Benchmarks 11C (Common Themes: Constancy and Change)
Grades 9-12, page 275
Graphs and equations are useful (and often equivalent) ways for depicting
and analyzing patterns of change.
Benchmarks 2C (The Nature of Mathematics: Mathematical Inquiry)
Grades 9-12, page 38
Much of the work of mathematicians involves a modeling cycle, which
consists of there steps: (1) using abstractions to represent things or
ideas, (2) manipulating the abstractions according to some logical rules,
and (3) checking how well the results match the original things or ideas.
If the match is not considered good enough, a new round of abstraction
and manipulation may begin. The actual thinking need not go through these
processes in logical order but may shift from one to another in any order.
Benchmarks 11B (Common Themes: Models)
Grades 9-12, page 270
The basic idea of mathematical modeling is to find a mathematical relationship
that behaves in the same ways as the objects or processes under investigation.
A mathematical model may give insight about how something really works
or may fit observations very well without any intuitive meaning.
Benchmarks 11B (Common Themes: Models)
Grades 9-12, page 270
The usefulness of a model can be tested by comparing its predictions
to actual observations in the real world. But a close match does not necessarily
mean that the model in the only "true" model or the only one that would
work.
Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades 9-12, page 29
Theories and applications in mathematical work influence each other.
Sometimes a practical problem leads to the development of new mathematical
theories; often mathematics developed for its own sake turns out to have
practical applications.
Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades 9-12, page 29
New mathematics continues to be invented, and connections between different
parts of mathematics continue to be found.