### Standard 4: Mathematical Connections

In grades 9-12, the mathematics curriculum should include investigation of the connections and interplay among various mathematics topic and their applications so that students can:
• Recognize equivalent representations of the same concept
• Benchmarks 11C (Common Themes: Constancy and Change)
Graphs and equations are useful (and often equivalent) ways for depicting and analyzing patterns of change.

• Relate procedures in one representation to procedures in an equivalent representation
• Benchmarks 2C (The Nature of Mathematics: Mathematical Inquiry)
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)
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)
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.

• Use and value the connections among mathematical topics
• Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)