development of set theory as a
comprehensive background for mathematics.
These two aspects of algebra—as a powerful
representational tool and as a vehicle for
comprehensive concepts such as function—form the
basis for the expectations spanning grades 4, 8, and
12. By grade 4, students should be able to recognize
and extend simple numeric patterns as foundation
for a later understanding of function. They can begin
to understand the meaning of equality and some
of its properties, as well as the idea of an unknown
quantity as a precursor to the concept of variable.
By grade 8, representation of functions as
patterns, via tables, verbal descriptions, symbolic
descriptions, and graphs, can convey the idea of
function. Linear functions receive special attention.
They connect to the ideas of proportionality and
rate, forming a bridge that will eventually link
arithmetic to calculus. Other means of fnding
solutions, including graphing by hand or with a
calculator, reinforce symbolic manipulation in the
relatively simple context of linear equations.
By grade 12, students should appreciate the rules
of algebra as a basis for reasoning. Nonlinear
functions, especially quadratic, power, and
exponential functions, are introduced to solve
real-world problems. Students should become
accomplished at translating verbal descriptions of
problem situations into symbolic form. Students
at grade 12 also should encounter expressions
involving several variables, systems of linear
equations, and solutions to inequalities.