- Use, with increasing confidence, problem solving approaches to investigate and understand mathematical content
- Apply integrated mathematical problem-solving strategies to solve problems from within and outside mathematics
- Recognize and formulate problems from situations within and outside mathematics
- Apply the process of mathematical modeling to real-world problem situations

*Benchmarks* 1C (The Nature of Science: The Science Enterprise)

Grades 6-8, page 17

Important contributions to the advancement of science, mathematics,
and technology have been made by different kinds of people, in different
cultures, at different times.

*Benchmarks* 1B (The Nature of Science: Scientific Inquiry)

Grades 3-5, page 11

Scientists' explanations about what happens in the world come partly
from what they observe, partly from what they think. Sometimes scientists
have different explanations for the same set of observations. That usually
leads to their making more observations to resolve the differences.

*Benchmarks* 1B (The Nature of Science: Scientific Inquiry)

Grades 3-5, page 11

Scientific investigations may take many different forms, including
observing what things are like or what is happening somewhere, collecting
specimens for analysis, and doing experiments. Investigations can focus
on physical, biological, and social questions.

*Benchmarks* 2B (The Nature of Mathematics: Mathematics, Science,
and Technology)

Grades 9-12, page 33

Mathematical modeling aids in technological design by simulating how
a proposed system would theoretically behave.

*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* 4A (The Physical Setting: The Universe)

Grades 9-12, page 65

Mathematical models and computer simulations are used in studying evidence
from many sources in order to form a scientific account of the universe.

*Benchmarks* 8E (The Designed World: Information Processing)

Grades 9-12, page 203

Computer modeling explores the logical consequences of a set of instructions
and a set of data. The instructions and data input of a computer model
try to represent the real world so the computer can show what would actually
happen. In this way, computers assist people in making decisions by simulating
the consequences of different possible decisions.

*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

Computers have greatly improved the power and use of mathematical models
by performing computations that are very long, very complicated, or repetitive.
Therefore computers can show the consequences of applying complex rules
or of changing he rules. The graphic capabilities of computers make them
useful in the design and testing of devices and structures and in the simulation
of complicated processes.

*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.